A STUDY IN PERSPECTIVE DRAWINGby DAVID C. OPHEIM

VP1

OLYMPUS

OLYMPUSOM

-SYS

TEM

B1

234

0.5

0.7

0.45

OM-1ON

ROLYMPUS

A STUDY OF PERSPECTIVE DRAWING

COPYRIGHT 1992 DAVID OPHEIM

All rights reserved. No part of this work covered by the copyright hereon may be reproduced or used in any form or by any means - graphic, electronic, or mechanical, including photocopying, re-cording, taping, or information storage and retrieval systems - without written permission of the author.

PREFACEPREFACE������

���

Perspective is the most common method of depicting three dimensional forms realistically. Perspective is found in all types of printed matter. It is used widely in the profes-sional areas: Architecture, Interior Design, Industrial Design, Commercial Art, Graphic & Environmental Design. In all instances drawings depicting ideas or real objects are com-municated to a wide variety of viewers.

Perspective has been with us for a long time. 15th Century painters used aerial perspective by incorporating a gradual change of intensity of colors and light. Adding overlap and a single vanishing point gave the illusion of objects gradu-ally getting smaller as they went farther away. Two and three point perspective brought about new possibilities as it became necessary to show what objects and buildings would look like before they were manufactured or constructed.

Designers then began to find better ways to develop their perspectives that would use less construction - resulting in faster solutions. Also, new tools such as ellipse guides and perspective grids have been introduced to help quicken this process. There have been several good textbooks which have each made a contribution to help us understand these pro-cesses. Unfortunately, they are now out of print.

I have taught perspective for many years and have not found a comprehensive textbook. I solved this by writing a supplement in syllabus form. The syllabus eventually grew until it replaced the text. Students often encouraged me to put these notes in book form. I thank them for their encour-agement, and to them, I dedicate this book.

My greatest challenge was to make the subject matter ap-pealing to people with different interests and needs. Therefore, I have tried to make the examples very basic, so that they will relate to whatever your area of interest might be. I have also struggled with how technical it needs to be. What looks simple to some might seem to complex to others. So, I have kept the complexity to a minimum. It may not always be necessary to understand the underlying structure of some methods, if you are able to apply it. If you feel burdened by them, skip those sections and concentrate on the short-cuts. The structure is there though, for reference and background. The short-cuts will work without deeper understanding. It is like using a computer without knowing what actually makes it work. Those who know the inner workings can do much more with it, but the computer will work for the novice, if the right buttons are pushed. And, in the beginning, a computer can be pretty scary too.

Personally, I think perspective is the greatest thing since ice cream. If you are skeptical of your ability to learn perspec-tive, just remember, it is like reading and writing. Perspective is a learned skill. Once you have practiced and put it to use, you will be using perspective as easily as other skills you now have.

The intent of this text is to give a comprehensive look at most of the approaches used today including shadow, reflec-tions, rotations & surface development - hopefully, a useful text that teaches methods that will give you accurate and fast solutions. Lets get to it . . . .

TABLE OF CONTENTS

CHAPTER 1 / INTRODUCTIONMULTI-VIEW DRAWINGS 1 - 2SINGLE-VIEW DRAWINGS 1 - 3PERSPECTIVE RELATIONSHIPS 1 - 4 1 & 2 POINT PERSPECTIVE 1 - 5EYE LEVEL (HORIZON) 1 - 6CONE OF VISION 1 - 7PERSPECTIVE USES 1 - 8

CHAPTER 2 / PLAN ELEVATIONUSING ORTHOGRAPHIC VIEWS 2 - 1PLAN/ELEVATION METHOD 2 - 2WORKING BELOW HORIZON 2 - 6WORKING ABOVE HORIZON 2 - 7CONTROLLING VANISHING PTS. 2 - 8PLAN/ELEVATION IN USE 2 - 9LIMITATIONS & RELATIONSHIPS 2 - 10

CHAPTER 3 / MEASURING SYSTEM2 POINT MEASURING SYSTEM 3 - 2HOW TO USE MEAS. SYSTEM 3 - 3SHORT CUTS 3 - 4MEAS. POINTS BY PREDICTION 3 - 5YARDSTICK PERSPECTIVE 3 - 6SELECTIONS 3 - 7APPLICATIONS OF HML 3 - 7CUBE CONSTRUCTION 3 - 8SYSTEM EXTENDED 3 - 9MEAS. INSIDE BASE LINES 3 - 10ELEVATION USING DIAGONALS 3 - 11HORIZONTAL FLAPS 3 - 12VERTICAL FLAPS 3 - 13MEAS. OUTSIDE BASE LINES 3 - 14MEAS. INSIDE BASE LINES 3 - 15LAYOUT TABLET MEASURES 3 - 16FREEHAND SKETCH 3 - 17MOLDED FORMS 3 - 18ORTHOGRAPHIC VIEWS 3 - 19

CHAPTER 4 / 1 PT. PERSPECTIVECONVENTIONAL METHOD 4 - 2DEFINITIONS AND DISTORTION 4 - 3THREE SHORT CUTS 4 - 41 POINT FLOOR GRID METHOD 4 - 5SHORT CUT USING ELEVATION 4 - 6COMBINATION 1 & 2 POINT 4 - 8LARGE SCALE DRAWINGS 4 - 9SMALL OBJECT SKETCHES 4 - 10

CHAPTER 5 MODULAR PERSPECTIVESQUARE TO CUBE 5 - 2MULTIPLICATION OF VIEWS 5 - 4DIVIDING LINES & RECTANGLES 5 - 6ENLARGEMENT AND REDUCTION 5 - 7VERTICAL SURFACE MULT. 5 - 8HORIZONTAL SURFACE MULT. 5 - 9REDUCTION OR ENLARGEMENT 5 - 102 POINT INTERIOR GRID 5 - 11DEVELOPMENT/HORIZ. PLANE 5 - 12DIAGONAL VANISHING POINT 5 - 13 2 POINT GRID - VERTICAL MEAS. 5 - 141 POINT GRID - VERTICAL MEAS. 5 - 15PERSPECTIVE TRACING GRIDS 5 - 16

CHAPTER 6 MEASURING PLANEPERSPECTIVEGRID CONSTRUCTION 6 - 2INTERIOR/ARCH APPLICATION 6 - 6PRODUCT APPLICATION 6 - 7

CHAPTER 7CIRCLES IN PERSPECTIVECIRCLES IN PERSPECTIVE 7 - 2ELLIPSES DEFINED 7 - 28 POINT METHOD 7 - 312 POIINT METHOD 7 - 4USING ELLIPSE GUIDES 7 - 5ELLIPSE ALINEMENTS 7 - 7ELLIPSE ANGLE MEASUREMENT 7 - 8ELLIPSE SIZE MEASUREMENT 7 - 12ELLIPSE GALORE 7 - 13

CHAPTER 8CYLINDERS AND SPHERESCYLINDERS FROM BOXES 8 - 2VERTICAL CYLINDER W/GUIDES 8 - 3CYLINDER CONSTRUCTIONS 8 - 4CYLINDER ROTATION 8 - 5CAMERA CONSTRUCTION 8 - 6CAMERA ILLUSTRATION 8 - 7SPHERE CONSTRUCTIONS 8 - 8SPHERE STUDIES 8 - 9SPHERE TO SCALE 8 - 10SPHERE SHORT CUT 8 - 11

CHAPTER 9 / SHADOWS LIGHT LOCATION 9 - 14 SHADOW TYPES 9 - 1PLOTTING BASICS 9 - 2PARALLEL METHOD 9 - 2VERTICAL PLANE 9 - 4HORIZONTAL PLANE 9 - 4BOX FORMS / EXTERIOR 9 - 4BOX FORMS / INTERIOR 9 - 5FLAGPOLE RULE 9 - 5MULTIPLE BOX SOLUTIONS 9 - 6WALL VARIATIONS 9 - 7

TABLE OF CONTENTS (continued)

CHAPTER 9 (CONTINUED)SHADOW SAVVY 9 - 7DOORWAYS & WINDOWS 9 - 8BOX WITH FLAPS 9 - 9PYRAMID CONSTRUCTION 9 - 9STANDING CYLINDERS 9 - 10HORIZONTAL CYLINDERS 9 - 13CYLINDER INSIDE SHADOW 9 - 15SPHERE SHADOW 9 - 16SHORT CUT SPHERE SHADOW 9 - 17SPHERE SHADOW ON WALL 9 - 18BOX FORM ON CONE 9 - 19CAST SHADOWS ON FORMS 9 - 20FLOATING FORMS 9 - 22CONVERGING LIGHT SHADOW 9 - 24POSITIVE LIGHT SHADOW 9 - 27NEGATIVE LIGHT SHADOW 9 - 30

CHAPTER 10 / REFLECTIONSFORMS INTO VERTICAL MIRROR 10 - 2REFLECTIONS INTO WALL 10 - 3REFLECTIONS OF FORMS 10 - 4CONVEX MIRROR 10 - 7HORIZONTAL CYLINDER 10 - 10SPHERE REFLECTIONS 10 - 11HIGHLIGHT & REFLECTIONS 10 - 12

CHAPTER 11 / FORM ROTATIONCUBE AROUND HORIZ. AXIS 11 - 2MULT. OF ROTATING CUBE 11 - 3MEASURING METHOD 11 - 490° ROTATION 11 - 61 POINT DRAWN ROTATED 11 - 8ROTATION OF CUBES 11 - 9

CHAPTER 123 POINT PERSPECTIVECONVENTIONAL CONSTRUCTION 12 - 2TRIANGULAR METHOD 12 - 33 POINT EFFECT 12 - 4

INTRODUCTION TO PERSPECTIVE

GETTING STARTED

In order to define perspective it is necessary to be aware of several different types of drawings that give visual information about objects such as size, scale, and details. We need to take a quick look at 2 types that are the most common. . . the Multi-View and the Single-View.

MULTI-VIEW DRAWINGS

Multi-views show what an object looks like from several different directions at the same time. Objects are visually described in three dimensions (width, depth and height) by orthographic projection. Orthographic drawings give us true length visuals of the object by using several views arranged around a front elevation. All lines parallel to the edge of the object are shown

in true length. This means that the actual dimension of the object is used either at full scale (actual size) or a scale representation of the size. These views are called projections because each view is projected from one of the other views. A good understanding of orthographic projections will help you understand the relationships of various surfaces.

PLAN VIEW

FRONT ELEVATIONEND VIEW

FRONT ELEV

ATION

SIDE ELEVAT

ION

TOP VIEW

ISOMETRICall angles equal

all sides in true lengthand parallel

ELEVATION OBLIQUE(DIMETRIC)

2 angles equal, sidesare true length and parallel

PLAN OBLIQUE90 degree cornerssides true length

or fraction and parallel TRIMETRICno angles equal sides true length

or fraction and parallel

1 - 3

SINGLE-VIEW DRAWINGS

Axonometric drawings also show all dimensions in true length, but use a single view showing three surfaces. Different sides are shown in direct relation to each other and are seen as parallels at different angles to the horizontal. Each type of drawing takes a different approach in communicating what the object looks like, but are all basically the same each emphasizing a different aspect of the object. Axons give a mental picture of what the object might look like, but will play visual tricks on you. Each

veiw appears to get larger as it gets farther away.

To get a more realistic looking drawing, a distortion called fore-shortening is used. This is the basis of perspective drawing.

Examples of different ways that perspective gives us more realistic and "believable" drawings and a fuller definition of perspective will follow . . . . .

AXONOMETRIC VIEWS

OBSERVER

GROUND PLANE CENTER OF VIS

ION

HORIZON LINEEYE LEVEL @ INFINITY

LINE DRAWING

ACTUALOBJECT

PICTURE PLANE

PERSPECTIVE DEFINEDPERSPECTIVE is a system of drawing by which an ob-ject of three dimensions is represented on a flat (two-di-mensional) surface to appear real by means of distorting the drawing in a controllable manner. Perspective is derived from a relationship of OB-SERVER, PICTURE PLANE, OBJECT and HORIZON at INFINITY.

PERSPECTIVE RELATIONSHIPS

The PICTURE PLANE is a transparent surface through which the OBSERVER sees the OBJECT. The OBJECT touches the PICTURE PLANE at the CENTER OF VISION. This is the only edge that will be seen in (true length). The PICTURE PLANE can be thought of as representing the piece of paper on which you are drawing. As the OB-SERVER looks at different corners or details, the position of the corresponding point on the PICTURE PLANE is in

a straight line from the detail to the OBSERVER'S eye. This point on the PICTURE PLANE is the (forshortened) position on the drawing. The HORIZON is the elevation of the eye (earth's horizon at infinity) and is represented by a horizontal line. This means that the HORIZON is at the farthest visual distance possible from the OBSERVER and is seen as a horizontal line on the drawing either below, at or above the OBJECT.

OBSERVER, PP, OBJECT & HORIZON

1 - 4

OBSERVER

GROUND PLANE CENTER OF VIS

ION

HORIZON LINEEYE LEVEL @ INFINITY

LINE DRAWING

ACTUALOBJECT

PICTURE PLANE

VANISHINGPOINT

VANISHINGPOINT

HORIZON LINE

OBSERVER

TOP VIEW

CENTEROF VISION

OBJECT

PICTURE PLANE

OBSERVER

TOP VIEW

CENTEROF VISION

OBJECT

PICTURE PLANE

VANISHINGPOINT

HORIZON LINE

1 - 5

ONE-POINT PERSPECTIVE RELATIONSHIP

The One-Point view is what the OBSERVER sees when the object touches the picture plane along one side. This pro-duces a view of the side perpendicular to the observer and appears to have depth in one direction only.

This results in a single VANISHING POINT on the HORIZON LINE at INFINITY.INFINITY is the farthest distance possible from the object.

TWO-POINT PERSPECTIVE RELATIONSHIP

The Two-Point view above is what the OBSERVER sees when the object touches the picture plane at one corner. This produces a view that is turned from the observer and appears to have depth in two directions, to the left and to the right.

This will result in two VANISHING POINTS on the HORIZON LINE. Locations of VANISHING POINTS & HORIZON LINES will be discussed in CHAPTER 2.

TWO-POINT VIEWONE-POINT VIEW

EYE LEVEL (HORIZON LINE)

The EYE LEVEL is the same as the HORIZON LINE. All eye levels are the same if every viewer is relatively the same height as the OBSERVER. This is true whether you are standing on the ground plane . . or several stories above the ground.

1 - 6

The HORIZON LINE is used in all perspective drawings to place the object relative to the OBSERVER'S eye level. The object location can be above, below or at the eye level.

CONE OF VISION DETERMINES OBSERVER DISTANCE

CONE OF VISION

The CONE OF VISION is a cone shape with its single apex at the OBSERVER'S eye and the circular base on the PIC-TURE PLANE. This cone can be thought of as your visual perception as you look at an object 90 degrees to your line of sight. The correct distance the OBSERVER should be from the OBJECT is determined by this CONE. The CONE gives a way to establish a distance , i.e. the observer can be too close and too far away. The object should be well within this 60 degree cone circle. In practice distortions are corrected by (1) increasing the distance between the Vanishing Points or (2) reducing the scale (size) of the drawing. They both have the effect of increasing the OBJECT to OBSERVER distance.

1 - 7

CV

OBSERVERSMINIMUM DISTANCE

TOO CLOSE

PICTURE PLANE

60° 60°

PERIPHERAL VISION LINE PERIPHERAL VISION LINE

OBSERVER

30° 30°

60° CONE OF VISION

90° FIELD OF VISION LINE

CV

ELEVATION VIEW

PLAN VIEW

PICTURE PLANE

CUBE WITHINCONE IS NOTDISTORTED - *LEADINGCORNER IS GREATERTHAN 90°

90° FIELD OF VISION

60° CONE OF VISION LINE

CV

CUBE OUTSIDEFIELD IS FAROVER DISTORTED& UNACCEPTABLE.

VP VP

CUBE WITHINFIELD APPEARSSLIGHLTY DISTORTED(STRETCHED). *LEADINGCORNER APPROACHES 90°

*IMPORTANT: LEADING CORNERMUST ALWAYS BE GREATER THAN

90°

*

*

*

PERSPECTIVE . . . . . . . . . . . . .

OPHEIM’89

. . .visually foreshortens dimensions until they vanish at the horizon.

. . . . develops forms in different locations from the eye.

. . . . allows objects to be rotated in many directions and angles.

LEFT CENTER RIGHT

BELOW EYE LEVEL

AT EYE LEVEL

ABOVE EYE LEVEL

. . . . is much like drawing what is seen on a sheet of glass.

1 - 8

PERSPECTIVE . . . . . . . . . . . . .

. . . gives solutions for reflections.

Look through magazines and newspapers for pictures of buildings, cars and products that have strong 3-dimensional qualities. Now find their vanishing point(s) and horizon line(s) by drawing lines on these pictures which extend outward to points of intersection.

Also find an example of the eye level of a person in a photo that is level with the distant horizon.

. . . . gives shadow solutions of objects.

. . . enables us to draw a wide variety of forms at different scales to represent objects of any shape and size.

1 - 9

Try this!2H

1 & 2-POINT PLAN/ELEVATION PERSPECTIVE PLAN/ELEVATION METHOD - 1-POINTPLAN/ELEVATION METHOD - 2-POINT

VIEWS ABOVE EYE LEVEL

PERSPECTIVE FROM ORTHOGRAPHIC VIEWS

The Plan/Elevation method uses a plan and elevation view to represent height, width, depth & details of the object. This method is most useful when there are existing plan and elevation drawings available. That is often the case in Architectural or Interior Design applications, but not usu-ally available in the early stages of the design process i.e. ideation stage. Often ideas must be presented in perspective before any refinement is possible. However, this method does

The Plan/Elevation Method can be applied to both 1-Point & 2-Point Perspective Drawings.

serve to give us a good example of the relationship of the observer, picture plane, object and horizon. Even though you may not use this method in everyday practice, it is important to understand its basic theory. It will help you grasp the many other variations.

2 - 1

LIMITATIONS

Unfortunately, there are certain limitations to this method.1. Too time consuming. Time must be taken to construct the

plan and elevation views unless they already exist. Multiple projections are needed to find any one dimension.

2. Drawing must be done on a very large drawing surface as the location of the vanishing points are dictated, thus robbing us of any practical control of the distance be-tween them.

3. Vanishing Points locations are often at an unusual dis-tance apart, causing them to be off the drawing or work surface.

4. Space is needed above and beside the drawing to give room for the plan and elevation views which makes the perspective view small when compared to sheet size.

The following exercises can be done in 1 or 2-point perspective or both. Show variations by using different horizon lines, so that the objects will appear to be above or below the horizon.

BASIC FORMUsing the Plan/Elevation Method, construct a drawing of a cube measuring 6" on a side, the base being 12" below the eye level. (Cubes have the same height, width and depth).

INTERIORDraw a floor plan and elevation of a room measuring 25" x 12' deep and 8' high including a window or door on each wall to your specifications and furniture using box forms. Construct a 2 point Plan Elevation perspective drawing of this room favoring the longest wall. Use a scale that will allow room for all drawings. Portions of near walls may be removed to allow vision into the room.

ARCHITECTURAL Construct a perspective of a house measuring 40' x 20'. Exterior walls are 10' high with a center apex at 15' on the short side. Use plan and elevation views to establish the system. Place doors and windows anywhere you wish in these views and project them into the perspective drawing.

PRODUCTDraw a plan and elevation of a toaster, based on a box. Show this in a perspective drawing using Plan/Elevation method. Measurements can be taken off an existing toaster. Otherwise design one of your own.

GRAPHIC Take any letter in the alphabet that has all straight lines. Do a plan and elevation of the letter form. Make sure that the proportions are as they should be for the type face that is se-lected. Construct the letter using the Plan/Elevation method.

IMPORTANT RELATIONSHIPS OF ELEMENTSWe can, however, learn from the Plan/Elevation method certain relationships and locations that are true, whatever method of perspective is used:1. Vanishing points always are on the horizon line (elevation

of your eye, called eye level).2. This line is always horizontal for objects resting on or

parallel to the earth's surface.3. All Vertical lines of the form are vertical in the drawing.4. All Horizontal lines of the form go to vanishing points and

are not measured directly, but are foreshortened.

Try this!

2H

2 - 10

PICTURE PLANE (PP)

PLAN VIEW

3. Draw a horizontal Ground Line (GL*).4. Draw an Elevation View of the object and place it on

the Ground Line directly beneath the Plan View.

ELEVATION VIEW

GROUND LINE (GL)

PP

GL*

SIGHT POINT (SP)(OBSERVER)

HORIZON LINE (HL)VANISHING POINT (VP)

ELEVATION VIEW

PLAN VIEW

PLAN/ELEVATION METHOD FOR ONE-POINT PERSPECTIVE

* The Ground Line can be placed anywhere you want. It works best near the bottom edge of the paper, leaving room for the view above. This line is not always on the ground as its name suggests. It can be above the ground and is used to define the bottom edge of the object.

EYE LEVEL

STEPS

This method is derived by positioning a Plan View over the Elevation View. Elevations in depth are projected from the Elevation View to a single Vanishing Point. Depth measure-ments are determined by the foreshortened projections from various details on the Plan View to a Sight Point which represents the position and distance the Observer is from the Plan View. Try this exercise using illustrated steps 1-10 in the examples following.

1. Establish a Picture Plane (PP) using a horizontal line. Leave space above the PP for a Plan View.

2. Draw a Plan View of the object and place it with one side touching the PP.

5. Establish the Horizon Line or Eye Level (HL) by drawing a horizontal line at a measured vertical distance above the ground line. This Horizon will determine the eye level.

6. Position the Observer (SP) by drawing a vertical line from the horizon where you wish. It is best to be closer to one side than the other. Establish the single Vanishing Point where this vertical line intersects the HL.

2 -2

7. Establish the Sight Point (SP) near the bottom on a vertical line from VP. This shows the distance from the Object to the Observer in Plan View.

8. Find all (foreshortened) depth measurements by taking a line of sight from the Sight Point to all corners or details in the Plan View. Also project lines from all corners of the Elevation View to the Single Vanishing Point.

PP

HL

GL

SP

VP

STEPS

ELEVATION VIEW(FRONT PLANE ONLY)

PLAN VIEW

PP

GL

Though shown here, it is not necessary to draw this SP lines below the Picture Plain.

SIGHT POINT (SP)(Observer)

9. Drop vertical lines from all sight lines where they cross the Picture Plane. This will give the locations for all lines in depth.

10. Complete the drawing by connecting all found points of intersection, where the vertical lines cross the lines to the Vanishing Point. The view will appear open, transparent, or solid depending on what lines are visible.

VP

PLAN VIEW

ELEVATION VIEW

FINISHED VIEW

2 -3

HL

5. Establish the Eye Level or Horizon Line (HL) by draw-ing a horizontal line at a measured vertical distance above the GL.

6. Establish the Sight Point (SP) by placing a point near the bottom of the sheet on the TL.

1. Establish a Picture Plane (PP) near top of sheet.2. Draw or attach a Plan View with one corner touching.3. Draw a vertical True Length (TL) line through

touch point.4. Near sheet bottom, draw a horizontal Ground Line (GL).

Draw an Elevation View placed on the GL to one side.

ELEVATION VIEW

GROUND LINE (GL)

PLAN VIEW

TRUELENGTH(T)

SIGHT POINT (SP)

TL

Horizon can be raised and loweredto any eye level desired.

PP

GL

HORIZON LINE (HL)

STEPS

PLAN/ELEVATION FOR 2-POINT PERSPECTIVE

The Elevation View in this case is not used as part of the drawing, so must be placed far enough to the side to not overlap the perspective view.

PICTURE PLANE (PP)* *

* Here we see that the distance between the Vanishing Points and their location depends on the angle the Plan View makes with the Picture Plane. This angle is determined by how much you wish to see of that side, i.e. the smaller the angle - the more prominent that side will be & the less prominent the adjoining side will be.

2 -4

.

7. Draw two lines from the SP to the PP that are parallel to the sides of the Plan View.

8. Draw two vertical lines where the parallels hit the PP.9. Establish Vanishing Point Left (VPL) and Vanishing

Point Right (VPR) at the intersection of the verticals and the HL.

PARALLEL

TL

VPL VPR

GL

VPL

PP

SP

Though shown here, it is not neces-sary to draw this SP lines below the Picture Plain.

VPR

STEPS

PP

SP

TL

GL

PARALLEL

Elevations of all points on the Object must be projected from the Elevation View to the TL and back to one of the Vanishing Points. The detail is on this elevation line at the location where the vertical from the Sight Point crosses the Picture Plane.

10. Establish all vertical dimensions by projecting from the elevation to the TL and back into the drawing using lines to the VP's.

11. Find depth dimensions by taking a line from SP to Plan View details, dropping a vertical line where it crosses the PP.

2 -5

HL HL

TRUE LENGTH (TL)

VPL

Make the side that is most important the lesser angle to the PP.

PLAN VIEW

GROUND LINE (GL)

HORIZON LINE (HL) (EYE LEVEL) VPR

PICTURE PLANE (PP)

Elevation & Plan views should include hidden lines or details not seen so that the view shows every detail.

ELEVATION VIEW

SIGHT POINT (SP)

The Plan/Elevation Method for 2-Point Perspective is derived by positioning overlapping plan and elevation views. Two Vanishing Points are used. Elevations are projected from the elevation view. Depth measurements are determined by the foreshortened projections from various details on the plan view.

For a better understanding, do this exercise using the 11 steps on the preceding page.

VIEW BELOW HORIZON LINE

Horizon will determine whether you are looking under, at, or above the object. Think of it as moving the eye up and down and the object remaining stationary.

2 - 6

Now draw the same object with a horizon line that is below the object and ground line.

HLVPL VPR

TL

PP

PLAN VIEW

GL

ELEVATION VIEW

SIGHT POINT (SP)

VIEW ABOVE HORIZON LINE

2 - 7

CONTROLLING VANISHING POINT DISTANCE

The Vanishing Points are often too close or too far apart as they are dependent on the location of the Sight Point and the angle that the Elevation View makes with the Picture Plane. One way to avoid this is to work backwards from the Vanishing Point distance you want to use. Above is an example of how the locations can be controlled. Start out with the Picture Plane and find the Sight Point by using angles which total 90 degrees. These angles are

located at the outer edge of your sheet and will be the distance between Vanishing Points. Draw a vertical True Length line through the Sight Point to locate the touch point of the Elevation View. Draw the Elevation View par-allel to the lines to Sight Point. You have now controlled the width between Vanishing points.

Continue as usual.

60° 30°

PLAN VIEW

SP

90°

VPL VPR

NOTE: A 30°/60° triangle wasused here, but any combinationof angles adding up to 90° is OK.

TL

VP DISTANCE

HL

BEGIN WITH THEPICTURE PLANE

2 - 8

SP

VPL

HL

TL

PP

PLAN

ELEVATIONS

VPR

PLAN/ELEVATION IN FULL SWING

Below is a drawing of a small Condo Unit. Note that some of the dimensions in the plan view have to be extended to the edge of the roof line to find the distance the walls are set in from the roof edge.

2 - 9

LIMITATIONS

Unfortunately, there are certain limitations to this method.1. Too time consuming. Time must be taken to construct the

plan and elevation views unless they already exist. Multiple projections are needed to find any one dimension.

2. Drawing must be done on a very large drawing surface as the location of the vanishing points are dictated, thus robbing us of any practical control of the distance be-tween them.

3. Vanishing Points locations are often at an unusual dis-tance apart, causing them to be off the drawing or work surface.

4. Space is needed above and beside the drawing to give room for the plan and elevation views which makes the perspective view small when compared to sheet size.

The following exercises can be done in 1 or 2-point perspective or both. Show variations by using different horizon lines, so that the objects will appear to be above or below the horizon.

BASIC FORMUsing the Plan/Elevation Method, construct a drawing of a cube measuring 6" on a side, the base being 12" below the eye level. (Cubes have the same height, width and depth).

INTERIORDraw a floor plan and elevation of a room measuring 25" x 12' deep and 8' high including a window or door on each wall to your specifications and furniture using box forms. Construct a 2 point Plan Elevation perspective drawing of this room favoring the longest wall. Use a scale that will allow room for all drawings. Portions of near walls may be removed to allow vision into the room.

ARCHITECTURAL Construct a perspective of a house measuring 40' x 20'. Exterior walls are 10' high with a center apex at 15' on the short side. Use plan and elevation views to establish the system. Place doors and windows anywhere you wish in these views and project them into the perspective drawing.

PRODUCTDraw a plan and elevation of a toaster, based on a box. Show this in a perspective drawing using Plan/Elevation method. Measurements can be taken off an existing toaster. Otherwise design one of your own.

GRAPHIC Take any letter in the alphabet that has all straight lines. Do a plan and elevation of the letter form. Make sure that the proportions are as they should be for the type face that is se-lected. Construct the letter using the Plan/Elevation method.

IMPORTANT RELATIONSHIPS OF ELEMENTSWe can, however, learn from the Plan/Elevation method certain relationships and locations that are true, whatever method of perspective is used:1. Vanishing points always are on the horizon line (elevation

of your eye, called eye level).2. This line is always horizontal for objects resting on or

parallel to the earth's surface.3. All Vertical lines of the form are vertical in the drawing.4. All Horizontal lines of the form go to vanishing points and

are not measured directly, but are foreshortened.

Try this!

2H

2 - 10

1 109876SCALE

VPRHL

TL

VPL

NEW METHODS

TWO-POINT MEASURING SYSTEMMEASURING SHORT CUTS

PREDICTED MEASURING POINTSCUBE CONSTRUCTION

DIFFERENT HML POSITIONSSELECTION OF SYSTEMS

MEASURING INSIDE & OUTSIDE BASE LINESDEPTH ELEVATIONS

INCLINED PLANES

Plan-Elevation Perspective gives us these three relation-ships which are also true for all other perspective methods:

1. The horizon line is a horizontal line and determines the eye level.

2. Vanishing points are always on the horizon line.3. The true length line is a vertical line that can be

used for measurement on the sheet of paper and is located anywhere between the Vanishing Points.

3 - 1

TWO-POINT PERSPECTIVE - A MEASURING SYSTEM

2 POINT PERSPECTIVE MEASURING SYSTEM

This measuring system will eliminate the use of plan and elevation views. This means any object can be drawn from its known dimensions and makes spontaneous change pos-sible. See the steps above.

Note: The Measuring Point locations are dependent on the VML location and change with each different location of the VML. MP labels are on opposite sides than the VP's., i.e. MPR on the left and MPL on the right.

VML (TL)

VPL HL VPR

P

FIND MEASURING POINTS AS FOLLOWS:

1. Establish Horizon Line, VML (TL)* where you want and both Vanishing Points at a good distance apart.

*TL is changed to the (VML) Vertical Measuring Line.2. Draw half circle with center as radius through VP'S.3. Place the VML anywhere you wish between the VP's. Find

point "P" where the VML intersects the half circle.

4. Using VP'S as pivots, make an arc with distance VP to point P as radius from both Vanishing Points.

5, Measuring Points (MPL & MPR) are found where these arcs intersect the HL If a compass isn't large enough, measure the distance VP-"P" (dashed line) and transfer this to the same distance along HL to find the MPs.

HOW TO FIND MEASURING POINTS

VML

VPL HL VPR

P

MPR MPL

3 - 2

The True Length Line is now used as a Vertical Measuring Line (VML). A Horizontal Measuring Line (HML) is also used to find foreshortened depth measurements. The HML is a hori-zontal line which give measurements to the left and right of the VML. Measurements are taken from locations on the HML to Measuring Points. The depth is found where each line to the Measuring Point crosses the Base Line (see circled points). Heights are projected from the VML to the VP's along the Base

HOW TO USE THE MEASURING SYSTEM

Line plane, All other measurements inside the Base Lines are found by projecting back to both Vanishing Points.

THE MEASURING POINT SYSTEM

VMLVPL HL VPR

P

MPR MPL

36L 78R

EYE LEVEL: 48"DIMENSIONS: 36"x78"x30" HIGH

IMPORTANT: Never use Point "P"as the leading corner of a box. TheHML is determined by Eye Level andmust be well above Point "P".

HORIZONTAL MEASURING LINE (HML)

MEASURED HEIGHTOF EYE LEVEL

BASE LINE (BL)

BL

0

30H

Accessariesgive the drawing scale.

Note: The dashed lines from the HML to the MP's may ap-pear to cross several lines, but actually only cross the BL plane at the first line on the "ground" surface. This will always be the first line the measurement crosses.

3 - 3

If a circle is bisected by a hori-zontal line forming two VP's which are connected to any point on the perimeter of a circle, a 90° corner is formed. Therefore, any method that produces a 90° cor-ner from the VP's will work as well. Following are examples of 3 different ways to find MP's using angles rather than a circle. These examples incorporate the use of 45° & 30°/60° angles drawn from VP's creating in-tersecting lines which form 90° corners. This point would be on a circle had it been drawn. The distance of Point P from the Vanishing Points is then taken to the Horizon Line as always.

2-POINT CENTER CONSTRUCTION ABC

AB

C

2-POINT RIGHT CONSTRUCTION

2-POINT LEFT CONSTRUCTION

Gives equal orientation of sides "A & B"

30° - 60°

ABC

30° - 60°

SHORT CUT

Gives orien-tation favor-ing side "B"

45° - 45°

Gives orientation favoring side "A"

EQUAL 1/2 EQUAL 1/2

P

VPL MPR

VML

VPRMPL

EQUAL 1/3EQUAL 1/3EQUAL 1/3

HL45° 45°

90°

EQUAL 1/2 EQUAL 1/2EQUAL 1/4 EQUAL 1/4

EQ. 1/8

P

VPL MPR

VML

VPRMPL

HL

EQ. 1/8

30° 60°

90°

EQUAL 1/2EQ. 1/4EQ. 1/4

EQUAL 1./2

EQ. 1/8 EQ. 1/8

P

VPLMPR

VML

VPRMPL

HL

60° 30°

90°

3 - 4

EQUAL 1/2

All 90° corners

VPHLVP

The idea is to predict the location along a HL of these three variations to measuring point. This will save the time it takes to draw circles or arcs to find these MP's. Above are samples of how this might look Establish the VP's as far apart as practical and divide the HL into these relationships by division.

MEASURING POINTS FOUND BY PREDICTION

Try this!

2H

NOTE: Since locations are in known positions for these systems - CIRCLES AND ANGLES ARE NO LONGER NECESSARY.

The VML and MP's are found the same way as in the circle method. Beyond getting rid of the circle we also find that the MP locations are predictable in these circumstances.

MPLVMLMPRVPR1/8 1/8 1/4 1/2VPL

2-POINT LEFT

2-POINT RIGHT

MPR MPLVPL 1/41/2 1/8 1/8 VPR

VML

MPLVMLVPVPL 1/31/3

MPR

1/2 1/21/6 1/6

2-POINT CENTER

3 -5

18 24 36120 6 30VPL

MPL/R

VML MPLMPR

MPL

VPRMPR

CENTER CENTER

LEFT RIGHT

2-POINTLEFT

2-POINTCENTER

2-POINTRIGHT

MOVE YARDSTICK LEFT OR RIGHT

TO CENTER DRAWING

VML-L

EFT

VML-CENTE

R

VML-RIGHT

"YARDSTICK" PERSPECTIVE

Try the use of a yardstick to measure distances. This gives 36" between Vanishing Points and is easily divided to find halves and thirds to find MP's & VML's. Simply tape the yardstick to the top of the drafting board or table. Push pins can also be at MP and VP locations to act as stops for the straight edge. This eliminates looking every time to see if the line goes to these points.

3 - 6

BA

C BAC

ABD

I POINTCHAPTER 4

MEASURING SYSTEM

2 POINT LEFT2 POINT CENTER

C

C

I POINTCHAPTER 4

P

BAC

BAC

2 POINTRIGHT

MEASURING SYSTEM

DBA

HL

D D

ABD

VPL VPR

B

B

A

A2 POINT RIGHT

2 POINT CENTER

I POINTCHAPTER 4

I POINTCHAPTER 4

P

HML

MPLMPR

MPLVPL

HML

MPR

BLBL

BL

HL

Working above the HL will give a view that "floats" above the eye level. Here the solution is the same for three locations of the HML, on top and bottom of form and on the surface. Note that the BL changes for each view. The BL is always the first line that the measurement from HML crosses.

BL

BL

HL

VPL MPR MPL

BL

HML

VPR

VPR

VPR

VPL

TL

TL

TL

Draw a box that is 1 x 3 x 2 high that is above eye level.

APPLICATIONS OF DIFFERENT HML LOCATIONS

Each system or method will give a different location of the object from the viewer. This varying vantage point is a choice to make for each drawing. Coupled with many choices in eye level, these many variations become possible. Above is an example of the locations for different Measuring Systems.

SELECTION OF MEASURING SYSTEMS

3 - 7

HL

NOTE: THESE MP'S WERE FIRST FOUND BY USING A POINT "P" ON A CIRCLE.

CONSTRUCTION OF CUBES USING THE MEASURING SYSTEM

MPLHL

BL

MPRVPL VPR

HML

BL

VML

LEFT 55

EYE LEVEL: 8 UNITSDIMENSIONS: 5 X 5 X 5 UNITS

RIGHT10 15100

Above diagram shows the use of two side views (see shaded squares) of a cube to demonstrate which dimensions are used along the vertical and horizontal measuring lines. It is not necessary to draw these views when their dimensions are known. Note that the height dimension is on the VML with its base at the HML. The depth dimensions are on the HML and are to the left and right of the VML. It is im-portant to remember that the projection from the measured

distance along the HML to either MP stops at the first base line that it comes to. Any dimension then must go to either vanishing point to get into the view In practice, it is not necessary to draw the line to the MP beyond the BL. Example shown at 10 & 15 right. The line to MP stops at the BL and then goes to VPL. This eliminates unnecessary lines and clutter.

3 - 8

2-PT. CENTEREYE LEVEL: 5'DIMENSION: 12'X14'X13' HIGH

Here is a more extensive use of the measuring points to construct a drawing of a small house. This shape requires several different measurements to develop the form.

The best approach is to develop the drawing on the ground plane and then work vertically to develop the vertical planes and details.

2-POINT MEASURING SYSTEM EXTENDED

Elevations or plans were not needed. Each dimension of the house was measured along the HML or VML respectively. Depth dimensions are projected to the measuring points stopping at the base lines and then vertically for heights and to VP's for depths. Height dimensions are all taken from the VML and projected into the drawing using both Vanishing Points.

VPLMPR

VML

MPL VPR

BLBL

HML

LEFT MEASUREMENTS TO MPL STOP AT BL RIGHT MEASUREMENTS TO MPR STOP AT BL

HL

3 - 9

HML

BL

MPL

VML

HLVPL

MPR

BL

5

9

55

99

9

9

5

5

3

3

2

2

35

5

3

0LEFT RIGHT

It is sometimes necessary to draw the view set back from either or both base lines. Nothing really changes. The vacant space is measured first and the box dimensions are added to it. This moving of an object left or right of the TL line results in a visual shift of the object. The more practical application of this method is to leave room for possible ad-ditions to the surface such as overhanging roof lines, pro-truding knobs on products or lenses on cameras.

MEASURING INSIDE THE BASE LINES (VISUAL SHIFT)

VPR

Notice how the second box appears to be shifted to the left This shift can also go far enough right to be very near center.

3 - 10

HML

MPLMPR

VML

B

VP1

C

D

F

E

G

H

BL

VP2HL VPR

BL

VPL

A

3 - 11

ELEVATION MEASUREMENTS USING DIAGONALS

Usually the elevations are found by taking elevations from the VML to each VP taking the vertical distance to the surface first and then along that surface to the needed position. Sometimes it is more expedient to take the mea-surement directly to that position. Line AB shows how the measurement can be taken diagonally to the HL along the ground from the VML ground point through Point B which locates a VP1 on the HL Then a line is drawn back to

the needed height on the VML. A vertical between points A and B gives the right height for that point. Height CD is transferred to EF & GH using the same method. See vertical C - D transferred E - F & G - H.

Any known vertical dimension can be transferred to another vertical in the same manner. ALL LINES REPRESENT THE SAME HEIGHT IN PERSPECTIVE.

HML

VPL MPR

VML

HL

MPL

BL BL

FLAPS HINGED ALONG HORIZONTAL EDGES

Y

ELEVATION VIEW

X

Since the flap angles cannot be measured directly, a side elevation view is needed for each flap to give the X & Y di-mensions of the flap edge. See the side elevation below which shows the measured flap at the wanted angle and the resulting X & Y dimensions.

138°

3 - 12

find the next edge. Going to the wrong side for the VP is a common mistake. When this happens the flap will not appear to be get larger as it comes toward you or smaller as it goes away. When they look wrong, vanish to the opposite side.

All Vanishing Points are located along a vertical at VPL and VPR. Lines parallel to sides still go to the same VP's. Find this VP by taking one flap edge out to a point above or below the VP's on the HL and then back to the opposite corner to

BL

VML

VP2

HL

BL

MPR MPLVPL

TO DISTANT VP4ON HORIZON LINE

VP1 VP5 VP3

For vertical flaps all Vanishing Points are located along the Horizon Line. The VP is found using a ground line in the flap direction to the HL. This is then taken back to the top edge of the hinge constructing a flap using a

FLAPS HINGED ALONG VERTICAL EDGES

TO DISTANT VP4ON HORIZON LINE

VPR

vertical line at outer edge. Flap measurement uses the X & Z measurements of a flap in plan view which may be drawn to scale on a separate sheet. This is the most ac-curate way to find flaps at any angle.

HML

X

z

PLAN VIEW

147°

3 - 13

CONSTRUCTION OF UPHILL & DOWNHILL PLANES

ROAD CONSTRUCTION USING MULTIPLE VANISHING POINTS

The same technique used for boxes with flaps also works for surface development such as roads or uneven terrain. Measurements can be tricky, but as long as you stick to measuring locations and their heights, you can find the slope of any road Notice that all level roads vanish at the horizon line and all down and uphill roads vanish above or below the horizon line. Once details and terrain features are added, it can be quite convincing.

VPL

VP-UP

VP-DOWN

VP-TURNVPR

VP-DOWN

VP-DOWN

VP-UP

HLHL

LEVEL

LEVEL

LEVEL

LEVEL

LEVEL

LEVEL

UP

UP

DOWN

DOWN

DOWN STOP

3 - 14

MEASURING OUTSIDE THE BASE LINES

MEASURING OUTSIDE OF BASE LINE

It is sometimes necessary to measure forward of the BL. This should be avoided, but can be done if the leading corner remains within a reasonable Cone of Vision. This often is necessary when additions are made or details are extended forward after the initial surface is established.

ADD 3 UNITS TO THE FRONT:1. Extend Left BL forward of O.2. Measure 3 along HML to the right. This is a measure to the right of 0, but it is still a Left measurement.3. Draw a line from the MPL through 3R crossing the ex tended BL to find +3 forward. Take this measurement to the VPR and project other details forward to that line.

VPLHL

VPRMPR MPLVM

L

0

+3

3R

BL

BL

CONE OF VISION

Add-on section is"outside" the originalBase Line.

Original boxwithin the Base Line.

3 - 15

4. Measure heights by taking the VML base through X to the HL at VPX and back to the height on VML. The height of the box is where it crosses the vertical from point X.

5. Now measure to new BL's as usual for depths by using the HML measurements to the MP's starting with each corresponding O point.

VPL HL

BLBL

O 7

MPR

HML

VML

O2

MEASURING INSIDE OF BASE LINE

You may also want to use the measuring system directly to a view inside the Base Lines. This method can be used to shift a view to the left or right and back into the view where you want without changing the Measuring Points or the Measuring Lines. Follow the 5 steps above.

VPR

1. Pick a point X where you want the front corner.2. Draw lines from MPR & MPL through X to the HML

to find O's location.3. Place a VML anywhere along the HML.

MPL

7

2

X

VPX

4

4

3 - 16

A MEASURING SYSTEM USING A LAYOUT TABLET

When you are sketching or laying out a line drawing for a new drawing or rendering, it is a good idea to work on a sketch pad or layout tablet. You can quickly set up a system by folding the page a couple of times to divide the sheet into MP halves, quarters and eighths. Verticals can be drawn by using a triangle or a small T-square. You might try using your thumbs along the bottom edge of the tablet when using a triangle.

4 steps to the right show how this is done. This automati-cally gives you enough distance between VP's to give a 60° Cone of Vision.

1. Open sketch pad and establish Horizon Line & Vanishing Points. Find MPR @ centerline.

2. Find VML by folding top sheet to centerline, crease @ top & fold back..

3. Fold to VML, crease @ HL & fold back to find MPL.

4. Finish drawing as usual.

MPR

MPR

MPR

HLVPL VPR

HLVPL

HLVPL

VML

VML

VML

MPR MPLHLVPL VPR

HML

3 - 17

Good sketches can be drawn first with straight line to VP's and then traced on a clean sheet with straight edges or free-handed as shown below Box forms can be easily drawn below, above and at the Eye Level.

In this case the boxes were drawn randomly overlapping them with eyeball measurements. As long as VP's are used, the perspective holds its own. It is interesting to see how the different boxes will appear to be in front or behind depending on which lines are used in the final view. Several variations to the same drawing can be shown this way. Now, all you have to do is add details. Heavy outlining gives impact to objects and shows which surfaces are open when looking through box openings.

FREEHAND SKETCH USING TABLET & VP'S

3 - 18

MEASURING SYSTEM - CONTOUR DRAWING

This is a simplified version of a boat hull that has all curved surfaces. Two elevations are used along HML to help measure each section. Station Plains (10 here) are selected either at key locations or equally spaced. These are used to draw section

views on each station plane in the perspective view. These can be measured point for point or a grid can be used for plotting the section onto each plane. The object is then found by con-necting the common section points horizontally. Add details.

HL VPL VPR

7 8 9 101 2 3 4 5 6

2

3 4 56

789 10

ç

ç

HML

N7705

3 - 19

5

8

FRONT END

FRONT END

2

TOP

1

TOP

END

FRONT

TOP

END

7

END

4

TOP

END

3

FRONT

6

FRONT END

TOP

FRONT

TOP

FRONT

TOP

FRONT

TOP

END

This layout shows orthographic views of different objects. Use these forms to practice the short cut Measuring System. After selecting which system you want to use, develop the objects using a light construction line and then heavy-up the object lines to make them stand out. Do at least one for each system, i.e. 2-Pt. Right or 2-Pt. Left, 2-Pt. Center & 2-Pt. Measuring System.

Do These

3 - 20

��

4 - 1

ONE-POINT PERSPECTIVE OF BOX FORMS - MEASURING SYSTEMMETHODS FOR ONE-POINT PERSPECTIVE

CONVENTIONAL METHOD TO FIND MPONE-POINT THEORY & PRACTICESEVERAL SHORT CUT METHODS

HLVP MP

RECORD/PLAY REW/REV FF/CUE STOP/EJECT PAUSE

TAPE

60% CONE CIRCLE

4 - 10

There is a fib advantage to use One-Point for sketches of objects that have most detail on the front face. This allows you to draw in full size or scale and measure everything in true length. A great deal of depth is still possible. Another advantage is that circles are still circles and can be easily

represented. In later chapters we will cover the circle as it turns from our line of sight and becomes elliptical. The draw-ing below was constructed using a modular application of a cube that was multiplied to double the proportion. Details were measured in true length on the front surface.

SMALL OBJECT SKETCHES

TRY THESE EXERCISES:

1. Before creating your finest drawing ever, become familiar with the 3 different ways to find the MP. Then try to draw several different sized boxes on the same floor plan. Make some large and small. They can represent rooms or objects like furniture.

2. Develop a Floor Grid of any size. Place several vertical planes representing walls that are perpendicular to your line of sight and at 90° to you. Give 6 inch thickness to these walls. Place several pieces of furniture on the grid

and at least 1 piece turned at 45°. Find its VPs on the HL. Hint: It is always best to draw a plan view of your room first.

3. Draw a cube and multiply to a large structure. Make details for product applications similar to the drawing above.

4. Construct or trace a large letter facing you. Take all edges to a VP. Pick a letter depth and measure remaining depth details.

30°

HLVP MP

1

2

3(1.732)

BA

C

D

60°

ELEVATION VIEW

C

PP

This distance represents how far the Observer is from the Picture Plane in Plan View.

What you get is a room that appears to be square, i.e. as deep as it is wide.

VP

CONVENTIONAL METHOD TO FIND MP

One-Point Perspective also uses a Measuring Point to find depth measurements. A MP is found by taking the longest diagonal from the VP and rotating this distance to the HL to find point D. This is taken down to point B. A 30/60 degree triangle is drawn using corner ABC. The distance of VP to point C on the triangle is rotated to the HL to find

the MP. As you can see from this construction, the MP is approx. twice the distance from the VP (1.732 + BD) as the length of the longest diagonal. This technique involves strenu-ous construction and uses a large area of space below the view which is usually unavailable on a small sheet. SHORT CUT methods follow . . . . . . .4 - 2

VP

60° CONE OF VISION CIRCLE. R = 60% x VP-MP DISTANCE

MP

ALL SQUARESWITHIN CONE CIRCLEAPPEAR TO BE VISUALLYCORRECT.

STRETCHED DISTORTIONDISTORTION BEGINS

HL

DEFINITIONS & DISTORTION

n One-Point Perspective the near surface of the object touches the picture plane. This creates a single vanishing point and all other lines are either horizon-tal or vertical. Unlike Two-Point Perspective where the Vanishing Points are distant from the view, this Vanishing Point is now inside the view or its proximity.

A Measuring Point is also used, much like Two-Point Perspective.

DEFINITION: The Measuring Point is a point along the HORIZON LINE that is the same distance from the VP as the OBSERVER is from the PICTURE PLANE and is then an arbitrary chosen distance.

This distance must be far enough to give a 60° Cone of Vision. Without this quideline, distortions will appear at the outer edges of the view. Drawing shows how squares distort as they pass be-yond the circle.

CONE OF VISION CIRCLE

There are 3 short cut ways to find a Measuring Point They are as follows . . . . . . .

4 - 3

MP

0

VP

HML

HL

Estimated Square DepthLooks deep

Looks close

12

MP

Found Sq. Depth

HLVP

HML 07

R=0.6 x

MP-VP

MP

015

VP

HML

HL

Found Square Depth

Longest Diagonal

1 2THREE SHORT CUTS TO FIND MEASURING POINT

DIAGONAL METHOD1. Draw HML in size and scale needed.2. Establish HL and extend to right or left

side.3. MP is found by taking the longest diago-

nal (VP-O) up to HL and doubling that distance to find MP.

3.

SQUARE DEPTH METHOD1. Draw HML in size and scale needed.2. Establish HL and extend to right or left.3. Move a horizontal line up and down until it

appears to represent a square lying on a horizontal surface. Draw a diagonal to find MP. This moves the observer forward and back until it looks visually correct and gives a way to vary the depth.

CONE OF VISION METHOD1. Pick a MP at a random distance from the

VP.2. Draw a circle with a radius 60% of the

MP-VP distance to represent a 60° Cone of Vision.

3. Draw the HML anywhere within this circle. The object will not be distorted.

4 - 4

MP

012

VP

HML

24

12

3648

17

MP

012

VP

HML

HL

12

MP

012

HL

HL

VML

VP

HML

ONE-POINT FLOOR GRID METHOD

1. Find the VP, HL & MP by using one of the methods just discussed. Be careful at this stage because the looks of the entire drawing will be effected by this depth and the location of the VP.

2. Draw a line to MP from the most distant point on the grid (12 here). This will cross the VP line from O to give 12 deep.

3. Begin to grid the square by draw-ing depth lines using the equal measurements along the HML to the VP.

4. Finish the grid by drawing horizon-tals where these depth lines cross the diagonal to MP at dots.

5. Increase the depth to 17, 24 or any additional depth by using di-agonals wherever the measurement is needed.

6. Also add vertical planes where they are needed using VMLs along the HML.

NOTE: It is not necessary

to grid the square if these measurements are not needed.

4 - 5

VML

HML

HLVPMP

VML

HML

HLVPMP

VML

HML

6'-4" HL

SHORT CUT STEPS USING ELEVATION VIEW

Following steps show how elevation and details can be used to develop a floor grid.

1. Construct an elevation with Horizon Line at desired height (6 feet and 4 inches here)

2. Establish a Vanishing Point (VP) off center and con-nect ground lines to front two corners.

3. Establish MP at twice the distance of long diagonal (VP-A) this will give a short cut solution for a 60° Cone of Vision and will not be distorted. If the room is made wider, a new MP should be found.

4. Connect all other corners to VP.5. Construct a grid by connecting measurements along

HML to the VP and drawing horizontal depth lines through their 15 foot back wall.

6. Construct back at 15 feet deep using verticals at corners.

7. Project fireplace, stairs and table using grid and measured heights along each wall from the elevation view.

NOTE: Only 1 MP is shown here, but there are always 2 MPs. 1 to the left and 1 to the right, each the same distance from the VP. Either or both MPs can be used for depth measurements.

.4 - 6

HML

VP6'-4" HL

VML

HML

6'-4" HL

4 - 7 DETAIL DEVELOPMENT

Different types of views can be developed by using the VP and HL at various positions.

VP

ELEVATION VIEWPLAN VIEW

TO MPR

5' HL

TO MPL

VMLVML

TO MPRVML

4 - 8

COMBINATION ONE AND TWO-POINT PERSPECTIVEObjects are positioned using the grid for plan view and measured vertically using the front elevation plane (VML).

Two MPs equidistant from the VP are used as VPs for a box rotated 45°. For other degrees of rotation each corner of the object is located on the perspective grid first and then lines are drawn through the respec-tive corners to locate their VPs.

LINCOLNPLACEMALL

PRO GOLF�TROY’S PETS

MERV'S CIGARSOPE�

Music

Roger's�Fabrics

THEBAYCOMPANY

10' HL

To MP

4 - 9

LARGE SCALE DESIGN DRAWINGS

This drawing is projected from elevation views and uses a floor grid. This semi-detailed view of a shopping mall shows how a 1-point view might be used to develop a design solution. Depths and elevations are changed at randomuntil the ultimate solution is found.Eye level here is at 10 feet.

This process allows you to draw as you think using modular forms or building blocks much like an erector set. A single block is multiplied to gargan-tuan proportions, if necessary, or can be divided down into minuscule increments.

MODULAR PERSPECTIVE

SQUARE TO CUBE METHODMULTIPLICATION & DIVISION OF FORMS

VERTICAL SURFACE MULTIPLICATIONHORIZONTAL SURFACE MULTIPLICATION

REDUCTION & ENLARGEMENT

MODULAR PERSPECTIVE

5 - 1

be placed in the right scale and the correct distance from the HL for desired eye level. To avoid distortion it works best if the front corner of the cube is near the center between the VPL and VPR or placed within a 60° cone circle. The system works by assuming that the Horizontal Diagonal HD is a true horizontal.This is only true at the center of VP's, and is not always the case in other methods.

The 2-Point Center, Left or Right Method discussed in Chapter 3 can be used also to construct a cube if measuring points are used. Measuring Points defeat the purpose of Multiplication or Division since measurements of any size can be made directly. Note that this method works for a cube only.

DEVELOPMENT OF CUBE USING SQUARE TO CUBE METHOD

SQUARE TO CUBE PERSPECTIVE METHOD

Up to this point we have used dimensions given to us either by plan and elevation views, written dimensions or measure-ments from actual objects. Many times it is necessary to construct a drawing of an object that does no yet exist and whose dimensions are not yet known. In this case it is a good idea to construct forms from building blocks called cubes. These are the same dimension in height, width and depth. If the proper number of cubes can be placed together in the right numbers, any proportion in height, width and depth can be constructed.

The following steps show how a cube can be constructed in 2 Point Perspective from its square elevation. This elevation can

HL VPL VPR

HD STEPS FOLLOW

5 - 2

1. Construct HL with both VP's as wide apart as practical for your paper size and work surface. Use a 60° Cone of Vision Circle at center.

2. Place side elevation square of needed scale within the Cone of Vision and near center of the VP's and at desired distance below eye level.

4. Find front corner of cube by projecting lines from VP's through the 4 corners of the Horizontal Diagonal Plane.

5. Draw a vertical at the found front corner.

3. Find Horizontal Diagonal (HD) by taking the vertical diagonal of the square to the horizontal base line. This gives the Horizontal Diagonal Plane in true length.

Note: This works at this location because the Horizontal Diagonal is a horizontal line at center of VP's only.

6. Take outside corners to both VP's.7. Finish the cube by taking a vertical through the found

back corners.8. Now the cube can be multiplied to other sizes. .

HL VPL VPR

60° Cone of Vision

HL VPL VPR

HD

HL VPL VPR

HL VPL VPR

5 - 3

VPLVPRHL

DIAGONALS HALVES MULTIPLY

MUL

TIPL

Y

MULTIPLY AGAIN

MULTIPLICATION OF ORTHOGRAPHIC AND PERSPECTIVE VIEWS

HD

MULTIPLICATION OF ORTHOGRAPHIC AND PERSPECTIVE VIEWS

Once the cube is developed, it can be used as a building block to construct more complex forms of different propor-tions. In orthographic views the divisions are made by using diagonals to find the center of each surface and then lines parallel to the sides will divide the surface in half. The same method works in perspective.

5 - 4

FULL AND HALF DIAGONALS CROSS TO FIND THIRDS

VP

USE CENTER LINE TO MULTIPLY A DISTANCE BETWEEN VERTICALS

MULTIPLICATION/DIVISION IN ORTHOGRAPHIC AND PERSPECTIVE VIEWS

A

D

B

C

Rectangle ABCD is divided in half, then into thirds (three equal divisions).

Once a multiplied space is used, it can be repeated to infinity.

5 - 5

DIVIDING LINES AND RECTANGLES

5 - 6

HL VPL VPR

Example above shows how to divide a line AB into seven equal spaces using parallels.

A

B

A

BAny Angle

A

B7 equal units Join 7th to A Draw parallels

A

B

A

B7 divisions

A

B

A

BIf a rectangle is divided vertically into equal spaces, a diagonal will divide those spaces horizontally,

A

B

VERTICAL DIVISIONSThis concept can be applied to a staircase construction. The height is divided into the number of equal steps needed and the diagonals do the rest for you.

VP5

0 5

5

HORIZONTAL DIVISIONSDivisions along foreshortened horizontal lines are similar except they are not equal. The divisions are taken on a horizontal & fan from found VP on the HL by taking 5 through 5 to VP5 & back to each division.

Radiate lines from a chosen Radiation Point near center of form. Pick an enlargement point and draw lines parallel to original view.

RP

RP

RP

RP

ENLARGEMENT & REDUCTION BY USING A RADIATION POINT

Radiate with diagonals through corners and use parallels.

5 - 7

Radiate lines from a chosen Radiation Point near center of form. Pick an enlargement point and draw lines parallel to original view.

RP

RP

RP

3. The cube can be multiplied to 4 one cube at a time or by using full diagonals of the doubled height.

HD

VPLVPR

HL

VERTICAL SURFACE MULTIPLICATION OF A 3 X 4 X 2 HIGH OBJECT.

The cube is used as a building block for the larger object. Once the cube is divided by diagonals to find its center, it can then be multiplied into any proportion wanted. This can be accomplished in different ways depending on which multiplication is done first. What makes this method so great, is the option of changing the size quickly, without having to start all over again.

1. The cube side is divided by crossing diagonals. A line through midpoint is taken to VPL and used to multiply the cube to the left two times using half diagonals making the width 3 deep.

2. Cube is multiplied in height by taking the half diagonal to the extended front vertical line of the original cube. Be careful about using exact points, as noticeable error can occur. Check the doubled height, by using a ruler or similar device.

MULTIPLICATION AND DIVISION IN PERSPECTIVE

5 - 8

HL

HORIZONTAL SURFACE MULTIPLICATION OF A CUBE TO A 3 X 4 X 2 HIGH OBJECT.

The horizontal plane can be multiplied also by using horizontal diagonals as first constructed on the original cube.

This is the best approach if the object covers a large hori-zontal surface. An overall grid can be constructed in this manner and will be discussed in the following section

HD

HD

HD

HD

MULTIPLICATION AND DIVISION IN PERSPECTIVE

1. Draw additional Horizontal Di-agonals through the far edge of each found square.

2. In this method the height of the doubled cube is found tak-ing the full diagonal (dashed line) from the second horizontal square.

5 - 9

VPL VPR

REDUCTION OR ENLARGEMENT OF RECTANGLES

The common corner picked will remain stationary. This means that if you want the top to remain the same distance below HL, pick the top corner instead of the bottom.

HL VPL VPR

HL VPL VPR

X

X

Y

Y

DIAGONAL ENLARGEMENT

Z

Z

If you draw a box shape and decide later to enlarge by some unknown scale. You can enlarge each side in proportion by drawing a diagonal from a common corner (Z) across each side. Pick the new size any-where along this diagonal (X) and draw a new line from VPL through it and extend to the other VPR as well. The second diagonal will give you the depth of the second side at Y.

5 - 10

5' HL

ORIGINAL SQUARE

2 POINT INTERIOR GRID

This grid was developed from a large square using the Square To Cube Method. Once the back two walls are developed and divided equally in the back corner, the wall heights are projected outwards and diagonals give the vertical divisions. Develop the grid on the walls first and then the floor. This grid has a good application for an interior showing a corner of a room. The figure adds scale with eye level at 5'.

5 - 11

DEVELOPMENT OF HORIZONTAL PLANE

The horizontal plane can be developed in several ways from the Square to Cube Method. Once the cube is developed, there is a square in 2-Point Perspective on the horizontal plane. This square can be multiplied as in previous exercise by using the HD of many multiplied squares.

VERTICAL SQUARE

HORIZONTAL SQUARE

VPRHL

HORIZONTAL MULTIPLICATION TO DEVELOP GRID

VPL

5 - 12

HL DVP VPR

DEVELOPING THE DIAGONAL VANISHING POINT

If the depth diagonal of each of these squares is also pro-jected to the horizon line, we find that they will all converge at the same point which is exactly midway between the VP's. This point is called the Diagonal Vanishing Point (DVP). What appears here is a horizontal plane that is divided into squares using 2 different methods. One method is to use the Left and Right Vanishing Points, and the second method uses the Diagonal Vanishing Point.

DIAGONAL VANISHING POINT

VPL

Following are two grid variations for 1-point perspective & 2-point perspective that is derived from this process.

5 - 13

VML2

This Grid could be used to make many different drawings with objects placed in many different locations.

This makes it possible to use any corner as the leading corner of a view or a detail within a larger drawing.Figure below shows the development completed.

VPR

VML1

VML3

2

3

1

ORIGINAL SQUARE

VPL HL

TWO-POINT GRID

If the lines to the DVP are removed, we are left with a 2-Point Perspective Grid. Each multiplication represents a square in 2-point perspective. The vertical measure-ments are taken along the VML which is the vertical multiples of the original square used to set up the grid. Other VML's can be found by using a ground line to the HL and back to different heights.

TWO-POINT GRID USING VERTICAL MEASURING POSITIONS

5 - 14

VML2

D

DVP VPRVPL HL

ONE-POINT GRID USING VERTICAL MEASURING POSITIONS

Now the process can be reversed. All lines to the VP's are removed and those to the DVP and horizontal remain. This gives a grid that's slightly smaller, and is in 1 Point Perspective. Each VML is the same dimension as the base at any depth in the grid. Multiply that distance vertically by measuring or 45° construction shown.

Figure below shows the 1 Point Perspective Grid with a stack of smaller squares using the height the same dimension as its base at that depth in the drawing.

Any diagonal (D-VPR) to either VP will give more grid hori-zontals where needed.

45°

ONE-POINT GRID

Note how distortion increasesoutside the Cone of Vision Circle.

D

45°

5 - 15

PERSPECTIVE TRACING GRIDSDifferent grids have interior or exterior orientations. These are examples of tracing grids that were developed for small consumer product drawings and large interior layouts. There are many "ready-made" grids available. They can be very helpful, but in many cases are not very accurate and allow for over distortion. It is much better to develop your own trace grids that meet your specific needs. Any of the Measuring Systems will work in both 1 and 2 Point Systems.

5 - 16

DRAWING FIGURES INTO YOUR PERSPECTIVES

Many times figures are needed in drawings to give warmth, show how something functions, and give scale. Figure drawing scares many people needlessly. You might begin by tracing figures from pictures in newspaper and magazine ads. Learn to simplify features and show relaxed stances. Once you have a style that works, try these steps for more originality. Each step is traced from the other.1. Establish eye level.2. Block in main body parts and line in arm

and leg positions, keeping good proportion.3. Outline main head, arm and body features.4. Tighten details, using simplified face and hands.

This might take several steps.5. Add accessories to meet the requirements of

the drawing and increase interest.6. Figures can easily be changed, so keep your

originals on file. Enlarge or reduce using photo copies.

5.4b. 6.1. 2. 3. 4a.

1. 2. 3. 4a. 4b. 5.

5 - 17

+16

DVPVP

8

5

8

0

+8

1624

UNIVERSITY PLAZAUNIVERSITY PLAZA

DEVELOPMENT OF MEASURING PLANE GRID

OPHEIM '92

+16

DVPVP

8

5

8

0

+8

1624

UNIVERSITY PLAZAUNIVERSITY PLAZA

6 - 1

VPR

DRAWING SHEET SIZE

FIELD OF VISION LINE

VPL

CONSTRUCTION OF A MEASURING PLANE GRID

Measuring Plane Perspective is much like any two-point system except it is based on the use of a very large Field of Vision Circle. The view is moved to the outermost VP (either side) and allowed to distort slightly beyond the Field. The resulting system looks very similar to a One-Point Perspective except that all horizontal lines are not parallel and go to a distant VP.

This is appealing because the resulting view is more real-istic as long as the distortion is kept under control and not over done. One-Point Perspective is often thought of as being rather static and uninteresting and not the way we ordinarily see things.

0

5

8

HL HL

08

5

8

45°

Draw a Horizon Line (5' here) and near one margin measure above and below in even increments making the height 8' (10' is OK). On the other margin make the same measurements in slightly smaller increments. They should be about 1/8" less per 1". Connect each measurement line across to its corresponding measure-ment on the other side.

Draw a line 45° from the vertical at the lower corner. Connect a vertical where this angle reaches the top line. This will give the first vertical measuring square representing 8' x 8'.

STEP 2STEP 1

6 - 2

STEP 3 STEP 4

Multiply the first 8' x 8' using half diagonals through the 4' height. Take these multiples out as far as your scale allows. Three is considered to be best. Make smaller if necessary.

Now draw diagonals for each square and place verticals where the diagonals cross the horizontals. Pick a VP near the center of the first square. This is now a completed Vertical Measur-ing Plane.

Establish a Diagonal Vanishing Point (DVP) on the HL at the left border. This is a chosen point representing the distance of the observer from the picture plane as in many other systems.

081624

5

8

VP

081624

5

8

DVP VP

081624

5

8STEP 5

HL

HLHL

This means that moving the DVP to the left is the same as backing away from the view. This makes the floor appear shal-lower. Moving to the right makes the floor appear deeper.

6 - 3

STEP 6 STEP 7

Draw lines from VP through 0, 8, 16, & 24 Draw lines from DVP through 0, 8, 16 & 24. Connect crossing points from +8 and +16. You now have outlines of 8' x 8' squares on the horizontal plane.

STEP 9STEP 8

DVP VP

081624

5

8

DVP VP

081624

+8

+16

5

8

DVP VP

081624

+8

+16

5

8

Draw all depth lines through all points on the ground plane forward from the VP. This gives the width lines of the grid and begins the wall on the right hand side (also a vertical measuring plane).

Using the DVP crossing points with the depth lines, draw the horizontal lines of the grid. Portions here were left out to show how each line is referenced. They can be drawn to the full width as well as all verticals drawn to full height.

DVP VP

081624

+8

+16

5

8

6 - 4

DVP VP

081624

+8

+16

5

8

LAST STEP

This shows the entire grid completed. This would be a lot of work for just one drawing. The idea is to do a solid job, even in ink, and use it over and over again as an underlay for drawings. It can be flopped to make the near "wall" on the left side. Actually, these are not walls, but measuring

planes. It is possible to measure be-hind the Vertical Measuring Plane by counting where the DVP crosses each depth line just like the 1-Point Perspec-tive floor grid.

HL

6 - 5

DVPVP

8 16

+16

5

8

0

24

+8

INTERIOR & ARCHITECTURAL APPLICATION - LARGE SCALE FORMS

The grid was flopped and used as an underlay for this in-terior study. Each grid square represented 1'-0". Notice that the walls do not necessarily fall at the vertical measuring plane and can be in front or in back of this plane.

J.J.ATTORNEYJ.J.ATTORNEY

J.J.ATTORNEYJ.J.ATTORNEY

OPHEIM '92

6 - 6

DVP VP

816

+16

5

8

0

24

+8

PRODUCT APPLICATION - SMALL SCALE FORMS

In this case the grid square represented 1" increments. There is much distortion as the form gets closer to the right measuring wall. If this is a problem, move farther inward. This will give less distortion to the right hand side.

A OPHEIM '92

A

6 - 7

CIRCLES IN PERSPECTIVECIRCLES/ 8-PT. & 12-PT. METHOD

CIRCLES CONSTRUCTED W/ ELLIPSE GUIDESELLIPSE ANGLE MEASUREMENTS

DEVELOPMENT OF CIRCLES IN PERSPECTIVE

'92

OPHEIM

7 - 1

CIRCLES IN PERSPECTIVE

Circles are usually thought of as being perfectly round. This would mean that they are constructed with a consistent radius of a specific size about a center point. Actually we seldom see a circle this way. The only time would be when the circle is at eye level and perpendicular to our line of sight.

Our perception of a circle, then, is not a true circle at all, but an ellipse that varies considerably from a perfect circle to an ellipse that is so tight that it becomes a straight line. This can happen on the horizontal and vertical plane as well as any other plane at any angle.

HORIZONTAL PLANE

VERTICAL PLANE

MINOR AXIS

ELLIPSES DEFINED

The definition of an ellipse is more mathematical than it is perceptual. What we need for our purposes is the relationship of the Minor Axis (diameter of the smaller circle) and the Major Axis (diameter of the larger circle). Every ellipse has a Major and Minor Axis. It is their variations that give the ellipse its perceptual difference. The above construction is very reliable for constructing large ellipses on an orthographic plane. Other methods are necessary to find ellipses in perspective.

ELLIPSE CONSTRUCTION

One of the most important skills in perspective drawing is the ability to construct circles (ellipses) in perspective. The circle is so commonplace when you are working with drilled holes in surfaces, circular knobs protruding from a surface, radius edges, rounded corners, cylinders of various types, cones and circular lines on spheres.

DEVELOPMENT OF ELLIPSE FROM 2 CIRCLES

MAJOR AXIS

7 - 2

ELLIPSES IN PERSPECTIVE USING 8-POINT METH-OD

It is possible to project information from a circle onto a perspective plane. First make a square that joins a vertical side of the perspective square. Construct a true circle within the square and draw its diagonals. Then project the lines where the circle crosses the diagonals into the perspective view. This will give 4 points around the circle in addition to the 4 midpoints of the square. This is done on a vertical plane first and then projected to a horizontal plane if needed.

8-POINT CIRCLES ON CUBE

A square and circle are attached to the vertical side. Horizontals through the diagonal/circle in-tersections are projected onto perspective surface from the VPL (not shown).

Projections can be taken to the top horizontal sur-face in the same manner. Construction square and circle can now be removed.

3

5 6

7

12

8

4

STEPS:

A cube composed of 3 squares in perspective. The problem is to place a full circle on each surface.

The projections can be taken to the next side using VPR (not shown) and repeated.

Note: Only half the circle and square is actually needed.

7 - 3Perspective circle is then drawn using the eight points found.

8

9

7

10

11

12

12

3

4

5

6

The same constructions can be done in perspective. Use your VP's and diagonals to divide into 16 squares. The rest is just the same. The 12 points connected will give an ellipse that represents a circle within the perspective square on both horizontal and vertical planes.

IMPORTANT

Always make certain that you are drawing a square in per-spective. If it is rectangular, you will be constructing an ellipse in perspective - not a circle.

Orthographic construction is as follows: Divide the square into 16 smaller squares following the constructions above. Then draw the diagonals of the outside sets of four squares. Draw circle through points found at the crossing of the first grid line from each corner.

ELLIPSES IN PERSPECTIVE USING 12-POINT METH-OD

Since ellipses are representing circles, it is first necessary to construct a square that is the same size as the diameter of the circle. This can be accomplished on an orthographic view as well as a view in perspective.

12-PT. CIRCLE CONSTRUCTION

7 - 4

CIRCLE CONSTRUCTION WITH ELLIPSE GUIDES

You don't have to construct very many circles using 8-point and 12-point to realize that it takes a great deal of time. That explains the reason that constructions of this type are done infrequently. Their best application is for circles that are large. For smaller circles it is more convenient to use ellipse guides. If done correctly, this method is quite accurate and becomes very easy with a certain amount of practice.

Ellipse guides are available individually and in sets of 4 or more depending on how many ellipse angles you want. Below is a set of 4 with ellipse angles of 15, 30, 45 and 60 degrees. Each angle has a series of different ellipse sizes. Combination ellipse guides have all 4 angles on one guide.

The more Ellipse Angles you have to work with the better. Complete sets include every 5° increments from 15° to 60°, & 10° with 65° thru 85°. Trace templates are available.

The ellipse guide has a series of elliptical holes stamped into a plastic sheet with a MAJOR and MINOR AXIS marks dividing the ellipse into 4 equal quadrants.

III

III IV

This ellipse varies from a perspective ellipse, because the 4 quadrants are not the same in perspective. Below is a circle construction on a horizontal plane using 1 point perspective. The MAJOR AXIS is not half way between the top and bot-tom edge of the circle. The MINOR AXIS is still dividing the ellipse in half vertically. Therefore it is important to use the MINOR AXIS in ellipse alignment. The MAJOR AXIS cannot be used because it isn't where it should be on the ellipse guide.

MINOR A

XIS

15 degree

30 degree

45 degree

60 degree

MAJOR AXIS

MAJOR AXIS

MINOR A

XIS

NOT EQUAL

EQUAL

7 - 5

CIRCLE CONSTRUCTIONS USING ELLIPSE GUIDES

Constructing a circle on a flat plan with an ellipse guide involves three choices:1. ELLIPSE ALIGNMENT (minor axis to opposite VP or vertical).2. ELLIPSE ANGLE (fullness or tightness).3. ELLIPSE SIZE (perspective box measurement).

ELLIPSE ALIGNMENT ON FLAT PLANES.

The rules for ellipse alignment can be quite simple. Shown below is the 8-point circle construction. If you overlay the drawing with an ellipse guide of the correct size and angle, you can determine the major and minor axis of the ellipse used. When the actual minor and major axis of each ellipse is drawn, you will discover several relationships of the ellipse to the surface it is resting on.

The MAJOR and MINOR AXIS do not . . . . .1. . . . lie on the diagonals of the square.2. . . . relate to the corners of any side.3. . . . cross at the center of the square.

If the MINOR AXIS is extended to the horizon line, we dis-cover that it goes to the vanishing point of the side that is perpendicular to it. On the top the MINOR AXIS is found to be a vertical line.

The MINOR AXIS . . . . . .1. . . . goes through the center of the square.2. . . . extends to the opposite Vanishing Point or is vertical.3. . . . is always perpendicular to the surface is rests on.

7 - 6The MINOR AXIS alignment is predictable.

MAJOR AXISMINOR AXIS

VPRHL

HL2

VP2

VPLVP1

OPHEIM '92

ELLIPSE ALINEMENTS

Almost everyone has the ability to perceive what objects should look like when they see a drawing. They can't be fooled. Circles must look like circles lying on whatever plane they are on. The ellipses below were constructed using correct alinement and angles. They show the use of minor axis alignments perpen-dicular to the surface they are resting on. Ellipses on the inclined plane use HL2. This inclined horizon line is found by connecting VP2 and VPL.

Finding correct sizes and angles follow . . . .7 - 7

VERTICAL PLANE ELLIPSE RELATIONSHIPS

ELLIPSE ANGLE MEASUREMENT

This theory can be put to use in practice . . . . . . . . . . . . . . . . .

When in a search for a way to measure ellipse angles, one must look for clues from what is known. We have used our best judgement up to this point as to align-ment and angle. From these observations we can see that all the ellipses are the same vertically and change gradually as they move horizontally across the plane. Therefore, the only thing they have in common is a vertical line running through the ellipse center. This vertical line must somehow give us a measured ellipse angle. Since the HL crossing will only give the same angle anywhere within the circle, the only other crossing point is at the Field of Vision edge. Once this is taken to VPL (for circles on planes facing left) we discover that the angle the vertical centerline makes with the line to VPL is the same as theellipse angle on the ellipse guide.

This is true for surfaces facing right as well. Their angles use the VPR.

Here we see that the four commonly used ellipse angles make their corresponding angles on each protractor.

0

1020

3040

50

60

70

80

90

100

110

120130

140 150 160 170

180

0

1020

3040

50

60

70

80

90

100

110

120130

140 150 160 170

180

0

1020

3040

50

60

70

80

90

100

110

120130

140 150 160 170

180

0

1020

3040

50

60

70

80

90

100

110

120130

140 150 160 170

180

HLVPL VPR

15°

30°

45°

60°

60°

60°

60°

60°Ellipses

45°

45°

45°

30°

30°

30°

15°

15°

45°Ellipses

30°Ellipses

15°Ellipses

'92

OPHEIM

FIELD OF VISION

7 - 8

The ellipse angle of any circle on a vertical plane is found by taking a vertical center-line (ç) to the edge of a Field of VisionCircle (with VP's as diameter) then to the same vanishing point that the surface goes to. The angle formed with that vertical is the ellipse angle of the circle any-where along the vertical line.Now, align the ellipse using minor axis to opposite VP.

*Use a protractor to measure the anglesbetween the verticalline and the line toVP.

E = Ellipse Guide

Hot Stuff

ELLIPSE ANGLE MEASUREMENT-VERTICAL PLANES

MEASURING ELLIPSE ANGLES

'92

OPHEIM

*60°

*30°

*45°

60°E

30°E

*15°

15°E

ç

ç

ç

ç

VPL VPR

*38°

*47°

40°E

*47°

Field of Vision

60°E

45°E

45°E

45°E30°E

7 - 9

40°

ç

0

10

20

3040

50

60

708090100

110

120

130

140

150

160

170

180

40°Ellipse

0

10

20

3040

50

60

708090100

110

120

130

140

150

160

170

180

15°

0

10

20

3040

50

60 70 80 90

100

110

120130

140

150

160

170

180

30°

15°Ellipse

30°Ellipse

VPL VPRHL

HORIZONTAL PLANE ELLIPSE RELATIONSHIPS

A search for a way to measure ellipses onhorizontal planes leads us to the onlypoint that is common to all the ellipses that are the same distancebelow HL. This is a horizontal line at center. If this point istaken to either VP, it will givethe angle of the ellipse. Thisworks for any location within the Field of Vision.

This diagram shows 3 different angles andhow they were found.

It also points out avery interesting factabout any angles over45°. It appears to suggest that when lookinghorizontally, we cannot see angles that are greaterand stay within the Field ofVision. To get larger we mustlook downward or rotate thesurface.

ELLIPSE ANGLE MEASUREMENT This is how you do it . . .

'92

OPHEIM

FIELD OF VISION

7 - 10

ELLIPSE ANGLE MEASUREMENT-HORIZONTAL PLANES

The ellipse angle of any circle on a horizontalplane is found by first drawing the squareand finding its center. Take the horizon-tal centerline to the vertical ç of thecircle, then to either VP. The anglethat this line makes with the HLis the ellipse angle used within the square. Fit by size the square and then rotate toa vertical minor axis align-ment to correct distortion.

Squares below HL were drawn using the DVP. But, any 1 or 2 PointMeasuring System will work to find squares.

*Measure angles withprotractor.

E = Ellipse Guide

'92

OPHEIM

*30°

*15°

15°E

30°E

*35°

35°E

ç

*30°

If size is not important, you can also do it without thesquare.

*30°

30°E

35°

*15°

15°E

*15°

ç

VPL VPR

Field of Vision(not needed here).

DVP

15°E

7 - 11

The ellipse size can be measured by drawing a box at the circle location by using any of the measuring systems. This box will give the height and depth of the circle. One would think that all you would have to do is put an ellipse inside this box and that would be it. The problem is that each box requires a certain angle to fit it exactly. This angle might be 32 degrees or 53 degrees. You are limited to the size and scope of the set of ellipse guides you are using. Fitting each perspective square thus becomes impossible in many cases. Since the ellipse is also a perfect ellipse and not a perspective ellipse, the fit is not exact because the ellipse on the guide is heavier or fuller on the back half. To make matters seem worse, the fore-shortening of the square many times makes the ellipse look turned into the surface or flat vertically. This often occurs on plotted ellipses.

ELLIPSE SIZE MEASUREMENT

One alternative in placing an ellipse on a surface in a certain location and size is to forget about the square that surrounds it and rely only on the vertical axis in the correct location and measurement. The correct ellipse angle will take care of the circle being as wide as it is high.

TO VPRTO MP

VML

TO VPR

Find the height locations along the VML and then find the center vertical axis using MP. Construct the perspective circle using the ellipse size and alignment that best fits the require-ments making sure that the minor axis goes to VP. Visually pick or measure the ellipse angle. If it looks indented or turned into the surface, try a fuller (wider) ellipse. If it appears to be turned outward, try a tighter (thinner) ellipse.

So, the square gives a hint, but does not give exact angle needed. Ellipse guide circles rarely touch thesquare in the correct locations - that isthe center of each side. Hold the height at center and allow the ellipse to go outside the square if necessary so that it will visually ap-pear to lie on the surface. 7 - 12

ELLIPSE GALORE

All the ellipses on this side are wrongly chosen and aligned. All the ellipses on this side are correctly chosen and aligned.

Each problem is noted.

OK

OKOK

Wrong ellipse angle and correct alignment looks indented into

surface.

Wrong ellipse angle with correct alignment.

Wrong ellipse angle and align-ment. Appears to float.

Right ellipse - Wrong VP.

Wrong alignment and ellipse angle

Wrong ellipse angle & alignment - hangs outward at top.

7 - 13

DEVELOPMENT OF CYLINDERS AND SPHERESSTANDING CYLINDER CONSTRUCTIONS

HORIZONTAL CYLINDER CONSTRUCTIONSCONSTRUCTION OF SPHERE

'92

OPHEIM

8 - 1

VPL HL

HL

VP

Vertical Minor Axis

Minor Axis to VP

Full circle within a square

1 and 2-point perspective can also be used to construct cylinders lying on a horizontal plane. Once the boxes of square cross-section are drawn in the location needed and to scale, the ellipses or circles at each end are drawn within the box and then connected with tangent lines.

HL

This time the Minor Axis goes to the opposite Vanishing Point. Use full circles in 1-point perspective where the cylinder is pointed toward the viewer. This extreme foreshortening can give very exciting views, but should be close to the VP to avoid over distortion.

STANDING CYLINDER CONSTRUCTION USING SQUARE BOXES

Cylinders are no more than circles extended through a third dimension. The illusion that is necessary to convey a cylinder is very similar to circles on flat planes. Any circle can be taken through this third dimension by using vanishing points or verticals, just like rectilinear forms.

Cylinders are seen in almost any position relative to the viewer. They can be above, at or below eye level, and turned in any of 360 degrees.

Notice below that the same cylinder can be constructed by using either 1 or 2-point perspective. Either method works equally well. Each square horizontal plane gives an easy alignment of the ellipses using ellipse guides. If the ellipse is slightly too full, allow the ellipse to extend beyond the back line at top and bottom always keeping the minor axis vertical. VPL

VPLVP1 HL

8 - 2

VERTICAL CYLINDERS WITH ELLIPSE GUIDES

We learn from the box constructions that both top and bottom ellipses share the same minor axis. We also can see that both ellipses have the same major axis measurement and that the ellipse gets tighter as it nears the horizon line. Using a center vertical axis line makes it possible to construct a convincing cylinder without boxes. In this case measurements are eyeballed.

larger and tighter ellipse at near end

HORIZONTAL CYLINDERS WITH ELLIPSE GUIDES

In the same manner it is possible to construct ellipses on a horizontal plane using the minor axis to VP.

fuller and smaller ellipse at far end

If you need measurements to give the right proportions, use a rectangle representing the height and diameter.

Diameter

This method is an exception to the rule, because it uses the major axis. The minor axis is still on a vertical line though the center. Both methods require a fuller ellipse at the bottom end. The distance below eye level determines the fullness of both ellipses. You can always use Chapter 6 Ellipse Angle Measurement if you need to be exact.

Height

VPR

VPR

VPR

VPR

8 - 3

CYLINDER CONSTRUCTIONS

Drawings of the cylinders below used "eyeball" as a basis for constructions with ellipse guides. Note the various positions of the cylinders and their ellipse angles. Measured ellipse angles and boxes can also be used. As cylinders go above the eye level, the ellipse angles need the same distance above the HL as they do below. For standing cylinders be careful not to work too close to the HL as very tight ellipses are needed. Even the 15° ellipse needs some distance below the HL.

15°

30°

15°

45°

60°

45°

30°

45°

30°

45°

30°

30°

15°

60°

45°60°

HL VP

90°

30°

90°

90°

45°

OPHEIM '92

8 - 4

ELLIPSTONETIRES185X

ELLIPSTONETIRES

185X

The tire stack was created with a single ellipse angle of 45°. The sizes get smaller along the tread line to give fore-shortening to the depth. The drawing construction was done using a minor axis to a distant VP. The VP will give the correct tilt for any distance below HL. The same drawing was repeated with a 90° rotation for th e horizontal tire. The overlap gives the illusion that one is above the other. It is hard to perceive these as the same drawing.

CYLINDRICAL FORMS & DRAWING ROTATION

'93

OPHEIM

HL VPR

ALL 45° ELLIPSES

8 - 5

A progression of ellipse angles from 45°, 50° & 55° in different sizes are used to draw the lens portion. Minor axis goes to VPR or is vertical.

All 20°.All 25°.

Lens cap is all 45°.

CONSTRUCTION OF CAMERA DRAWING

The Measuring System is used to first "block-in" the main parts. A super way to do this is by using elevation views along the bottom edge of the HML. This makes measurements easy. If you don't have these views or you don't want to take time to draw them, you can take the measurements from an actual camera.

LENS DEPTH CASE DEPTH

VML

HML

CAMERA AND LENS FACE

USE MP'S

8 - 6

CAMERA ILLUSTRATION COMPLETED

It is always important to finish off all de-tails as exact as possible. Then you can add other objects to create interest and scale. Overlap forms and develop a strong composition by making the negative areas into interesting shapes. Draw all the inner detail lines using a medium line weight and then trace around the outside edge with a heavier line.

OLYMPUS

OLYMPUSOM

-SYS

TEM

B1

234

0.50.7

0.45

OM-1 ON

ROLYMPUS

OPHEIM

8 - 7

DIVISION OF A CIRCLE INTO ANY NUMBER OF EQUAL PARTS

VPL VPRHL

MPR MPL

HML

ELEVATION

Construct the circle ellipse using any method. Draw the elevation of the circle with any number divisions wanted. Project these to the HML and to MP, then vertically on the circles face to find location points on circles front half. Each point is taken through center to locate points on the back half.

APPLICATION DRAWING

Details are now added to make spokes and hub or pinwheel.

8 - 8

Construct the circle ellipse using any method. Draw the elevation of the circle with any number divisions wanted. Project these to the HML and to MP, then vertically on the circles face to find location points on circles front half. Each point is taken through center to locate points on the back half.

APPLICATION DRAWING

Details are now added to make spokes and hub or pinwheel.

PICTURE PLANE

HLVP1

VML

MP

1

2

3

4

5

6

7

8

9

10

1 1

12

13

14

15

16

1

2

3

4

5

67 8 9

10

1 1

12

13

1415

16

16

15

14

1

2

3

4

PLAN VIEW

CONSTRUCTION OF SPIRAL STAIRCASE

STEPS:1. Construct a Plan View showing

how many steps there are in one revolution. This can vary depending on height requirements.

2. With the use of 1-Point Perspec-tive, transfer the Plan View onto the horizontal plane and number stations for each step.

3. Add a VML to one side for each step riser.

4. Project lines to the MP to find the riser height at the center of the cylinder.

5. Taking one step at a time. Trace the heights to their respective positions above the plan view. Add center cylinder support and trace their respective heights for each step.

6. Draw in each step from measure-ments found. Use a heavier outline to separate visually from construc-tions. It can get very busy.

Diagram at left shows 8 of the 16 steps finished. Complete the drawing by adding those that are missing.

8 - 9

1

2

3

4

5

6

7

8

1923 4 568 10 7

9

10

1

2

3

4

5

6

7

8

9

10

HLVP1

PICTURE PLANE

PLAN VIEW

CONSTRUCTION OF STANDING CONTOURED CYLINDERS

STEPS:1. Construct an elevation view of the

cylinder and place to one side.2. Choose Station Points (here numbered

from 1 - 10) at key locations where the form changes abruptly and also along soft curves.

3. The different Station Points are projected horizontally to the Picture Plane, then to their square depth using 1-Point Perspective methods. The measuring point is not shown here.

4. Draw diagonal lines on each square.

5. Construct a Plan View below the Pic-ture Plane composed of half circles which correspond to the same radii as each Station Point is from center in the elevation.

6. Take each Station Point on the Plan View vertically up the Picture Plane to the Station Point level and then in depth to the Vanishing Point.

7. Draw an ellipse that crosses where that depth line crosses the diagonal on that plane.

8. Finish by drawing a curve that touches all of these found ellipses.

Try doing this with several variations of cross section.

8 - 10

1

2

3

4

5

6

7

8

1923 4 568 10 7

9

10

1

2

3

4

5

6

7

8

9

10

HLVP1

PICTURE PLANE

TOP VIEW

CONSTRUCTION OF DIVISION LINES ON CYLINDERS

STEPS:1. Beginning with a constructed cylinder

using a Plan View. Add lines on Plan View that will divide the cylinder into any number of divisions. They can be equal angles or varied.

2. Project lines from each Station Point vertically to the Picture Plane and continue vertically to the Station Level.

3. Continue on that plane to the VP1. 4. This line will cross the ellipse on

that plane in two places. This is where that division line will cross that ellipse.

5. Draw a smooth curved line through all found points.

6. Continue for each division.

Only one division is shown here for clar-ity. Try adding several more and plotting their locations on the cylinder.

8 - 11

Smaller circles are placed onto the sphere surface by po-sitioning the minor axis through center of circle and mov-ing the ellipse outward until it looks to be in the correct location. These can represent many things, but in this case a portion of the sphere has been removed leaving a circular flat plane. The tight ellipses, 15° thru 45° usually touch the circle unless they are very small. The 60° will be just inside the circle unless it is very large. Then of course the Circle (90°) is at center. Other angles can be used between these locations.

DIFFERENT ELLIPSE ANGLE LOCATIONS

CONSTRUCTIONS OF SPHERES

Spheres are nothing more than circles until they are given detailing to give them the third dimension. They can be shaded or textured. When you are using only line work, there must be details added to give the 3D effect. The easiest of these is a line dividing the sphere in half. This could be called an equator, terminator or horizon line depending on which context you are referring.

Begin by drawing an ellipse of any ellipse angle with its minor and major axis through the center and with the ma-jor axis the same length as the circle's diameter. This can be placed at any desired slant or angle from the horizon. Various effects can then be created depending on which line you make visible. They can be seen as line around the form or a line cut through the center.

Next, try to use two such ellipses in the same way. Here the sphere may appear to have slices taken out and por-tion of the outer sphere are taken away. The angle between each ellipse can be measured directly. Example being, if you want the angle between the two ellipses to be 45°, you can measure 45° between their minor or major axis lines.

15° 30°

45°60° 90°

0°

8 - 12

align each ellipse. All objects were drawn using only 4 ellipse angle templates plus a circle template. Using more angles will make it possible to place circles in about any location which should make them even more convincing..

Constructions on this page all use ellipse angles aligned to the center of the circle. Most of the ellipse choices are made by visual trial and error (eyeball). PRACTICE is the key. Work on a few products or forms of your own and visually choose and

to VPSPHERE STUDIES

15°E

15°E

60°E

60°E

30°E

15°E

ALLEY-K

ING

8 - 13

DRAWING A SPHERE TO SCALE

Follow these easy steps to construct a sphere using a cube which measures the same as the sphere's diameter.

1. Construct a cube using any measuring system.

2. Find the center of one side where the diagonals of that side cross.

3. Using lines to both VP's, construct a vertical plane at the cubes center.

4. In the same manner construct a second vertical and a horizontal plane at cubes center.

5. Draw a circle on each plane using ellipse guides. Draw a vertical line at the cubes center to find touch point (T).

6. Draw a circle about the cubes center that runs tangent to the three ellipses. This is the sphere to scale.

T T

8 - 14

SPHERE SHORT CUT

Obviously the construction using a cube to find the true size of a sphere is not something you would want to do on a daily basis. The short cut employs our visual findings from the cube construc tion. Using the diagonals from the cube construction we can find the true center line of the sphere and the point where this line touches the surface. This is the point where the sphere touches the surface. This center line is always slightly smaller than the final circle encompassing the sphere. The idea is to first find the center line of the sphere to scale and in the right location and then draw a circle slightly larger. This gives a fast and fairly accurate solution. Any error would not be readily noticed and hard to prove wrong.

HL

VML

Measured vertical line can be transferred to any other location and then the circle is drawn slightly larger with the line as its center. See page 3-11

8 - 15

SHADOW PLOTTING OF FORMS SHADOW BASICS

SHADOWS ON BLOCKS, CYLINDERS & SPHERESPARALLEL SHADOWS

CONVERGING SHADOWSNEGATIVE & POSITIVE LIGHT SHADOWS

LIGHT LOCATION & SOURCE

It is always best to have a single light source. Think of the light as being a single point at a certain direction and height above the object. Shadows that are cast to either side or behind an object usually reflect a good choice of light position. This shadow will not dominate. The most advantageous location for the light source is one that is behind the viewer (in front of the object), is fairly high and is over the left or right shoulder. This location will give relatively short shadows behind and to the right or left side of the form. There are 4 different types of light used in shadow plotting. Light source choices depend on the subject matter, distance and direction.

PARALLEL - Fictitious distant light source producing shadows in any direction using parallel light rays.

4 SHADOW TYPES COMMONLY USED:

CONVERGING - Artificial near light source producing shadows in any direction using converging light rays.

POSITIVE LIGHT - Sun light source producing realistic shadows in front of forms using converging light rays.

NEGATIVE LIGHT - Sun light source producing realistic shadows behind forms using converging light rays.

-

SHADOW DETAILS

Shadows are important additions to drawings that help give the illusion that objects are resting on a surface. It is a good way to establish a strong contrast around forms, set moods and to separate the forms from their background.

9 - 1

SHADOW OF AIRPLANE ON THE GROUND

SHADOW PLOTTING BASICS PARALLEL METHOD OF PLOTTING SHADOWS

If a series of fence posts, all the same height and in a line to VP, the shadow of each post can be plotted in the same manner as the first one. Each light angle is parallel to the others and the ground angles are all parallel as well.

The shadow of the rail (D-E) added to the top goes to the same VP as the rail itself. This means that the shadow of any horizontal edge will go to the same VP as the horizon-tal line defining that edge, i.e. the cast shadow is actually parallel to the surface edge that is casting the shadow. It is represented as a line to the same vanishing point in perspective.

The shadow location of a point in mid-air (airplane) can be found if the distance above the surface is known. A vertical line from the point to the surface beneath will give that height. A line, called the LIGHT ANGLE (LA) through that point is used to show the light ELEVATION. A second line, called GROUND ANGLE (GA through the ground point beneath is used to show the light DIRECTION. A post (A-B) casts a shadow in the direction of the ground angle. When plotting this shadow, the LA passes through a point on the top (A) and the GA passes through the base of the post (B) and continues to the light angle intersection (C). The shadow is connected back to the base of the post (B-C).

LIGHT ANGLE (LA)Chosen angle indicates the ELEVATION of the light as it comes from the light source.

GROUND ANGLE (GA)Chosen angle indicatesthe DIRECTION thelight is coming from.

Shadow is located where the LA & GA crossGA

GA

GA

GA

GA

GA

LA LALA LA

LA

VP

LA

GA

LALA

GA

LAA

BC

D

E

9 - 2

SHADOWS OF VERTICAL PLANES

The shadow will go in the Ground Angle direction until it reaches the Light Angle and then will go to the same vanishing point as the top edge. Not having to find the location of the shadow of the far corner by using the SHADOW RULES will save time without the loss of accuracy. These same rules can be used on all solid box forms.

SHORT CUT

LA

GA

LA

GA

GA

LA

Arrows go to VPR

Hot Stuff

A

B

C

DF

E

If a solid vertical wall casts a shadow behind and to one side, the shadow is found by using 2 sets of LA's & GA's. Once this is done, several observations become apparent that can be used to create rules which always apply.

SHADOW RULES:1. VERTICAL LINE RULE All vertical edges cast

shadows in the Ground Angle direction. See A-B.2. HORIZONTAL LINE RULE All horizontal edges

cast shadows that are parallel to them and go to the same VP. See C-D.

3. ANGLED LINE RULE Shadows of angled edges are plotted by finding the shadow of both end points using GA's & LA's. See E-F.

9 - 3

LA

LA

GA

This method used 3 LA's and 1 GA. The shadow lines are projected to VP's.

SHADOWS OF HORIZONTAL & VERTICAL PLANES

LA

LA GA

GALA

GA

GA

LA

This method used 4 LA's and 4 GA's

LA

PARALLEL SHADOWS OF BOX FORMS / EXTERIOR

First, determine a light source. Decide what surfaces will get direct light and which side of the box should get the most light. There are many possible combinations of light and ground angles. Choose one of the corners and place a point on the ground where you want the shadow of that corner to be. This gives you complete freedom of choice. Now, working in reverse connect the point to the top and bottom of the corner line. This gives you the light angle and ground angle used to find that point. Find the shadow of the other corners by using parallel light and ground angles to the original ones. Shorten the time by using the SHADOW RULES whenever possible.

LA

LA

LA

GA

GA

VPHL

LA

Minimal use of GA's and LA's to develop shadow using VP's to find shadow parallel to edges.

ORANGE�CITY�

GA GA

LA

POINT

LA

GA

9 - 4

This rule can be applied to the partition inside the box below. The shadow goes to the partition and upwards until it reaches the light angle, then horizontally from vanishing point to the next wall and then (not seen) up to the diagonal corner.

LA

GA

PARALLEL SHADOWS OF BOX FORMS / INTERIOR

The shadow into the box interior is found in much the same manner as the shadow outside. Interior shadows differ from exterior shadows as they pass over both horizontal and verti-cal surfaces. The exterior shadow of the box has shadows on horizontal surfaces only. The shadow of the nearest corner is found to be inside the box. This corner shadow will connect to the front exterior shadow and will have a ground angle into the box at the same angle as the exterior ground angle. There is only one corner shadow inside the box. The shadow must now connect from the interior shadow corner to the wall, in the direction of the exterior corner shadow, and then angled up the wall to the opposite diagonal corner from which it started. This angled line on the wall is not always seen. It depends on which surface is visible.

HL

LA

GA

VP

LA

GA

If a shadow is interrupted by a vertical plane, such as a wall, it will go to the wall along the ground line and then go vertically until it reaches the light angle. At this point is the location of the shadow of the pole tip on the wall.

FLAGPOLE RULE

9 - 5

HL

MULTIPLE BOX SHADOW SOLUTIONS

1

23

4

4

3

2

1

1

23

4

2

1

4

1

23

4

2

1TOP REMOVED

4

1

23

4

2

1TOP & FRONT REMOVED

4

1

23

4

2

1FRONT & SIDES REMOVEDSOLID BOX

4

1

23

4

2

1FRONT REMOVED

FRONT, BACK &TOP REMOVED

4

1

23

4

2

1

3

SHADED

FRONT & BACK REMOVED

4

1

23

4

2

1

3

3 SIDES REMOVED

4

1

23

4

2

1

3

3 SIDES REMOVED

4

1

23

4

2

1

3

TOP ONLY

4

1

23

4

2

1

3

FRONT & BACK ONLY

4

1

23

4

2

1

3

3 SIDES ONLY

4

1

23

4

2

1

3

CONSTRUCTION

MULTIPLE SHADOW SOLUTIONS FROM ONEPLOTTED CONSTRUCTION.

All shadows shown here are taken from this single construction.

9 - 6

LA

GA

LA

GA

Shadows of horizontal lines cast angled shadows down wall.

LA

GA

LIGHT ANGLES ARE NEVER SHADOWS. It is tempt-ing to use LA as a shadow line down a wall, but it is impossible for a light beam to be a shadow.

A

B

C

An awareness of which edges on a form cast the shadow will always help determine which step or rule to follow. Shadow plotting always involves the use of a light angle through a point in the air and a ground angle through a point directly beneath it. A certain amount of connecting of points must be done to complete any shadow. As the shadow solutions become more complex it is necessary to develop a more sophisticated shadow logic to complete the solution. This may mean simpli-fying more complex forms to something that you understand and work back, one step at a time, until the final shadow is found. It may also require that vertical planes might need to be added or extended temporarily and then removed (AB). If there is doubt about a solution, check it by using a point along an edge at (C) and plot the shadow of that point. If it is found on the line in question it will prove it correct, or show where it should be.

SHADOW SAVVY

WALL VARIATIONS

Flagpole Rule in action

9 - 7

SHADOW PLOTS OF DOORWAYS AND WINDOWS

Below are two ways to plot the same shadow of a door or window top (header) and the side upright (jamb). Version 1 is probably the easiest and is done by placing a solid wall or door on the inside wall thickness and find the flagpole shadow of the jamb and header on the door at B and then project that to VP. Where this line crosses the walls at C, an angled line CD is drawn to the front corner.

This solution uses the crossing shadow pattern on the ground of the back jamb and the leading header. A light angle AB is drawn to the edge of the jamb showing where the angle shadow must go. Then a shadow line is drawn to the front corner.

VPHL

GA

LA

A

B

A

B VP

VP VPHL

GA

LA

A

CC

B

DD

VERSION 2

VERSION 1

9 - 8

BOX WITH FLAPS CONSTRUCTION & SHADOW

CONSTRUCTION & SHADOW OF PYRAMID

Since a pyramid doesn't have vertical walls, you use the center point beneath the apex for GA. This gives you the shadow of the apex on the surface. Connect this point to all corners. Shade all surfaces within visual shadow.

All flaps are drawn to VP1. The flap shadows are found by put-ting the GA through a point on the surface below the flap corner. This is always the case for any-thing floating off the surface. The shadow of the corner is found inside the box also and is pro-jected to the upper right back corner for inside shadow. LA

GA

VPL

VP1

HL

LA

GA

9 - 9

CONSTRUCTIONFAST METHOD

LA

GA

ALTERNATE METHOD

1

1

1

2

2

23

3

3 4

4

45

5

5

6

6

6

7

7

78

8

89

9

910

10

10

GA

LA

PARALLEL SHADOWS OF STANDING CYLINDERS

Shadows of curved forms can be more complex, but follow the same principles as box forms. The top is a circle on the horizontal plane and casts a shadow that is also a circle on the horizontal plane. Since there are no corners, several points around the top perimeter are used to plot the shadow. Each point is connected to a point directly beneath by a vertical line. Use parallel Light Angles through each point on the top and parallel Ground Angles through each point around the base. The shadow passes through were each set of lines cross.

Use a French Curve or ellipse template to connect the points in as smooth an ellipse as possible. Connect this shadow using ground angles, to the base of the cylinder to give the inclusive shadow of the sides. The core shadow is a vertical line located where this side shadow line runs tangent to the base ellipse.

SHORT CUT The shadow of the top is elliptical and relates to the ellipse that defines the bottom of the cylinder. The el-lipse might be slightly tighter, but for short shadows which are

half hidden behind the form, it is possible to use the same ellipse that was used to draw the bottom of the cylinder. The alignment of the ellipse is with the minor axis in the vertical direction. If the minor axis is turned along the ground angle line, it will distort the ground surface. The position of the ellipse can be determined by single points at the top and bottom center of the cylinder. GA's & LA's can be drawn through these points to find the center of the ellipse. Then it's just a matter of shifting the ellipse guide to the position and using a vertical minor axis through that found point, draw the shadow ellipse. This ellipse is then connected to the bottom ellipse by Ground Angle to give the shadow of the sides.

Sometimes it is helpful to put additional light angles at the outside edges. If the shadow gets longer due to low light angle, the ellipse angle used must gradually decrease (get tighter) as the ellipse gets closer to the horizon line. This can be done visually in the same way circles are positioned on horizontal planes.

LA

GA

CORESHADOW

9 - 10

LA

GA

Keep all Minor Axisvertical. Ellipses graduallytighten as they approach HLbut are the same diameteras the circle casting them.

PARALLEL SHADOWS OF CYLINDRICAL OBJECTS

For shapes with various diameters casting short shadows, use a Light Angle through the centers of each ellipse and a single Ground Angle through the base ellipse.. Then use the same diameters in the shadow ellipses, making them all the same ellipse angle with minor axis vertical. LA

LA

GALA

GA

a

Take GA back from ground shadow(a) to inside walledge (b). Go verticalto (c). Connecta-c with a slightlycurved line.

b

c

2•"-20°

2"-20°

1º" -25°

30°

1¡" -30°

30°

30°

30°30°

2"D

2•"D

1¡"D

1º"D

9 - 11

For longer shadows, ellipse angles gradually decrease (tighten i.e. minor axis de-creases) as they approach the HL, but are the same diameter (major axis stays the same) as the circle casting them.

PARALLEL SHADOWS OF CYLINDRICAL OBJECTS (CONTINUED)

Shadow on inner cylinder from overhang is plotted using several "flag poles" around edge and finding each shadow point on the cylinder wall. Then connect points with a smooth curve.

LA

GA

CORESHADOWS

TANGENT PT.

VERTICAL MINOR AXISALIGNMENTS

DRIP LINE

Flagpole Rule at work.

Flagpole Rule at work.

LA

GA 9 - 12

PARALLEL EXTERIOR SHADOW OF HORIZONTAL CYLINDERS

If a BASE LINE (BL) is drawn from the vanishing point, which was used to establish the end plane of the cylinder, through the touch point of the cylinder end, it can be used to find points beneath any point on the end ellipse. Add vertical lines across the form. Draw LA's & GA's through these points to construct the shadow. The shadow can go in front or behind the form depending on the ground angle direction. Note that there are two light angle lines for each ground angle. Connect the points found by the intersections of the light and ground angles with a curve or ellipse guide. This construction shows several aspects of this type of shadow. The shadow is a tight ellipse that touches both LA's & passes through the touch point "T", and doesn't cross the base line.

GA

BL

LA

GA

BL

LA

SHORT CUT VERSION -- See next page for details.

LIBERTY

1992D

IN GOD WE TRUST

9 - 13

SHORT CUT

Since we know that the shadow is an ellipse and that it passes through known points, a point at the center of the cylinder can be added to show how far in front or back the shadow is cast. Find this point A above the touch point (T) with a vertical line. A light angle through A and a GA through T will show at B how far the shadow is cast behind or in front of the cylinder. The shadow goes through B, touches both LA's and goes through T and cannot cross over the BL. This locates 4 points for the shadow ellipse. Repeat at the far end if necessary. This offers a fairly fast and accurate method of shadow plotting.

GA

BL

LA

A

B

T

GA

BL

LA

A

B

T

KALEID

OSCOPE

9 - 14

PARALLEL INSIDE SHADOW OF HORIZONTAL CYLINDER

The shadow on the inside is much more complicated as there seems to be an optical illusion as the convex surface casts it's shadow on a con-cave surface resulting in an almost straight line. It might be a good idea to look at a cylinder under different light conditions to get a visual idea of what the shadow looks like.

The inside shadow can be plotted from the exterior shadow on the surface be-neath the cylinder. Begin with the point above the touch point. The Light Angle and Ground Angle from this point will give a point P on the surface. From the far vanishing point construct a line to the base line B. From point B draw a Light Angle to the edge of the cylinder C. From C draw a line back to the vanishing point to where it crosses the Light Angle at L. Point L is the shadow of point P inside the cylinder. Repeat this using 2 additional points along the top edge. This will give you enough points to see how straight the line is, how far it is from the front edge and which direction it will slightly curve. Shading will diffuse the shadow at the top and bottom edges, but the shape is similar to a "pancake" laying against the curved concave surface of the cylinder.

GA

LA

C L

P

B

T

BL

9 - 15

angles from each outer edge and is slightly tipped upward as it goes behind the sphere or downward when it goes in front. The shadow is also seen under the form and would be going around the touch point (T) of the sphere. This touch point is near the bottom on a vertical line through the center of the sphere. Shadows that don't look right are usually constructed using poorly chosen terminators and drip lines.

CORE SHADOW (TERMINATOR)

T

TOP VIEW

T

TOP VIEW

LA

GAT

LA

GA

T

T

T

Core Drip Line

CORE SHADOW (Terminator)

CORE SHADOW (Terminator)

CORE SHADOW (Terminator)

A sphere shadow is constructed using a CORE SHADOW (terminator of light). This core line is an ellipse about the center of the sphere and is perpendicular to the LA.. If points around the core are used with their corresponding points on the ground (drip line) in the same fashion as on the cylinder, the shadow can be plotted. Once plotted it can be seen that the shadow is an ellipse that touches the light

SPHERE SHADOW CONSTRUCTION

9 - 16

LA

LA

LIGHT FROM FAR RIGHT -SPHERE IS NEAR HL

LA

LA

LIGHT FROM BEHIND -SPHERE IS NEAR HL

LIGHT FROM FAR RIGHT -SPHERE IS LOWER

LIGHT FROM BEHIND -SPHERE IS LOWER

LA

LA

LA

LA

SHORT CUT takes practice, but its much faster.The BAD NEWS is that there is no proven method to find the core shadow or terminator. Any plotted shadow will be no better than your best guess as to the core ellipse angle. It is always found arbitrarily. Even if there was, it would probably be too time consuming and complicated to warrant every day use. The GOOD NEWS is that we don't need a method to be convincing. We are free to pick our own ellipse angles and determine the shadow using known points and observations found by plotting with a sample sphere.

The short cut version of a sphere shadow plot depends on a core ellipse of your choice (usually 30° or 45°) which is perpendicular to the LA. The shadow ellipse angle (like a cyl-inder shadow) is determined by how far the shadow is below the eye level. The shadow ellipse touches both LA's, is slightly tilted into the form for light in front and out from the form for light from behind. Avoid too much tilt or it will look like it's rolling down hill. COMMON VARIATIONS:

SHORT CUT METHOD & STEPS

LA

LA

T

1 2

3 4

5

DRAW CIRCLE & TOUCH POINT

DRAW CORE ELLIPSE - LA AS MINOR AXIS

USE 2 LA's @ CIRCLES EDGE

CHOOSE SHADOW ELLIPSE AND POSITION TOUCHING BOTH LA'sAND MINOR AXIS VERTICAL.

MOVE UP AND DOWN UN-TIL THE SPHERE LOOKS NEITHER FLOATING OR INDENTED AND TILT INTO OR OUT OF FORM A FEW DEGREES

9 - 17

GA

LA's

For any floating object you mustknow where the ground is. This isthe only way to be able to use aGA.

GA's

LA's

DRIP LINE

Core

GA

LA

Here the shadow is found on bothsurfaces and connected together bythe LA's giving the location of the portionof the top shadow that overhangs the edge.

When spheres float above a surface, the shadow plot uses the center point and Drip Line of the form on the ground and plotted using the GA's and LA's similar to before. The shadow on both the vertical and horizontal surfaces is an ellipse.

SPHERE SHADOW ON WALLS AND FLOOR

9 - 18

SHADOW SOLUTION OF A BOX FORM ONTO A CONE.

The shadows of the box and cone are first found on the ground. Several lines are added on the face of the cone adjacent to the box and their shadow determined by connecting them to the apex shadow point of the cone. LA's are then used to transfer the points where the two shadows cross to the surface of the cone on each of the lines added on the cone face.

SHADOW SOLUTION OF A BOX FORM ONTO A CONE

9 - 19

GA

LALA

GA

GA

LA

CAST SHADOWS ONTO OTHER FORMS

A

B

C

Use the diagonal vertical plane ABC of the pyramid to find the direction of move-ment up the slanted surface. As the GA hits this plane and goes vertical to the outer edge of the pyramid and returns to where it strikes the bottom, the direction up the side is found. LA shows where the point is located.

Always find the shadow on the ground first. This helps determine which corners need to be found on the object.

Find the pyramid apex on ground first and then run a vertical at the wall and transfer to the LA.

9 - 20

Many times the shadows are cast onto other forms. There are thousands of possible shadows. Here are just a few possibilities. They all use the "flagpole" process of finding a shadow of a point on a wall. Slanted surfaces require addi-tional plotting to find the shadow direction up the plane.

Flagpole Rule in action.

LA

LA

LA

A

B

C

MORE CAST SHADOWS ON FORMS

LA

GAGA

Take a GA to A the vertical plane of the inclined surface. Point B above connected to C gives the direction of the shadow. The LA shows how far up the incline the shadow is.

All shadows should be found on the ground first and then plotted onto a surface. Here the flagpole shows the back corner location. Just angle up the wall to that point and then to VPL.

For cylinders use a series of flagpoles around near edge and plot each point. The result is an elliptical shadow.

Flagpole Rule in action.

9 - 21

SHADOWS OF FLOATING FORMS

The shadow of a box "floating" above a surface can be found by using single GA's through the "Drip Line" corners with LA's pairs through points above. Otherwise the same rules apply.

DRIP LINE

LA

GA

LA

LA

LA

LA

LA LA

LA

GA

GAGA

GA

LA

LA

VROOM..... VROOM

GA

LA

LA

GA

LA

GA

LA

LA

DRIP LINE

THIS END

UP

A

B

DRIP LINE

LA

GA

GA

GA

LA's

When a bench is supported by a block, the shadow may strike this block as well. Start by finding the shadow on the ground and then find the shadow on the block by extending the end wall to the outside of the bench and use line AB to define the shadow if the pedestal were to be flush with the side. Once this is found the pedestal shadow angle is used to the edge of the block and then taken to VP along the side. A fourth GA is used to connect the shadow at the far end of the block.

Finding the shadow of a toy truck is not much different than the bench except the cylindrical wheels add curved forms under the truck. Refer to the cylinder end shadow plots or the standing penny to help do this.

9 - 22

MORE SHADOWS OF FLOATING FORMS

The "floating" form below uses the same principle as the box for the ground shadow. The interior shadow cast by the underside of the front corner onto the inside surface is found by using a LA at the corner and by projecting the GA on the ground back to that plane surface and vertically up to the horizontal surface to show the path of the "GA" on that plane. GA's are only on the ground. The other horizontal planes must have a different angle. Another way is to take the GA to the HL and draw back to the wanted corner.

Not a GALA

GA

GA

GA

LA

LA

LA

LA DRIP LINE

Not a GA

A

B

VPL

VPLVPR

VPR

VPR

VP1

VPL

GA

DRIP LINE

LA

LA

LA

GA

GA

GA

This construction is developed from two separate drawings, one drawn using one VP the other using two VP's. It is important when doing this that both objects use the same Horizon Line. The first step is to find the Drip Line of the top box. This is found by using one of the points where both boxes cross. AB is such a line. Use VP lines through B and verticals at the corners to complete the Drip Line. Then use pairs of LA's and GA's to complete the shadow plot.

9 - 23

CONVERGING LIGHT SHADOW

A converging light shadow is used for existing light conditions most often within a room. Many varia-tions of light sources are possible and each casts its own type of light. This means that the LAVP location is important to the overall effect.

GAVP

LAVP

Converging Light shadows are cast by a light source that is close to the object. The shadow is spread out wider than the object and can be in any direction from the light. All Light Angles go to the LAVP and Ground Angles go to the GAVP. LA's are chosen points and GA's are directly beneath on the same surface as the shadow.

LAVP

GAVP

9 - 24

LAVP CP

VP

CP

CP(back wall)

GAVP

CP(shelf)

CP(bookcase end)

CONVERGING LIGHT SHADOW OF INTERIOR SURFACES & FORMS

on the ground beneath the light source or from the Conver-gence Point (CP). Lines from CP's are used just like Ground Angles on floors.

When shadows are cast by a known light source within a room, use converging light and ground angles. The light angles all go to the light source and the ground angles go to a point

When the light is within a room, it is best to think of the light as the center location of several light sources or possibly the point where light bounces off the ceil-ing from a table lamp below. Whatever the case, pick a point that will give a believable shadow from the light source being used.

Any light source will have Convergence Points (CP) located at the same height and perpendicular to the LAVP at each of the four walls and the floor. The CP on the floor in this case is called the GAVP. CP's are used to find the angle of shadow down walls and parti-tions. Once set up, the shadow plot is easier.

9 - 25

Set up Light Source and Convergence Points.

GAVP

HL

LAVP

VP

CP

CP

CP

CP

SKYLIGHT LIGHT SOURCE

GAVP

HL

LAVP

VP

CP

CP

CP

SKYLIGHT LIGHT SOURCE

9 - 26

POSITIVE LIGHT SHADOW STEPS

Shadows cast in front of forms can be done in all methods. Positive light method, however, gives a more realistic version as the shadow gets larger as it gets closer. This is not the case with Parallel Shadow plot.

1. Draw the form using any method. One Point Perspec-tive was used here.

2. Pick a corner and decide where you want the shadow of that corner to be.

3. Place a dot at that location.4. From the dot draw a line through the point below

the corner and extend to the HL. This is the lo-cation of all the ground angles GAVP.

5. Draw a vertical line from the GAVP.6. Draw a line from the dot through the chosen corner

to where it intersects the vertical from GAVP. This is the LAVP where all LA's go to. This is the ac-tual light source as in converging method except it represents a very distant light at the horizon such as sunlight.

7. Draw GA's and LA's through top and bottom cor-ners to complete shadow.

9 - 27

GAVP

LAVP

HLVP GAVP

LAVP

HLVP

HLVP

Cra

cker

s

Cra

cker

s

Cra

cker

s

POSITIVE LIGHT SHADOW INTO A ROOM

Here we get into some really interesting shadow possibilities. Light streams into the room from what is obviously the sun. This offers a back light to objects along with a very striking shadow effect. Details outside around the outside are not necessary to the shadow plot and are not usually shown.

VPGAVP

LAVP

LA

GA

GA

LA

LA

GA

9 - 28

POSITIVE LIGHT SHADOW INTO A ROOM

Light streams into the room. This time, however, the GAVP is at the VP. The effect is a more direct light direction. The LA can be a shadow line in this position only. Even though in reality this is impossible, the effect is believable, i.e. to have a shadow line on a wall the light must be in some other location than in line with the wall.

VP GAVP

LAVP

LA

GA

LA

LA

GA

9 - 29

7. Draw a vertical line from the GAVP to where it crosses the LA.

8. This is the Light Angles Vanishing Point (LAVP) for the object.

Shadow appears to foreshorten (smaller) as it approaches the HL.

1. Draw the form using any method,2. Pick a single corner and decide

where you want the shadow of that corner to be.

3. Place a dot at that location.

4. From the top corner draw a LA through that point.

5. Also draw a GA through the point from the corner beneath and extend to the HL.

6. This intersection is the Ground Angle

The LAVP is the opposite point from the actual light source. If you move this point, the light will move in the opposite direction. For example if the LAVP is lowered, it would have the effect of raising the light source.

9. Complete the shadow plot using the LAVP and GAVP much the same way as in Converging Light Shadow.

HLVP VP

HLVP VPGAVP

HLVP VPGAVP

LAVP

GAVP

LAVP

HL

Shadows of large forms need to be foreshortened to look right. This does not occur in the Parallel Method. Negative light shadow will give this illusion and the effect of sunlight casting a shadow behind the form.

NEGATIVE LIGHT SHADOW STEPS

9 - 30

GAVP

LAVP

HL

NEGATIVE LIGHT SHADOW APPLICATION

This method looks best when we can see the entire shadow under a form and loses its effect when our view is blocked. Once you get used to the idea that the LAVP is below the GAVP it becomes fairly easy.

Negative Light Shadow can be used on any form. It does seem to give the most realistic shadow effect.

Ground shadow cross-ing points (at dots) can be transferred by LA to find the shadow cast on the form.

GA

LA

Ground shadow cross-ing points (at dots) can be transferred by LA to find the shadow cast on the form.

9 - 31

NEGATIVE LIGHT SHADOWS FROM WINDOW WALL

The GAVP is placed at the far right. This gives a rather side-ways shadow effect across the floor and onto the wall. Works well to show exterior light coming into a room and illuminating certain details. The outlines and mullions need not show.

VP GAVP

LAVP

9 - 32

WINDOW WALL NEGATIVE LIGHT SHADOWS INTO ROOM

The GAVP is placed at the Vanishing Point. This gives a shadow on the wall and floor and is again the exception to the rule that a LA cannot be a shadow. The effect is quite believable, however, and can give an interior a terrific light effect.

VP GAVP

LAVP

9 - 33

DESIGN

DESIGN

A STUDY IN REFLECTIONS OF FORMSVERTICAL & HORIZONTAL MIRRORS

CONCAVE MIRRORSCONVEX MIRRORS

DESIGN

DESIGN

10 - 1

REFLECTION OF BOX FORMS INTO VERTICAL SURFACES

Mirror image reflections are based on multiplications in per-spective. When the object is touching the mirror or two mir-rors are touching in a 90 degree corner, the reflections are found by taking a line from the top leading corner through the mid-point of the side that is touching the floor. This gives the multiplied depth of the second box or mirror at the floor, thus producing the foreshortened mirror image.

Notice that when the object is parallel to the mirror, its reflec-tion is in the same direction and shares the same vanishing points. There is no change of angle or direction.

Below we have the same solution except two lines are used through a mid-point at the mirror. These give the location of the object reflection and also the reflection of the space between the object and the mirror.

ALTERNATE SOLUTIONHere is another way to find the same solution. There is not much advantage, but you might prefer using points outside the drawing.

VP

VP

MID-POINT

MID-POINT

VP

MID-POINT@ MIRROR

10 - 2

REFLECTIONS INTO WALL AND CORNER OF VERTICAL SURFACES

Mid-points again are used to find the reflections into a corner and along a partially mirrored wall. Shadows on the ground and objects reflect as well, but are not cast onto the mirror itself. Reflections overpower the cast shadow.

VP

MID-POINTS

MID-POINT

METRO TRUST

HL

ATM ATM

10 - 3

REFLECTIONS OF FORMS INTO VERTICAL SURFACES

Any form can be multiplied into a reflecting surface using diagonals. In this case the horizontal surface of the plate is used. Sizes are found using VP's. Usually values and line weight is lighter in the reflection.

ALTERNATE METHOD: Use vertical Centerline to find distance to mirror and double.

10 - 4

REFLECTIONS OF OBJECTS INTO A HORIZONTAL MIRROR

The construction uses distances that are doubled vertically on each object. Any point on a vertical line can be doubled using the point of ground contact.

HLVP1

EQUAL

EQUAL

EQ.

EQ.

EQ.

EQ.

EQ.

PARFUM

10 - 5

WaterLevelEqual

Equal

Equal

Equal

Equal

Equal

WaterLevel

Equal vertical divisions are used here as well, but in the water reflection, the water level under the form is used instead of ground level. This is the distance to the mirror plane.

10 - 6

WaterLevel

Equal

Equal

Equal

Equal

Equal

Equal

WaterLevel

REFLECTION ON CURVED CONVEX MIRROR

Any ground line that is taken from the center of the circle (mirror plane) to the HL and brought back to a point above will define the height of the reflection for any depth. This will give the corner points of a card placed in front of a convex mirror. As you would expect, the card appears larger in the mirror. Boxes and other forms require additional circular rings through all corner points. Find their positions using mid-points as discussed in 10-2..

HL

10 - 7

Following are two observations made using a standing cylinder as a convex mirror. HEIGHT is found observing a round circular disk. It reflects elliptically onto the surface where the edge of the disk passes behind the cylinder. This means that the reflection height of any ground point of an object can be found using a circle around the center of the cylinder and its reflection around the cylinder. DIRECTION is found observing

a series of circles and lines radiating out from center. Each straight line appears to curve slightly at first and then dra-matically up the side of the cylinder.

TOP VIEW

TOP VIEW

TOP VIEW

Find the squares reflec-tion using circles through square side centers and corners. If off center, use a circle through each cor-ner and use a line to cylinders center.

Card reflection is found us-ing circles through corners and lines from cylinders center to each card cor-ner. Reflections appear to be pinched together and stretched vertically.

TOP VIEW

10 - 8

CONVEX MIRROR REFLECTION THEORY

TOP VIEW

TOP VIEWTOP VIEW

Standing card is similar to hori-zontal card except the second circle is drawn above the other. This gives the height location for the top edge of the card. Top and bot-tom curves around cylinder following the surface.

The pencil shows how circles around center gives different height locations on the cylinder. Ends up as an ex-aggerated curve upwards as it gets farther away.

TOP VIEW

The box uses both the horizontal and standing card techniques.

Follow the same tech-niques as the box except use only one point on the top..

10 - 9

HORIZONTAL CYLINDER REFLECTION

Below is a rather simplified version ofrefections of a pole into a cylinder.

B

C

Reflections of other forms can be found by taking lines from corners or centerlines perpendicular to the cylinders surface i.e. to the opposite VP. Draw ellipses around the cylinder at these points. Lines from details are then drawn to the centers of the corresponding ellipses to find the location and size of the reflection.

ORTHOGRAPHIC VIEW

HORIZON LINE

REFLECTION OF HORIZON

REFLECTION OF POLE

PERPENDICULARTO CYLINDER

A

Several relationships are used here. They are based on the relationship of a Horizon to the cylinders surface. The horizon reflection is found to run around the center of the cylinder and can be traced around the form by using vertical lines at the edges of the ellipse at both ends. The horizon is located where this vertical is tangent to the el-lipse. See points A, B & C. All reflections of objects on the ground plane relate to the horizon line on the form and are found near it.

10 - 10

VIEWERS REFLECTION

POLE REFLECTIONREFLECTIONOF HORIZON

SHADOW REFLECTION

ORTHOGRAPHIC VIEW

HORIZON LINE

BALL REFLECTION

REFLECTIONS INTO SPHERES

Below is a simplified version of pole and ball refections into a sphere.

Several predictable reflection locations can be found in and around the horizon line. This is the reflection of the earths horizon at infinity. The ellipse angle used is dependent on how far below the eye level the sphere is.

The reflection of the viewer is always at dead center. The size of this shape and also how detailed it needs to be is dependent on the distance the viewer is from the sphere. This is somewhat ar-bitrary. You can make it any size you want. It is best to keep this somewhat non-descript and anyone viewing this drawing must imagine their own image in the reflection. So, keep it simple.

Everything else is found by taking lines to the center of the sphere.

The drawing on the following page shows shaded reflections on the basic forms. These have been somewhat simplified as well, but are intended to show locations of reflections and their different directions on each type of surface. The reflections can be seen as working perpendicular to or into all surfaces. Reflections should give a "see in" effect and appear to be into the surface and not on it.

10 - 11

HIGHLIGHT & REFLECTION OF BASIC FORMS

OPHEIM '92

10 - 12

HIGHLIGHT & REFLECTION OF BASIC FORMS

OPHEIM '92

A STUDY IN ROTATIONS OF FORMSVERTICAL & HORIZONTAL ROTATIONS

45° ROTATION90° ROTATION

11 - 1

ABC A

CBC

ABC

ABCAB

C

ROTATION OF CUBE AROUND A HORIZONTAL AXIS

This rotation is based on sound principles although the final view leaves a little to be desired. The cube has some distortion, but when multiplied into other proportions, it looks OK.

1. Construct a cube using any method.

2. Find and extend diagonal lines to their respective vanishing points located above and below the VPL.

3. Construct a circle on that side.

4. Draw tangent lines to the circle to where they cross - forming the rotated side.

5. Pick points A & R where both cubes cross the same point.

6. Complete by drawing lines through R

R

RA

A

VPL VPR

A

SIDE VIEW

More Stuff

11 - 2

aVPL VPR

MULTIPLICATION OF A ROTATED CUBE

The multiplication is the same as one which is on the vertical plane. This allows for any proportion to be found. Here the cube was multiplied to a 3 wide x 2 high x 1 deep box. The letter was first constructed in orthographic (usually found that way in type books) and enlarged using a grid similar to the multiplied box. You can use as many grid lines as you want. Find the shape on the front side and then again on the back edges. A back grid can be used or, as in this case, projections are made from key points on the front to the back using the box edge.

11 - 3

MEASURING METHOD FOR ROTATED FORMS IN ONE-POINT PERSPECTIVE

SPCV

HEIGHTS MP

ANYROTATIONANGLE

90°

HEIGHTS VP

HL

HML

ROTATION MPROTATION VP

HEIGHTWIDTHLENGTH

We're going to get a little heavy here with some rather fancy layouts for measuring a rotated form. This looks a bit complicated, but given a chance it will prove to be a great way to lay out a system on paper that can be reused as often as you want.

Do these steps:

1. Draw a HL and pick a CV where you want to see the ob-ject from.

2. Pick a SP at your distance from CV and also pick an angle of incline for the object and draw that line to a point above CV to find the Rotation VP.

3. Draw a line 90° from this line to a point below the CV to find the Heights VP.

4. With Heights VP as center draw an arc from SP to a horizontal position at Heights MP.

5. Find the Rotation MP by using the Rotation VP as center and drawing an arc from SP to a horizontal position.

6. Draw an HML at desired eye level.

7. Following the diagram, measure the length, width and height of the object. Continue the drawing as usual adding details, etc..

90°

11 - 4

MEASURING METHOD FOR ROTATED FORMS IN TWO-POINT PERSPECTIVE

WIDTH &HEIGHT

DEPTH

P

MPLMPRVPL VPR

HEIGHTS MP

ANYROTATIONANGLE

90°

HEIGHTS VP

HL

HML

CV

VMLNOTUSED

ROTATION MPROTATION VPHere again it is pretty much the same bailiwick. This time there are of course two vanishing points involved which means more plotting, but not much harder than in One-point.

Do these steps:

1. Draw a HL with both VPL and VPR.2. Establish a Standard Measuring System

of any type discussed in Chapter 3. The Measuring System Circle was used here, but predictions of MP locations will work just as well.

3. From MPL draw a line at the desired ro-tation angle to a point above VPL. This is the Rotation Vanishing Point.

4. Using that point as a center, draw an arc up to a point horizontal to the Rotation VP from the MPL. This is the Rotation MP.

5. With a line 90° from the chosen rotation angle at MPL draw a line to a point below VPL to find the Heights VP.

6. Using that line as a radius find a point horizontal from Heights VP to reach Heights MP.

7. Following the diagram, measure the length, width and height of the object. Continue the drawing as usual adding details, etc..

90°

11 - 5

HLVPL VPR

90¡

HL

VPL

VPR

90° ROTATION OF BOX FORMS

First construct a shape on its side by using any of the conventional methods. Then rotate . . .

Systems usually demand our perception of objects to be tied to this horizontal line of sight concept. One way to break away from this mo-notonous point of view is to take a standard construction and rotate it through a 90 degree rotation. The result is quite a surprise. We are given a view of a form that appears to be rotated into our horizontal line of sight - giving us a look into the top surface. This is very useful when seeing into the top of the form is important, The rotation can be used with any subject matter or object. You can see how different this box looks after it is rotated. Once the construction lines are erased, it is very difficult to tell now the object was drawn.

11 - 6HL

HL

VPL

VPL

900

As details are added the drawing becomes more believable and gives an interesting interpretation of an area that defines space. One can imagine moving about within that space. The difference here is taking out a ceiling instead of taking out a leading wall. Actually there is not much difference in concept when looked at from that point of view.

11 - 7

1 POINT VIEW DRAWN IN A ROTATED POSITION

Objects that are drawn in an unexpected way can at-tract more attention and make interesting drawings out of uninteresting subject matter.

This was accomplished by drawing the interior as it would appear with the ceiling removed and the point of view from above looking straight down from a point directly above the Vanishing Point 1. This is not difficult as the only difference is the trading of dimensions. All height dimensions become depth and the length or width is drawn as the height. The view can be changed dramatically by moving the VP1. It can even be outside the view giving a look at one of the outside walls making it similar to the 2-point version on the previous page.

Once drawn, you can rotate this drawing in any direc-tion and it will communicate. This is not true of any other drawing. This can be an advantage in layout as the drawing can be used vertically or horizontally.

ONE-POINT INTERIOR DRAWN IN 90° ROTATION

VP1

11 - 8

A A A A BBBB

CC

C

C

C

HL

A

C

BC

ABC

ABC

ABC

ACBC

ABC

ABC

AB

C

ROTATION OF CUBES ON HORIZONTAL & VERTICAL AXIS

Above Measuring System drawings show views that are drawn in different locations between the VP's. They are all parallel to each other as they share the same Van-ishing Points.

In the second row we see the same blocks, but they have been moved to different positions. This means they no longer share the same Van-ishing Points but do share the same Horizon. They appear to have been rotated along a vertical axis.

The third row shows the same blocks with a slight horizontal rotation. They are traced in this position giving an im-pression of floating blocks with tilted horizons. TILTED BOXES WITH HORIZONTAL AND VERTICAL ROTA-

ROTATION OF CUBES ON HORIZONTAL & VERTICAL AXIS

11 - 9

WORMS EYE VIEW

BIRDS EYE VIEW

'92

OPHEIM

AFT

AFT

CIRCLE METHODRANDOM CHOICE METHOD

CORRECTION METHODPROS & CONS

THREE-POINT PERSPECTIVE

12 - 1

VPL VPR

XY

Z

VP3

MPR

HML

MPLCV

VML

THREE-POINT PERSPECTIVE CONVENTIONAL CONSTRUCTION

This variation of the measuring system uses an HML at the top corner of the view at point-X. Lines are taken from both Vanishing Points through X to locate Y & Z on the Field of Vision Circle. Lines through Y & Z from their respective Vanishing Points intersect at VP3. Measure the same as usual and use VP3 instead of verticals. This gives the illusion of seeing a slight foreshortening in the vertical direction as we see them.

There is a problem of overdoing this phenomenon. The third vanishing point is actually at the center of the earth. This would take a very large sheet of paper. The only time we could see it is when we are looking downward at a steep angle. It is therefore much better to get the VP3 as far away from the view as possible.

VPL VPR

XY

Z

VP3

MPR

HML

MPLCV

VML

12 - 2

TRIANGULAR METHOD OF CONSTRUCTION

Using a triangle is a more arbitrary method to find VP3, but will usually give less distortion in the vertical direction as the conventional method often does - especially when the view is large within the circle. Here all three VP's are chosen as 3 points of a triangle. VP3 is located as far away as possible (depending on how large your work surface is and length of straight edges) and on a vertical line from the Center of Vi-sion. This can be anywhere between the VPL and VPR. It is usually best to have the CV somewhere center of the view.

VP2MID-POINTVP1

TO VP3

VP3

VP1 VP2C.V.

C.V.

The CV (Center of Vision)is at Mid-point here. It can be moved left or right.

12 - 3

THREE-POINT EFFECT FROM CONSTRUCTION

This method works well for any object. The idea is to draw it as usual and work with the base to make it smaller. The diagonal on the box, if drawn from opposite corners will make the bottom smaller and also slightly foreshortened. Just draw the diagonals and pick a point just inside the edge. Draw

that around using the VP's. The Cylinder is done the same way using a smaller ellipse size on the bottom. This method allows for more control than the others and the best part - it doesn't need the third VP. Just be careful not to overdo the effect. Remember that VP3 is at the center of the earth.

BOX CONSTRUCTION

CYLINDER CONSTRUCTION

12 - 4

AANGLED WALLS - ONE POINT 4 8AXONOMETRIC DRAWINGS 1 3

BBANK FRONT REFLECTIONS 10 3BASE LINES 3 9BASE LINES 3 15-16BENCH SHADOW 9 22-23BOWLING BALL 8 13BOX FLOATING SHADOW 9 22-23BOX FORM ROTATION 11 6-7BOX REFLECTIONS 10 2BOX SHADOW ONTO WALL 9 21BOX WITH FLAPS SHADOWS 9 9

CCAR SHADOW 9 22-23CIRCLE CONSTRUCTION USING ELLIPSE GUIDES 7 5-6CIRCLE DIVISION 8 8CIRCLES IN PERSPECTIVE 7 2 CONE 1 3CONE OF VISION 1 7CONE OF VISION METHOD - ONE POINT 4 4CONTOUR DRAWING 3 18CONVERGENCE POINT DEFINED 9 25-26CONVERGING LIGHT SHADOW 9 24-26CONVERGING SHADOW DEFINED 9 1CONVEX MIRROR REFECTIONS THEORY 10 8-9CONVEX MIRROR REFLECTIONS 10 7CORE DRIP LINE 9 16CORE SHADOW 9 16CUBE CONSTRUCTION - TWO POINT 5 2CUBE ROTATION 11 2CYLINDER CONSTRUCTION EXAMPLES 8 4CYLINDER CONSTRUCTION 8 2CYLINDER CONSTRUCTION - CAMERA 8 6-7CYLINDER CONSTRUCTION - CONTOURED 8 10CYLINDER DIVISION 8 11

CYLINDER REFLECTIONS OF OBJECTS 10 10CYLINDER ROTATION 8 5CYLINDER SHADOWS 9 10-15CYLINDERS - CONSTRUCTION WITH GUIDES 8 3

DDEFINITION OF PERSPECTIVE 1 7DIAGONAL METHOD - ONE POINT 4 4DIAGONAL MULTIPLICATION 5 4DIAGONAL VANISHING POINT 5 12DIRECTION OF SHADOW 9 2DISTORTION 1 3-4DIVIDING SHAPES 5 5DRIP LINE 9 16, 18

EELEVATION MEASUREMENTS 3 10ELEVATION OBLIQUE DRAWINGS 1 7ELEVATION OF LIGHT 9 2ELEVATION VIEW 2 1-4ELLIPSE ALIGNMENT 7 6-7ELLIPSE ALIGNMENT RIGHT & WRONG 7 13ELLIPSE ANGLE LOCATIONS ON SPHERE 8 12ELLIPSE ANGLE MEASUREMENT 7 9, 11ELLIPSE CONSTRUCTION 7 2ELLIPSE DEFINED 7 2ELLIPSE GUIDES 7 9, 11ELLIPSE RELATIONSHIP - VERTICAL PLANE 7 8ELLIPSE RELATIONSHIP - HORIZONTAL PLANE 7 10ELLIPSE SIZE MEASUREMENT 7 12ELLIPSE TEMPLATES 7 5ELLIPSES IN PERSPECTIVE - 12 PT. METHOD 7 4ELLIPSES IN PERSPECTIVE - 8 PT. METHOD 7 3ENLARGEMENT OF VIEWS 5 6EYE LEVEL 1 3EYE LEVEL 1 6EYE LEVEL 5 11

FFIELD OF VISION 1 7, 9FIGURE DRAWING TECHNIQUE 5 17FLAGPOLE RULE 9 5FLAGPOLE RULE AT WORK 9 7FLAGPOLE RULE IN ACTION 9 20-21FLAPS - HORIZONTAL HINGE 3 11FLAPS - VERTICAL HINGE 3 12FLASHLIGHT 8 13FLOOR GRID METHOD 4 5FORESHORTENING 1 2

GGRID - ONE POINT WITH VERTICAL MEASURING 5 14GRID - TWO POINT WITH VERTICAL MEASURING 5 13GROUND ANGLE 9 4GROUND ANGLE DEFINED 9 2GROUND ANGLE VANISHING POINT 9 24GROUND ANGLE VANISHING POINT 9 27GROUND ANGLE VANISHING POINT 9 30-33GROUND LINE 2 3-44

HHIGHLIGHTS OF BASIC FORMS 10 12HORIZON 1 2HORIZON 1 9HORIZON LINE 1 4-5HORIZON LINE 2 4-6HORIZONTAL DIVISIONS 5 6HORIZONTAL MEASURING LINE LOCATIONS 3 7HORIZONTAL PLANE DEVELOPMENT 5 11

IINFINITY 1 2, 4INTERIOR GRID - TWO POINT 5 10INTRODUCTION 1 5ISOMETRIC DRAWINGS 1 1

LLARGE SCALE DESIGN DRAWINGS 4 8LAYOUT TABLET 3 16LETTER FORM ROTATION 11 3LIGHT ANGLE 9 4LIGHT ANGLE - NEVER SHADOWS 9 7LIGHT ANGLE DEFINED 9 2LIGHT ANGLE VANISHING POINT 9 24LIGHT ANGLE VANISHING POINT 9 27LIGHT ANGLE VANISHING POINT 9 30-33LIGHT LOCATION & SOURSE 9 1LIGHT SOURCE DETERMINATION 9 4

MMAJOR AXIS 7 5-6MEASURING - INSIDE BASE LINES 3 16MEASURING - OUTSIDE BASE LINES 3 13MEASURING INSIDE BASE LINES 3 10MEASURING PLANE ARCHITECTURAL INTERIOR 6 6MEASURING PLANE GRID 6 2MEASURING PLANE GRID 6 5MEASURING PLANE PERSPECTIVE DEFINED 6 2MEASURING PLANE PERSPECTIVE STEPS 6 2-4MEASURING PLANE PRODUCT APPLICATION 6 7MEASURING POINT - CONVENTIONAL METHOD 4 2MEASURING POINT - ONE POINT 4 19MEASURING POINTS - PREDICTION 3 4MEASURING SYSTEM - CONST. OF CUBES 3 7MEASURING SYSTEM - EXTENDED 3 8MEASURING SYSTEM - HOW TO USE 3 2MEASURING SYSTEM - LAYOUT TABLET 3 17MEASURING SYSTEM - MOLDED FORMS 3 19MEASURING SYSTEM - ONE POINT METHOD 4 2MEASURING SYSTEM - ROTATED FORMS 11 4-5MEASURING SYSTEM - SELECTION 3 6MEASURING SYSTEM - SHORT CUT 3 3MEASURING SYSTEM - TWO POINT 3 9MEASURING SYSTEM - TWO POINT STEPS 3 1

MINOR AXIS 7 5-6MINOR AXIS ALIGNMENT 8 2MIRROR REFLECTIONS 10 2MODULAR PERSPECTIVE 5 10MULTI-VIEW DRAWINGS 1 3MULTIPLICATION OF CUBES 5 3MULTIPLICATION OF HORIZONTAL SURFACES 5 8MULTIPLICATION OF ROTATED CUBE 11 3MULTIPLICATION OF VERTICAL SURFACES 5 7

NNEGATIVE LIGHT SHADOW DEFINED 9 1NEGATIVE LIGHT SHADOW FROM WINDOW 9 32-33

OOBJECT 1 2OBJECT 1 4OBJECT 1 6OBJECT 2 5OBJECT 2 7OBSERVER 1 3-7OBSERVER 2 2ONE & TWO POINT PERSPECTIVE COMBINATION 4 7ONE POINT PERSPECTIVE 4 2ONE POINT PERSPECTIVE - DEFINITION 4 2ONE POINT PERSPECTIVE - DISTORTION 4 3ONE POINT PERSPECTIVE - FLOOR GRID 4 4ONE POINT PERSPECTIVE - MEASURING POINT 4 3-4ONE POINT PERSPECTIVE - OFFICE INTERIOR 4 8ONE POINT PERSPECTIVE - SHORT CUTS 4 3ONE POINT PERSPECTIVE - SHORT CUTS 4 5-6ONE POINT PERSPTECTIVE - LARGE SCALE 4 9ONE POINT PERSPTECTIVE - SMALL OBJECTS 4 10ONE POINT RELATIONSHIP 3ORTHOGRAPHIC DRAWINGS 1 5ORTHOGRAPHIC VIEW MULTIPLIED 5 4ORTHOGRAPHIC VIEWS 2 5ORTHOGRAPHIC VIEWS 3 20

PPARALLEL METHOD 9 2PARALLEL SHADOW DEFINED 9 1PERIPHERAL VISION LINE 1 2PERSPECTIVE DEFINED 1 7PERSPECTIVE VIEWS MULTIPLIED 5 4-5PICTURE PLANE 1 4PICTURE PLANE 1 5PICTURE PLANE 1 7PICTURE PLANE 2 2PICTURE PLANE 2 4-5PICTURE PLANE 2 7PINWHEEL CONSTRUCTION 8 8PLAN OBLIQUE DRAWINGS 1 3PLAN VIEW 2 3PLAN VIEW 2 1PLAN VIEW 2 3-4PLAN/ELEVATION LIMITATIONS 2 9PLAN/ELEVATION METHOD 2 8PLAN/ELEVATION METHOD TWO POINT 2 1PLAN/ELEVATION METHOD/ONE POINT 2 3POSITIVE LIGHT SHADOW DEFINED 9 1POSITIVE LIGHT SHADOW INTO ROOM 9 28-29POSITIVE LIGHT SHADOW STEPS 9 27PROJECTIONS 1 1PROTRACTOR MEASUREMENTS 7 8-9PROTRACTOR MEASUREMENTS 7 10PYRAMID SHADOW 9 9

QQUADRANTS 7 5

RRADIATION POINT 5 7RECTANGLE ENLARGEMENT & REDUCTION 5 9REDUCTION OF VIEWS 5 7REFLECTION OF BOX FORMS 10 2REFLECTIONS IN BANK FRONT 10 3REFLECTIONS INTO SPHERES 10 11

RREFLECTIONS OF FORMS INTO MIRRORS 10 4-6REFLECTIONS OF OBJECTS INTO POOL 10 6REFLECTIONS ON BASIC FORM SURFACES 10 12REFLECTIONS ON CURVED CONVEX MIRROR 10 7REFLECTIONS ON HORIZONTAL CYLINDER 10 10ROAD CONSTRUCTION 3 14ROTATED VIEW OF ROOM 11 7-8ROTATION 11 1-9 ROTATION OF BOX FORMS - 90° 11 6-7ROTATION OF CUBE - HORIZONTAL AXIS 11 2ROTATION OF CUBE MULTIPLIED 11 3ROTATION OF CUBES ON HORIZONTAL AXIS 11 9ROTATION OF CUBES ON VERTICAL AXIS 11 9ROTATION OF LETTER FORM 11 3ROTATIONS TILTED 11 9

SSCALE 1 2SHADOW - ANGLED LINE RULE 9 3SHADOW - BOX FORM ONTO CONE SHAPE 9 19SHADOW - BOX FORM ONTO PYRAMID 20SHADOW - BOX ONTO WALL 9 21SHADOW - BOX WITH FLAPS 9 9SHADOW - CONVERGING LIGHT 9 24-26SHADOW - CYLINDRICAL OBJECTS 9 11SHADOW - CYLINDRICAL OBJECTS 9 12SHADOW - EIGHT BALL OVER EDGE 9 18SHADOW - FLAG POLE RULE 9 5SHADOW - FLOATING FORMS 9 22-23SHADOW - FLOATING SPHERE 9 18SHADOW - HORIZONTAL CYLINDERS 9 13-14SHADOW - HORIZONTAL LINE RULE 9 3SHADOW - INCLINED PLANE ONTO CYLINDER 9 21SHADOW - INSIDE HORIZONTAL CYLINDER 9 15SHADOW - MATCH ONTO INCLINED PLANE 9 21SHADOW - NEGATIVE LIGHT SHADOW 9 32-33SHADOW - NEGATIVE LIGHT OF TABLE 9 31SHADOW - NEGATIVE LIGHT STEPS 9 30SHADOW - PARALLEL SHORT CUT 9 3

SHADOW - POSITIVE LIGHT INTO ROOM 9 28-29SHADOW - POSITIVE LIGHT STEPS 9 27SHADOW - PYRAMID 9 9SHADOW - PYRAMID ONTO WALL 9 20SHADOW - SPHERE CONSTRUCTION 9 16SHADOW - SPHERE ON WALL & FLOOR 9 18SHADOW - SPHERE SHORT CUT 9 17SHADOW - STANDING CYLINDERS 9 10SHADOW - VERTICAL LINE RULE 9 3SHADOW - WALL OF EDGE OF PLATFORM 9 20SHADOW - WALL VARIATIONS 9 7SHADOW OF BIX FORMS -EXTERIOR 9 5SHADOW OF BOX FORMS - INTERIOR 9 4SHADOW OF HORIZONTAL PLANES 9 4SHADOW OF VERTICAL PLANES 9 4SHADOW PLOTS - DOORWAYS & WINDOWS 9 8SHADOW PLOTTING BASICS 9 2SHADOW SOLUTIONS - MULTIPLE OF BOX 9 6SHADOW TYPES 9 1SHADOWS 9 1-33 SHADOWS OF VERTICAL PLANES 9 3SPHERE STUDIES 8 11SIGHT POINT 2 9SIGHT POINT 2 5SINGLE VIEW DRAWINGS 1 3SMALL OBJECT SKETCHES - ONE POINT 4 9SPHERE CONSTRUCTIONS 8 12SPHERE DRAWN TO SCALE 8 14SPHERE DRAWN TO SCALE - SHORT CUT 8 15SPHERE REFLECTIONS 10 11SPHERE SHADOW ON WALL & FLOOR 9 18SPHERE SHADOW SHORT CUT 9 17SPHERE SHADOW WHEN FLOATING 9 18SPHERE SHADOWS 9 16SPHERE TOUCH POINT 9 16SPIRAL STAIRCASE 8 9SQUARE DEPTH METHOD - ONE POINT 4 4SQUARE TO CUBE METHOD 5 11

SQUARE TO CUBE PERSPECTIVE - STEPS 5 2SQUARE TO CUBE PERSPECTIVE METHODS 5 1-2

TTENNIS BALL CONSTRUCTION 8 13TERMINATOR 9 16THREE-POINT EFFECT FROM CONSTRUCTION 12 4THREE-POINT PERSPECTIVE 12 1-3THREE-POINT PERSPECTIVE CONVENTIONAL 12 2THREE-POINT - TRIANGULAR METHOD 12 3TOUCH POINT OF SPHERE 9 16TRACE TEMPLATES 7 5TRACING GRIDS 5 15TRIANGULAR METHOD 12 3TRIMETRIC DRAWINGS 1 3TRUE LENGTH 1 3TRUE LENGTH LINE 2 4-5TWO POINT RELATIONSHI 1 2TWO-POINT CENTER CONSTRUCTION 3 4-5TWO-POINT LEFT CONSTRUCTION 3 4-5TWO-POINT RIGHT CONSTRUCTION 3 4-5

VVANISHING POINT 1 5VANISHING POINT 2 5VANISHING POINT LOCATION CONTROL 2 7VANISHING POINT, SINGLE 2 2VANISHING POINTS 2 5VERTICAL DIVISIONS 5 6VIEW ABOVE HORIZON 2 6VIEW BELOW HORIZON 2 4VISUAL SHIFT 3 10

WWAGON WHEEL CONSTRUCTION 8 8

YYARDSTICK PERSPECTIVE 3 5