2D ESSENTIALSInstructor: Laura Gerold, PE

Catalog #10614113Class # 22784, 24113, 24136, & 24138

Class Start: January 18, 2012 Class End: May 16, 2012

QUESTIONS?

Class Notes

Project Proposals are due in one week on February 22nd

Extra Credit is due in one week on February 22nd

Project – Have one view (out of six total) drawn on non-grid paper

Homework assignment is now on Blackboard in Class Materials/ Homework folder

CHAPTER 4 – GEOMETRIC

CONSTRUCTION

Small Errors . . .

Small errors during drawing construction can lead to big errors in your end product . . .

Errors applied to Land Surveying

Land Surveying is "The art and science of measuring angles and distances on or near the surface of the earth.“

Hunter House

Originally built cabin in 1819, but had to build a new one in 1822 due to surveying error.

Fort Blunder, New York

Construction started on the fort in 1816 to protect against invasions of the British/Canadians, but it was soon discovered the fort was entirely in Canada and not in the US due to a surveying error.

Milling Machine

Group Activity Break into groups of 2 or 3 and refresh yourselves

how to draw these three items, be prepared to explain how you did the work . . .

Triangle Draw a triangle with two sides given, R = 2”, S = 1”

Triangle Draw a triangle with sides 5”, 4.5”, and 4” Bisect the three interior angles The bisectors should meet at a point Draw a circle inscribed in the triangle with the point where

the bisectors meet in the center Drawing a point through a line and perpendicular to a

line Draw a line Draw a point on the line Draw a line through the point and perpendicular to the line Repeat process, but this time put the point not on the line

BISECTING AN ANGLE

1. Lightly draw arc CR 2. Lightly draw equal arcs r with radius

slightly larger than half BC, to intersect at D

3. Draw line AD, which bisects the angle

TRIANGLES Drawing a right triangle with hypotenuse

and one side given1. Given sides S and R2. With AB as diameter equal to S, draw a

semicircle3. With A as center, R as radius, draw an arc

intersecting the semicircle C.4. Draw AC and CB

TRIANGLES

Drawing a Triangle with Sides Given1. Draw one side, C2. Draw an arc with radius equal to A3. Lightly draw an arc with radius equal to

B4. Draw sides A and B from the intersection

of the arcs

DRAWING A LINE THROUGH A POINT AND PERPENDICULAR TO A LINE

When the Point is Not on the Line (AB & P given) From P, draw any convenient inclined line, PD on (a) Find center, C, of line PD Draw arc with radius CP, center at C Line EP is required perpendicular P as center, draw an arc to intersect AB at C and D (b) With C & D as centers and radius slightly greater than

half CD, draw arcs to intersect at E Line PE is required perpendicular

When the Point Is Not on the Line When the Point Is on the Line T-square Method

DRAWING A LINE THROUGH A POINT AND PERPENDICULAR TO A LINE

When the Point is on the Line (AB & P given) With P as center and any radius, strike arcs to intersect

AB at D and G (c) With D and G as centers and radius slightly greater than

half DG, draw equal arcs to intersect at F. Line PF is the required perpendicular

When the Point Is Not on the Line When the Point Is on the Line T-square Method

Protractors

For measuring or setting off angles other than those obtainable with triangles, use a protractor.

Plastic protractors are satisfactory for most angular measurements

Nickel silver protractors are available when high accuracyis required

Measuring Angles with a Protractor

Math Made Easy: Measuring Angles (part 1)

Math Made Easy: Measure Angles (part 2)

Many angles can be laid out directly with the protractor.

LAYING OUT AN ANGLE Tangent Method1. Tangent = Opposite / Adjacent2. Tangent of angle q is y/x3. Y = x tan q4. Assume value for x, easy such as 105. Look up tangent of q and multiply by x

(10)6. Measure y = 10 tan q

LAYING OUT AN ANGLE Sine Method1. Sine = opposite / hypotenuse2. Sine of angle q is y/z3. Draw line x to easy length, 104. Find sine of angle q, multiply by 105. Draw arc R = 10 sin q

LAYING OUT AN ANGLE Chord Method1. Chord = Line with both endpoints on a

circle2. Draw line x to easy length, 103. Draw an arc with convenient radius R4. C = 2 sin (q/2)5. Draw length C

LAYING OUT AN ANGLE

Draw two lines forming an angle of 35.5 degrees using the tangent, sine, and chord methods

Draw two lines forming an angle of 40 degrees using your protractor

DRAWING AN EQUILATERALTRIANGLE

Side AB given With A & B as centers and radius AB,

lightly construct arcs to intersect at C Draw lines AC and BC to complete

triangle

DRAWING AN EQUILATERALTRIANGLE

Draw a 2” line, AB Construct an equilateral triangle

DRAWING A SQUARE1. One side AB, given2. Draw a line perpendicular through point

A3. With A as center, AB as radius, draw an

arc intersecting the perpendicular line at C

4. With B and C as centers and AB as radius, lightly construct arcs to intersect at D

5. Draw lines CD and BD

DRAWING A SQUAREDiameters Method1. Given Circle2. Draw diameters at right angles to each

other3. Intersections of diameters with circle are

vertices of square4. Draw lines

DRAWING A SQUARE

Lightly draw a 2.2” diameter circle Inscribe a square inside the circle Circumscribe a square around the circle

DRAWING A REGULAR PENTAGON Geometric Method

1. Bisect radius OD at C2. Use C as the center and CA as the radius to lightly

draw arc AE3. With A as center and AE as radius draw arc EB4. Draw line AB, then measure off distances AB

around the circumference of the circle, and draw the sides of the Pentagon through these points

Dividers Method Geometric Method

DRAWING A REGULAR PENTAGON Lightly draw a 5” diameter circle Find the vertices of an inscribed regular

pentagon Join vertices to form a five-pointed star

DRAWING A HEXAGONEach side of a hexagon is equal to the radius of the circumscribed circle

Use a compass Centerline Variation

Steps

DRAWING A HEXAGON Method 1 – Use a Compass

Each side of a hexagon is equal to the radius of the circumscribed circle

Use the radius of the circle to mark the six sides of the hexagon around the circle

Connect the points with straight lines Check that the opposite sides are parallel

Use a compass

DRAWING A HEXAGON Method 2 – Centerline Variation

Draw vertical and horizontal centerlines With A & B as centers and radius equal to that

of the circle, draw arcs to intersect the circle at C, D, E, and F

Complete the hexagon

Centerline Variation

DRAWING A HEXAGON

Lightly draw a 5” diameter circle Inscribe a hexagon

Drawing an Octagon Given a circumscribed square, (the

distance “across flats”) draw the diagonals of the square.

Use the corners of the square as centers and half the diagonal as the radius to draw arcs cutting the sides

Use a straight edge to draw the eight sides

Drawing an Octagon

Lightly draw a 5” diameter circle Inscribe an Octagon

DRAWING A CIRCLE THROUGH 3 POINTS

A,B, C are given points not on a straight line

Draw lines AB and BC (chords of the circle)

Draw perpendicular bisectors EO and DO intersecting at O

With center at ), draw circle through the points

DRAWING A CIRCLE THROUGH 3 POINTS

Draw three points spaced apart randomly Create a circle through the three points

FINDING THE CENTER OF A CIRCLE

Method 1 This method uses the principle that any right

triangle inscribed in a circle cuts off a semicircle

Draw any cord AB, preferably horizontal Draw perpendiculars from A and B, cutting the

circle at D and E Draw diagonals DB and EA whose intersection

C will be the center of the circle

FINDING THE CENTER OF A CIRCLE

Method 2 – Reverse the procedure (longer) Draw two nonparallel chords Draw perpendicular bisectors. The intersection of the bisectors will be the

center of the circle.

FINDING THE CENTER OF A CIRCLE

Draw a circle with a random radius on its own piece of paper

Give your circle to your neighbor Find the center of the circle given to you

DRAWING A CIRCLE TANGENT TO A LINE AT A GIVEN POINT

Given a line AB and a point P on the line At P, draw a perpendicular to the line Mark the radius of the required circle on

the perpendicular Draw a circle with radius CP

DRAWING A CIRCLE TANGENT TO A LINE AT A GIVEN POINT

Draw a 4” long line Place a point P at the midpoint of the line Draw a 2” diameter circle tangent to the

line at P

DRAWING A TANGENT TO A CIRCLE THROUGH A POINT

Work as a 2-3 person group to figure out the following problems without a T-square Given a point on a circle, draw a line tangent

to the circle Given a point not on the circle, draw a line

tangent to the circle and through the point

DRAWING AN ARC TANGENT TO A LINE OR ARC AND THROUGH A

POINT Method 1(a)

Given line AB, point P, radius R Draw line DE parallel to given line and distance R from

it From P draw arc with radius R, cutting line DE at C C is the center of the required tangent arc

Tangents

DRAWING AN ARC TANGENT TO A LINE OR ARC AND THROUGH A

POINT Method 2 (b)

Given line AB, tangent point Q on the line, and point P Draw PQ, the chord of the required arc Draw perpendicular bisector BE At Q, draw a perpendicular to the line to intersect DE at

C C is the center of the required tangent arc

Tangents

DRAWING AN ARC TANGENT TO A LINE OR ARC AND THROUGH A

POINT Method 3 (c)

Given arc with center Q, point P, and radius R From P, strike an arc with radius R From Q draw an arc with radius equal to given arc plus

R The intersection C of the arcs is the center of tangent

arc

Tangents

DRAWING AN ARC TANGENT TO A LINE OR ARC AND THROUGH A POINT

Given line AB= 3” long, any point P not the line (similar to graphic a on page 4.28), and radius 2”, draw an arc tangent to AB through point P.

DRAWING AN ARC TANGENT TO TWO LINES AT RIGHT ANGLES

Two lines are given at right angles to each other With given radius, R, draw an arc intersecting the given

lines at tangent points T With given radius R again, and with points T as centers,

draw arcs intersecting at C With C as center and given radius R, draw the required

tangent arc

For small radii, such as 1/8R for fillets and rounds, it is not practicable to draw complete tangency constructions. Instead, draw a 45° bisector of the angle and locate the center of the arc by trial along this line

DRAWING AN ARC TANGENT TO TWO LINES AT RIGHT ANGLES

Draw two intersecting at right angles vertical and horizontal lines, each 2.5 inches long

Draw a 1.5 inch radius arc tangent to the two lines

DRAWING AN ARC TANGENT TO TWO LINES AT ACUTE OROBTUSE ANGLES

Given two lines not making a 90° Draw lines parallel to the given lines at distance R from

them to intersect at C the center From C, drop perpendiculars to the given lines to locate

tangent points, T With C as the center and with given radius R, draw the

required tangent arc between the points of tangency

DRAWING AN ARC TANGENT TO TWO LINES AT ACUTE OROBTUSE ANGLES

Draw two intersecting vertical and horizontal lines as an acute angle, each 2.5 inches long

Draw a 1.5 inch radius arc tangent to the two lines

DRAWING AN ARC TANGENT TO AN ARC AND A STRAIGHT LINE

Given arc with radius G and a straight line AB Draw a straight line and an arc parallel to the given straight line

at the required radius distance R from them. Will intersect at C, required center

From C, draw a perpendicular to the given straight line to find one point of tangency, T. Join the centers C and O with a straight line to locate the other point of tangency

With the center C and radius R, draw the required tangency arc between the points of tangency

DRAWING AN ARC TANGENT TO TWO ARCS Given arcs with centers A and B and required radius R With A and B as centers, draw arcs parallel to the given arcs and at a

distance R from them; their intersection C is the center of the required tangent arc

Draw lines of the centers AC and BC to locate points of Tangency, T, and draw the required tangent arc between the points of tangency

Drawing an Arc Tangent to Two Arcs and Enclosing One or Both

Required Arc to Enclose Two Given Arcs With A & B as centers lightly draw arcs HK-r (given

radius minus radius of small circle) and HK-R (given radius minus radius of large circle) intersecting at G, the center of the required tangent arc

Lines of centers GA and GB (extended) determine points of tangency, T

Drawing an Arc Tangent to Two Arcs and Enclosing One or Both

Required Arc to Enclose One Given Arc With C & D as centers, lightly draw arcs HK+r (given

radius plus radius of small circle) and HK-R (given radius minus radius of large circle) intersecting at G, the center of the required tangent arc

Lines extended through centers GC and GD determine the points of tangency, T

TANGENT ARCS

Group project - Work together to do the following: Draw an arc tangent to an arc and a straight

line using your own measurements Draw an arc tangent to two arcs using your

own measurements Draw an arc tangent to two arcs and enclosing

one or both

What the heck is an Ogee?

Ogee is a curve shaped somewhat like an S, consisting of two arcs that curve in opposite senses, so that the ends are parallel.

Used in: Molding Architecture Marine Timber Construction

DRAWING AN OGEE CURVE Connecting Two Parallel Lines (Method 1)

Let NA and BM be 2 parallel lines Draw AB and assume inflection point T (midpoint) At A and B, draw perpendiculars AF and BC Draw perpendicular bisectors at AT and BT Intersections F and C of the bisectors and the

perpendiculars are the centers of the tangent circles

Connecting Two Parallel Lines Connecting Two Nonparallel Lines

DRAWING AN OGEE CURVE Connecting Two Parallel Lines (Method 2)

Let AB and CD be two parallel lines with point B as one end of curve and R the given radii

At B, draw the perpendicular to AB, make BG = R, and draw the arc as shown

Draw SP parallel to CD at distance R from CD With center G, draw the arc of the radius 2R, interesting line

SP and O. Draw perpendicular OJ to locate tangent point J, and join

centers G & O to locate point of tangency T. Use centers G & O and radius R to draw tangent arcs

Connecting Two Parallel Lines Connecting Two Nonparallel Lines

DRAWING AN OGEE CURVE Connecting Two Nonparallel Lines

Let AB and CD be two nonparallel lines Draw the perpendicular to AB at B Select point G on the perpendicular so that BG equals any

desired radius, and draw the arc as shown (c) Draw the perpendicular CD at C and make CE=BG Join G to E and bisect it. Intersection F of the bisector and the perpendicular CD,

extended, is the center of the second arc. Join the centers of the two arcs to locate tangent point T,

the inflection of the curve

Connecting Two Parallel Lines Connecting Two Nonparallel Lines

DRAWING AN OGEE CURVE

Draw two parallel lines Draw an ogee curve using method 1 or 2

THE CONIC SECTIONS

The conic sections are curves produced by planes intersecting a right circular cone.

Four types of curves are produced: the circle, ellipse, parabola, and hyperbola, according to the position of the planes.

Irregular CurvesThe curves are largely successive segments of geometric curves, such as the ellipse, parabola, hyperbola, and involute.

ELLIPSE TEMPLATES

These ellipse guides are usually designated by the ellipse angle, the angle at which a circle is viewed to appear as an ellipse.

DRAWING A FOCI ELLIPSE Major axis = long axis of ellipse Minor axis = short axis of ellipse The foci of the ellipse are two special points E

and F on the ellipse's major axis and are equidistant from the center point. The sum of the distances from any point P on the ellipse to those two foci is constant and equal to the major axis ( PE + PF2= 2A ). Each of these two points is called a focus of the ellipse.

DRAWING A FOCI ELLIPSE Let AB be the major axis and CD the minor axis To find foci E and F, draw arcs R with radius equal to half

the major axis and centers at the end of the minor axis Between E and O on the major axis, mark at random a

number of points. Using a random point (point 3), with E and F as centers

and radii A-3 and B-3, draw arcs to intersect at four points 3’. Use the remaining points to find four additional points on the ellipse in the same manner.

Sketch the ellipse lightly through the points

DRAWING AN APPROXIMATE ELLIPSE For many purposes, particularly where a small ellipse is

required, use the approximate circular arc method. Given axes AB & CD Draw line AC. With O as center and OA as radius, draw arc

AE. With C as center and CE as radius, draw arc EF. Draw perpendicular bisector GH of the line AF; the points K &

J, where they intersect the axes are centers of the required arc

Find centers M & L by setting off OL = OK and OM = OJ. Using centers K, L, M, & J, draw circular arcs as shown. The points of tangency T are at the junctures of the arcs n the lines joining the centers.

Drawing an Ellipse

Draw a major axis 5” long and a minor axis 2.5” long. Draw an ellipse by the foci method with at least five points in each quadrant

Use the same axes and use the approximate ellipse method

DRAWING A PARABOLAThe curve of intersection between a right circular cone and a plane parallel to one of its elements is a parabola.

DRAWING A PARABOLA Given focus F and directrix AB Draw a line DE parallel to the directrix and

at any distance CZ from it With center at F and radius CA, draw arcs to

intersect line DE at the points Q & R (points on the parabola)

Proceed in the same manner to determine as many points as needed

DRAWING A PARABOLA

Draw a parabola with a vertical axis and focus 0.5” from the directrix.

Find at least 9 points on the curve

Group Activity

Look at Exercise 4.60 and talk through methods that you would use to create this drawing in pencil

Present these methods to class

CHAPTER 5 – ORTHOGRAPHIC

PROJECTION

UNDERSTANDING PROJECTIONS

To make and interpret drawings you need to know how to createprojections and understand the standard arrangement of views.

You also need to be familiar with the geometry of solid objects and be able to visualize a 3D object that is represented in a 2D sketch or drawing.

UNDERSTANDING PROJECTIONS

Vanishing Points: An Introduction to Architectural Drawing

Views of Objects

The system of views is called multiview projection. Each view provides certain definite information. For example, a front view shows the true shape and size of surfaces that are parallel to the front of the object.

Multiview ProjectionThe system of views is called multiview projection. Each view provides certain definite information.

The Six Standard ViewsAny object can be viewed from six mutually perpendicular directions,

Revolving the Object to Produce Views

Revolving the Object to Produce Views. You can experience different views by revolving an object.

Principal DimensionsThe three principal dimensions of an object are width, height, and depth.

The front view shows only the height and width of the object and not the depth. In fact, any principal view of a 3D object shows only two of the threeprincipal dimensions; the third is foundin an adjacent view. Height is shown inthe rear, left-side, front, and right-sideviews. Width is shown in the rear, top,front, and bottom views. Depth isshown in the left-side, top, right-side,and bottom views.

Projection Method

Projection of an Object

The outline on the plane of projection shows how the object appears to the observer. In orthographic projection, rays (or projectors) from all points on the edges or contours of the object extend parallel to each other and perpendicular to the plane of projection. The word orthographic means “at right angles.”

Horizontal and Profile Projection Planes

Specific names are given to the planes of projection. The front view is projected to the frontal plane. The top view is projected to the horizontal plane. The side view is projected to the profile plane.

What’s Next?

• Chapter 5 – Orthographic Projection• Project Proposal due next week (February

22nd)

Questions?

On one of your sketches, answer the following two questions: What was the most useful thing that you

learned today? What do you still have questions about?

Homework

Chapter 4 Exercises (note they are in mm)

4.11, 4.13, 4.15, 4.18, 4.20, 4.22, 4.30, 4.33, 4.35 (major axis should be 100 mm not 10 mm), 4.40, 4.58