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 two weeks on February 22nd

Laura’s office hour will be cancelled next week on February 15th as she will be driving back from a 1 PM meeting in Milwaukee.

Please email any questions you may have about the homework or class!

Extra Credit #1 For ten extra points, write a question for our

upcoming first exam (first exam is in one month on March 7th)

Question can be in any of the following formats Question with a drawing/sketch for an answer Essay Question Fill in the blank question True/False Multiple Choice

Question can cover any topics we covered in class so far. Also can include tonight and next week.

Please include your answer Question & Answer are due in two weeks on

February 22nd for extra ten points

Keeping up with Technology

Engineering Scale Common Errors

The wrong scale is used for measurement Architect Scale: 1/8 & ¼ scales are

similar in the middle and wrong numbers can be read off

Plans are printed out on a different size paper than was designed on and wrong scale used (check scales)

Plans don’t fit to scale as they were printed out wrong (check scales)

STANDARD SHEETSThere are ANSI/ASME standards for international and U.S. sheet sizes. Note that drawing sheet size is given as height width. Most standard sheets use what is called a “landscape” orientation.

* May also be used as a vertical sheet size at 11" tall by 8.5" wide.

Typical Sheet Sizes and Borders• Margins and Borders• Zones

Engineering Scale Common Errors

Units are not converted correctly Not understanding the relationship

between the different sides of the scale Not understanding on engineer’s scale

that 1”=20’ (etc.) and on Architect’s scale that ¼” = 1’ (etc.)

Engineer scale is divided by tens, Architect scale is divided by twelve

Architect 16 Scale – Sixteen divisions per inch

Exercise 2.2

Engineer Scale Actual Length

1. Measure actual length with 10 scale2. 1:2 is half size. Use 20 scale and measure out

length from step 1. Draw3. 2:1 is double size. Double measurement from

step 1, measure on 10 scale. Draw Using scale lengths to double

1. Measure length using 20 scale2. 1:2 is half size. Use 40 scale to measure out

length from step 1. Draw3. 2:1 is double size. Use 10 scale to measure out

length from step 1. Draw.

Exercise 2.2

Architect Scale Actual Length

1. Measure actual length with 16 scale2. 1:2 is half size. Divide length from Step 1 in 2.

Use 16 scale and measure out length. Draw.3. 2:1 is double size. Double measurement from

step 1, measure on 16 scale. Draw. Using scale lengths to double

1. Measure length using 1/4 scale2. 1:2 is half size. Use 1/8 scale to measure out

length from step 1. Draw.3. 2:1 is double size. Use ½ scale to measure out

length from step 1. Draw.

Extra Credit #2

Do the same exercise as 2.2 over again, using the line lengths from 2.1.

Maximum of ten points will be rewarded Some tips

Label lines 1, 2, 3, etc. Explain which scale you used and label on lines

(engineer or architect, 20, 1/2 ) Include your measurements of the original lines

Highly recommended for students who were deducted points on this exercise (and bonus for those scale experts who were not)

What Line Weight Should be used for Title Blocks and Borders?

THICK! Use your 7 mm mechanical pencil.

Ames Lettering Guide

How to Use the Ames Lettering Guide

How well do we need to know solids?

Very well! It is a course competency to be able to “describe solids”

You need to be able to describe the following: Prisms Cylinders Pyramids Cones Spheres Torus Ellipsoids

How well do we need to know solids?

On a test, you may be asked to do one of the following: Draw one of the solids Give the definition of one of the solids (essay,

fill in the blank, true/false, or multiple choice (like questions 2 & 3 on your technical sketching worksheet homework).

Solids

Three-Dimensional Geometry Geometry Basics: 3D Geometry Pyramids and Prisms Cylinders and Cones

Pop Quiz: What Solid?

Pop Quiz: What Solid?

Right Circular Cylinder

This is the rotunda in Birmingham, UK

Pop Quiz: What Solid?

Pop Quiz: What Solid?

Sphere Water Tower

Pop Quiz: What Solid?

Pop Quiz: What Solid?

Right Pentagonal Prism

The Pentagon

Pop Quiz: What Solid?

Pop Quiz: What Solid?

Ellipsoid Caravan Interior

Light

Pop Quiz: What Solid?

What object has a double curved surface and is shaped like a donut? A. Ellipsoid B. Torus C. Sphere D. Cylinder

Pop Quiz: What Solid?

What object has two triangular bases and three additional faces? A. Pyramid B. Torus C. Cone D. Triangular Prism

Solids Group Activity

Make a list of solids that you saw today on your way to class, in your house, or at work.

Sketch few of these shapes Present

CHAPTER 4 – GEOMETRIC

CONSTRUCTION

GEOMETRY REVIEW

• Triangles

• Quadrilaterals

• Polygons

• Circles

• Arcs

UNDERSTANDING SOLID OBJECTS

Three-dimensional figures are referredto as solids. Solids are bounded bythe surfaces that contain them. Thesesurfaces can be one of the following fourtypes:

• Planar (flat)• Single curved (one curved surface)• Double curved (two curved surfaces)• Warped (uneven surface)Regardless of how complex a solid may be, it is composed of combinations of these basic types of surfaces.

What are Plane Figures?

A two-dimensional figure, also called a plane or planar figure, is a set of line segments or sides and curve segments or arcs, all lying in a single plane. The sides and arcs are called the edges of the figure. The edges are one-dimensional, but they lie in the plane, which is two-dimensional.

Triangles

A triangle is a plan figure bounded by three straight lines

The sum of the interior angles is always 180 degrees

A right triangle has one 90 degree angle

Quadrilaterals

A plane figure bounded by four straight sides

If opposite sides are parallel, the quadrilateral is also a parallelogram

A Trapezoid is a quadrilateral which has at least one pair of parallel sides

Parallelograms

A Parallelogram is a four-sided shape with two parallel sides.

Parallelograms have the following characteristics:• The opposite sides are equal

in length.• The opposite angles are

equal.• The diagonals bisect each

other.

Examples are a rectangle, rhombus, square.

Trapezium

A trapezium is defined by the properties it does not have. It has no parallel sides. Any quadrilateral drawn at random would probably be a trapezium.

Polygon

A plane shape (two-dimensional) with straight sides.

Examples: triangles, rectangles and pentagon

Note: a circle is not a polygon because it has a curved side

Be prepared to define a polygon up to an eight-sided figure (Figure 4.3, page 125)

Group Activity

Come up with a list of Polygons you see in nature, at work, driving around, at home . . . Etc.

Sketch a few of these shapes. Present shapes

Circles A circle is a closed curve all points of which

are the same distance from a point called the center.

Circumference refers to the distance around the circle (equal to pi (3.1416 multiplied by the diameter)

A diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle

Circles A radius of a circle is the line from the

center of a circle to a point on the circle. The quadrant of a circle is a quarter of a

circle (made by two radiuses at right angles and the connecting arc)

A chord of a circle is a line that links two points on a circle or curve.

Circles

Concentric circles are circles that have their centers at the same point

Eccentric circles are circles that do not have their centers at the same point

Arcs

An arc is a portion of the circumference of a circle

An arc could be a portion of some other curved shape, such as an ellipse, but it almost always refers to a circle

Tangent

Tangent is a line (or arc) which touches a circle or ellipse at just one point. Below, the blue line is a tangent to the circle c. Note the radius to the point of tangency is always perpendicular to the tangent line.

Geometric Formulas

See Appendix for useful geometric formulas (pages A-32 to A-37)

BISECTING A LINE OR CIRCULAR ARC

From A ad B draw equal arcs with radius greater than half AB

Join Intersection D and E with a straight line to locate center C

Compass system

BISECTING A LINE OR CIRCULAR ARC

Draw line AB 2.3 inches long Bisect this line using demonstrated

method

TrianglesInclined lines can be drawn at standard angles with the 45° triangle and the 30° x 60° triangle. The triangles are transparent so that you can see the lines of the drawing through them. A useful combination of triangles is the 30° x 60° triangle with a long side of 10" and a 45° triangle with each side 8" long.

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

Angles

A Song about Angles

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

TRANSFERRING AN ANGLE

1. Use any convenient radius R, and strike arcs from centers A and A’

2. Strike equal arcs r, and draw side A’C’

Angles

Draw any angle Label its vertex C Bisect the angle and transfer half the

angle to place its vertex at arbitrary point D

DRAWING A LINE PARALLEL TO A LINE AND AT A GIVEN DISTANCE AB is the line, CD is the given distance Use CD distance as the radius and draw

two arcs with center points E and F near the ends of the line AB

Line GH (tangent to the arcs) is the required line.

For CurvesT-square Method

DRAWING A LINE PARALLEL TO A LINE AND AT A GIVEN DISTANCE Draw a line EF Use distance FH equal to 1.2” Draw a new line parallel to EF and

distance GH away

For CurvesT-square Method

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 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

DRAWING A LINE THROUGH A POINT AND PERPENDICULAR TO A LINE

Draw a line Draw a point on the line Draw a point through the point and

perpendicular to the line Repeat process, but this time put the

point not on the line

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

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

Draw a triangle with sides 3”, 3.35”, and 2.56.”

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

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 Dividers Method

Divide the circumference of a circle into five equal parts with dividers

Join points with straight line

Dividers Method Geometric Method

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 Octogon

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

Tangents

DRAWING AN ARC TANGENT TO TWO LINES AT RIGHT ANGLES

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 ACUTE OROBTUSE ANGLES

DRAWING AN ARC TANGENT TO AN ARC AND A STRAIGHT LINE

DRAWING AN ARC TANGENT TO TWO ARCS

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

DRAWING AN OGEE CURVE

Connecting Two Parallel Lines Connecting Two Nonparallel Lines

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.

DRAWING A FOCI ELLIPSE

DRAWING A CONCENTRIC CIRCLE ELLIPSE

If a circle is viewed with the line of sight perpendicular to the planeof the circle…

…the circle will appear as a circle, in true size and shape

DRAWING A PARALLELOGRAM ELLIPSE

The intersection of like-numbered lines will be points on the ellipse. Locate points in the remaining three quadrants in a similar manner. Sketch the ellipse lightly through the points, then darken the final ellipse with the aid of an irregular curve.

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.

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

DRAWING AN APPROXIMATE ELLIPSE

For many purposes, particularly where a small ellipse is required, use the approximate circular arc 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 HELIXA helix is generated by a point moving around and along the surface of a cylinder or cone with a uniform angular velocity about the axis, and with a uniform linear velocity about the axis, and with a uniform velocity in the direction of the axis

DRAWING AN INVOLUTEAn involute is the path of a point on a string as the string unwinds from a line, polygon, or circle.

DRAWING A CYCLOID

A cycloid is generated by a point P on the circumference of a circle that rolls along a straight line

Cycloid

DRAWING AN EPICYCLOID OR A HYPOCYCLOID

Like cycloids, these curves are used to form the outlines of certain gear teeth and are thereforeof practical importance in machine design.

What’s Next?• Chapter 5 – Orthographic Projection• Project Proposal due in two weeks

(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?