Class 4 presentation posted

Instructor: Laura Gerold, PE
Catalog #10614113
Class # 22784, 24113, 24136, & 24138
Class Start: January 18, 2012 Class End: May
16, 2012

 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!

 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

 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)

There 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.

• Margins and Borders
• Zones

 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

 Engineer Scale
 Actual Length
1. Measure actual length with 10 scale
2. 1:2 is half size. Use 20 scale and measure out length
from step 1. Draw
3. 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 scale
2. 1:2 is half size. Use 40 scale to measure out length
from step 1. Draw
3. 2:1 is double size. Use 10 scale to measure out length
from step 1. Draw.

 Architect Scale
 Actual Length
1. Measure actual length with 16 scale
2. 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 scale
2. 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.

 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)

THICK! Use your 7 mm mechanical pencil.

 How to Use the Ames Lettering Guide

 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

 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).

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

 Right Circular
Cylinder
 This is the rotunda in
Birmingham, UK

 Sphere
 Water Tower

 Right Pentagonal
Prism
 The Pentagon

 Ellipsoid
 Caravan Interior Light

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

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

 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

• Triangles
• Quadrilaterals
• Polygons
• Circles
• Arcs

Three-dimensional figures are referred
to as solids. Solids are bounded by
the surfaces that contain them. These
surfaces can be one of the following four
types:

• 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.

 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.

 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

 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

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.

 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.

 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)

 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

 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

 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.

 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

 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 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.

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

 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

 Draw line AB 2.3 inches long
 Bisect this line using demonstrated method

Inclined 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.

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 accuracy
is required

 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

 1. Use any convenient radius R, and strike arcs
from centers A and A’
 2. Strike equal arcs r, and draw side A’C’

 Draw any angle
 Label its vertex C
 Bisect the angle and transfer half the angle to
place its vertex at arbitrary point D

 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.

T-square Method For Curves

 Draw a line EF
 Use distance FH equal to 1.2”
 Draw a new line parallel to EF and distance
GH away

T-square Method For Curves

 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

 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

 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

 Drawing a Triangle with Sides Given
1. Draw one side, C
2. Draw an arc with radius equal to A
3. Lightly draw an arc with radius equal to B
4. Draw sides A and B from the intersection of
the arcs

 Drawing a right triangle with hypotenuse and
one side given
1. Given sides S and R
2. With AB as diameter equal to S, draw a semicircle
3. With A as center, R as radius, draw an arc
intersecting the semicircle C.
4. Draw AC and CB

 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

 Math Made Easy: Measuring Angles (part 1)
 Math Made Easy: Measure Angles (part 2)
 Many angles can be laid out directly with the
protractor.

 Tangent Method
1. Tangent = Opposite / Adjacent
2. Tangent of angle is y/x
3. Y = x tan
4. Assume value for x, easy such as 10
5. Look up tangent of and multiply by x (10)
6. Measure y = 10 tan

 Sine Method
1. Sine = opposite / hypotenuse
2. Sine of angle is y/z
3. Draw line x to easy length, 10
4. Find sine of angle , multiply by 10
5. Draw arc R = 10 sin

 Chord Method
1. Chord = Line with both endpoints on a circle
2. Draw line x to easy length, 10
3. Draw an arc with convenient radius R
4. C = 2 sin ( /2)
5. Draw length C

 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

 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

 Draw a 2” line, AB
 Construct an equilateral triangle

1. One side AB, given
2. Draw a line perpendicular through point A
3. 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

Diameters Method
1. Given Circle

2. Draw diameters at right angles to each other

3. Intersections of diameters with circle are
vertices of square
4. Draw lines

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

 Dividers Method
 Divide the circumference of a circle into five equal
parts with dividers
 Join points with straight line

Dividers Method Geometric Method

 Geometric Method
1. Bisect radius OD at C
2. Use C as the center and CA as the radius to lightly draw
arc AE
3. With A as center and AE as radius draw arc EB
4. 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

 Lightly draw a 5” diameter circle
 Find the vertices of an inscribed regular
pentagon
 Join vertices to form a five-pointed star

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

Use a compass Centerline Variation

Steps

 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

 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

 Inscribe a hexagon

 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

 Inscribe an Octogon

 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

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

 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

 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.

 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

 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

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

Connecting Two Parallel Lines Connecting Two Nonparallel Lines

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.

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

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

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.

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

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

For many purposes, particularly where a small ellipse is required, use the
approximate circular arc method.

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

A 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

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

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

Cycloid

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

• Chapter 5 – Orthographic Projection
• Project Proposal due in two weeks (February
22nd)

 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?

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