Class 5 presentation

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 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

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

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

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

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.

 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

 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

 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

 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

 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

 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

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

 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

 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 Octagon

 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

 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

 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

 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

 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

 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

 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.

 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

 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

 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

 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

 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

 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

 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

 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

 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

 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

 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

 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 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


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

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.

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

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

 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

 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.

 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

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

 Draw a parabola with a vertical axis and focus
0.5” from the directrix.
 Find at least 9 points on the curve

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

To make and interpret drawings you need to know how to create
projections 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.

 Vanishing Points: An Introduction to
Architectural Drawing

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.

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

Any object can be viewed from six mutually perpendicular
directions,

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

The 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 three
principal dimensions; the third is found
in an adjacent view. Height is shown in
the rear, left-side, front, and right-side
views. Width is shown in the rear, top,
front, and bottom views. Depth is
shown in the left-side, top, right-side,
and bottom views.

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.”

Projection of an Object

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.

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

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

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