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symphony
instrument
cd
movie
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R
Angle
Parallel
Triangle
Perpendicular
Radius
Square
Diameter
Centerline
CL
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2.
With a compass
With a triangle
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 Given line AB
 With points A & B as
centers and any radius
greater than ½ of AB,
draw arcs to intersect,
creating points C & D
 Draw line EF through
points C and D
 Given line AB
H
F
D
 Draw line CD from
endpoint A
 Draw line EF from
endpoint B
E
B
C
A
 Draw line GH through intersection
G
 Given arc AB
 With points A & B as centers
and any radius greater than ½
of AB, draw arcs to intersect,
creating points C & D
 Draw line EF through
points C and D
 Given angle AOB
 With point O as the center
and any convenient radius R,
draw an arc to intersect AO
and OB to located points C
and D
 With C and D as centers
and any radius R2 greater
than ½ the radius of arc
CD, draw two arcs to
intersect, locating point E
 Draw a line through points O
and E to bisect angle AOB
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2.
True
False
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A
 Given lines AB and CD
 Construct parallel lines
at distance R
B
O
 Construct the
perpendiculars to locate
points of tangency
 With O as the point,
construct the tangent arc
using distance R
C
D
 Given lines AB and CD
 Construct parallel lines
at distance R
A
 Construct the
perpendiculars to locate
O
points of tangency
 With O as the point,
construct the tangent arc
using distance R
B
C
D
 Given angle ABC
 With B as the point,
strike arc R1 equal
to given radius
 With D and E as the
points, strike arcs R2
equal to given radius
 With O as the point,
strike arc R equal to
given radius
A
O
D
B
E
C
 Given line AB and arc CD
 Strike arcs R1 (given radius)
 Draw construction arc parallel to
given arc, with center O
 Draw construction line parallel to
given line AB
 From intersection E, draw EO to
get tangent point T1, and drop
perpendicular to given line to get
point of tangency T2
 Draw tangent arc R from
T1 to T2 with center E
O
C
E
T1
R1
A
B
D
T2
 Given arc AB with
center O and arc CD
with center S
 Strike arcs R1 = radius R
A
 Draw construction arcs
O
parallel to given arcs,
using centers O and S
 Join E to O and E to S to get
tangent points T
 Draw tangent arc R from T to T,
with center E
E
T
BC
S
T
D
 Prism
◦ Right Rectangular
◦ Right Triangular
 Cylinder
 Cone
 Sphere

Pyramid

Torus
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2.
3.
4.
Torus
Sphere
Cylinder
Pyramid
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An alternative postion for the side view is
rotated and aligned with the top view.
Third angle projection is used
in the U.S., and Canada
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2.
3.
First Angle
projection
Second Angle
projection
Third Angle
Projection
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


The six standard views are often thought
of as produced from an unfolded glass
box.
Distances can be transferred or projected
from one view to another.
Only the views necessary to fully describe
the object should be drawn.
Review
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2.
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4.
RIFG
UZDP
IFBH
EBHB
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2.
3.
4.
peach
fig
apricot
prune
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4
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2.
3.
4.
Carthage
Rome
Jerusalem
Babylon
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The advantage of oblique pictorials like these over isometric pictorials is
that circular shapes parallel to the view are shown true shape, making
them easy to sketch.
Oblique pictorials are not as realistic as isometric views because the
depth can appear very distorted.
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2.
3.
30/30/120
60/60/40
90/60/30
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57%
7%
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Oblique drawings of objects having a lot of depth can appear very
unnatural due to the lack of foreshortening.



Perspective drawings produce the view
that is most realistic. A perspective
drawing shows a view like a picture taken
with a camera
There are three main types of perspective
drawings depending on how many
vanishing points are used.
These are called one-point, two-point,
and three-point perspectives.
Orient the object so that a principal face is parallel to the viewing plane
(or in the picture plane.) The other principal face is perpendicular to the
viewing plane and its lines converge to a single vanishing point.
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2.
3.
Where all the
lines converge
together.
Where the earth
ends.
Where the view
point comes
together.
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A review of some ideas, That are both
relevant to calculus and drafting.

The physical tools for drawing the figures are:
◦ The unmarked ruler (i.e., a ‘straightedge’)
◦ The compass (used for drawing of circles)


Given any two distinct points, we can use our
straightedge to draw a unique straight line
that passes through both of the points
Given any fixed point in the plane, and any
fixed distance, we can use our compass to
draw a unique circle having the point as its
center and the distance as its radius

Given any two points P and Q, we can draw
a line through the midpoint M that makes a
right-angle with segment PQ
P
M
Q

Given a circle, and any point on it, we can
draw a straight line through the point that
will be tangent to this circle

Step 1: Draw the line through C and T
C
T

Step 2: Draw a circle about T that passes
through C, and let D denote the other end of
that circle’s diameter
C
T
D

Step 3: Construct the straight line which is
the perpendicular bisector of segment CD
tangent-line
C
T
D

Any other point S on the dotted line will be
too far from C to lie on the shaded circle
(because CS is the hypotenuse of ΔCTS)
S
C
T
D

Given an ellipse, and any point on it, we can
draw a straight line through the point that
will be tangent to this ellipse
F1
F2

Step 1: Draw a line through the point T and
through one of the two foci, say F1
T
F1
F2

Step 2: Draw a circle about T that passes
through F2, and let D denote the other end of
that circle’s diameter
T
F1
F2
D

Step 3: Locate the midpoint M of the linesegment joining F2 and D
T
M
F1
F2
D

Step 4: Construct the line through M and T (it
will be the ellipse’s tangent-line at T, even if
it doesn’t look like it in this picture)
T
D
M
F1
F2
tangent-line

Observe that line MT is the perpendicular
bisector of segment DF2 (because ΔTDF2 will
be an isosceles triangle)
T
D
M
F1
F2
tangent-line

So every other point S that lies on the line
through points M and T will not obey the
ellipse requirement for sum-of-distances
S
T
D
M
F1
F2
tangent-line
SF1 + SF2 > TF1 + TF2 (because SF2 = SD and TF2 = TD

When we encounter some other methods that
purport to produce tangent-lines to these
curves, we will now have a reliable way to
check that they really do work!
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2.
Yes
No
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


A cone is generated by a straight line moving
in contact with a curved line and passing
through a fixed point, the vertex of the cone.
This line is called the generatrix.
Each position of the generatrix is called
element
The axis is the center line from the center of
the base to the vertex



Conic sections are curves produced by planes
intersecting a right circular cone. 4-types of
curves are produced: circle, ellipse, parabola,
and hyperbola.
A circle is generated by a plane perpendicular
to the axis of the cone.
A parabola is generated by a plane parallel to
the elements of the cone.


An ellipse is generated by planes between
those perpendicular to the axis of the cone
and those parallel to the element of the cone.
A hyperbola is generated by a planes between
those parallel to the element of the cone and
those parallel to the axis of the cone.
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2.
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4.
Circle
Ellipse
Parabola
Hyperbola
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Circle
Ellipse
Parabola
Hyperbola
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 If
a circle is viewed at an angle, it
will appear as an ellipse. This is
the basis for the concentric circles
method for drawing an ellipse.
 Draw two circles with the major
and minor axes as diameters.
 Draw
any diagonal XX to the large
circle through the center O, and
find its intersections HH with the
small circle.
 From
the point X, draw line XZ
parallel to the minor axis, and
from the point H, draw the line
HE, parallel to the major axis.
Point E is a point on the ellipse.
 Repeat for another diagonal line
XX to obtain a smooth and
symmetrical ellipse.

Along the straight edge
of a strip of paper or
cardboard, locate the
points O, C, and A so
that the distance OA is
equal to one-half the
length of the major axis,
and the distance OC is
equal to one-half the
length of the minor axis.

Place the marked
edge across the axes
so that point A is on
the minor axis and
point C is on the
major axis. Point O
will fall on the
circumference of the
ellipse.

Move the strip,
keeping A on the
minor axis and C
on the major axis,
and mark at least
five other
positions of O on
the ellipse in each
quadrant.

Using a French
curve, complete
the ellipse by
drawing a
smooth curve
through the
points.
1.
2.
3.
Splitting a line
in 3rd’s
Splitting a line
in 4ths
Splitting a line
in Half
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2.
Trammel
Concentric
Circles
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46%
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2.
Yes
No
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