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thespian : theater :: musician : . symphony instrument cd movie. Things to Review…. C. L. Geometric Symbols. Angle Triangle Radius Diameter. Parallel Perpendicular Square Centerline. R. What type of Bisect does this picture show?. With a compass With a triangle. - PowerPoint PPT Presentation
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thespian : theater :: musician :
1 2 3 4
25% 25%25%25%1. symphony2. instrument3. cd4. movie
Things to Review….
Angle
Triangle
Radius
Diameter
Parallel
Perpendicular
Square
Centerline
Geometric Symbols
R
CL
What type of Bisect does this picture show?
1 2
50%50%1. With a compass2. With a triangle
Bisect a Line w/ a Compass 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
Bisect a Line w/ a Triangle
A B
Given line AB
Draw line CD from endpoint A
E
F
Draw line EF from endpoint B
C
D
G
H
Draw line GH through intersection
Bisect an Arc 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
Bisect an Angle 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
Given angle AOB
Draw a line through points O and E to bisect angle AOB
Circumscribed is out side of circle
1 2
50%50%1. True2. False
Construct an Arc Tangent to Two Lines at an Acute Angle
A
B
C
D
Given lines AB and CD Construct parallel
lines at distance R Construct the
perpendiculars to locate points of tangency
With O as the point, construct the tangent arc using distance R
R
R
O
Construct an Arc Tangent to Two Lines at an Obtuse Angle
C
D
Given lines AB and CD Construct parallel
lines at distance R Construct the
perpendiculars to locate points of tangency
With O as the point, construct the tangent arc using distance R
R
A
B
R
O
Construct an Arc Tangent to Two Lines at Right Angles Given angle ABC
With D and E as the points, strike arcs R2 equal to given radius
A
B C
R 1
R2
R 2
With B as the point, strike arc R1 equal to given radius
O
E
D
With O as the point, strike arc R equal to given radius
Construct an Arc Tangent to a Line and an Arc
Given line AB and arc CD
A B
C
D
Strike arcs R1 (given radius)
R1
R 1
Draw construction arc parallel to given arc, with center O
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
ET1
T2
Draw tangent arc R from T1 to T2 with center E
Construct an Arc Tangent to Two Arcs Given arc AB with
center O and arc CD with center S
S D
C
O
B
A Strike arcs R1 = radius R
R1
R1
Draw construction arcs parallel to given arcs, using centers O and S
Join E to O and E to S to get tangent points T
E
T
T
Draw tangent arc R from T to T, with center E
R
Solids
Prism
◦Right Rectangular
◦Right Triangular
Solids
Cylinder
Cone
Sphere
Solids
Pyramid
Torus
Which solid is shown here as an orthographic?
1 2 3 4
25% 25%25%25%1. Torus2. Sphere3. Cylinder4. Pyramid
Position of Side Views
An alternative postion for the side view isrotated and aligned with the top view.
First Angle Projection
Symbols for 1st & 3rd Angle Projection
Third angle projection is usedin the U.S., and Canada
In class we use….
1 2 3
33% 33%33%1. First Angle
projection2. Second Angle
projection3. Third Angle
Projection
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.
Summary
Day TwoReview
D_A_T_N_
1 2 3 4
73%
18%
9%
0%
1. R I F G2. U Z D P3. I F B H4. E B H B
grape : raisin :: plum :
1 2 3 4
0%
64%
18%18%
1. peach2. fig3. apricot4. prune
Alexander : Macedonia :: Hannibal :
1 2 3 4
38%
31%
8%
23%
1. Carthage2. Rome3. Jerusalem4. Babylon
Oblique Pictorials
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.
Isometric Drawing is done at what angles?
57%7%
36%
1 2 3
1. 30/30/1202. 60/60/403. 90/60/30
Unnatural Appearance ofOblique Drawing
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.
Perspective Drawings
One Point Perspective
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.
What is the vanishing point?
1 2 3
40%
60%
0%
1. Where all the lines converge together.
2. Where the earth ends.
3. Where the view point comes together.
Tangents to CurvesA 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)
Straightedge and Compass
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
Lines and Circles
Given any two points P and Q, we can draw a line through the midpoint M that makes a right-angle with segment PQ
The ‘perpendicular bisector’
P QM
Given a circle, and any point on it, we can draw a straight line through the point that will be tangent to this circle
Tangent-line to a Circle
Step 1: Draw the line through C and T
How do we do it?
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
How? (continued)
C
T
D
Step 3: Construct the straight line which is the perpendicular bisector of segment CD
How? (continued)
C
T
D
tangent-line
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)
Proof that it’s a tangent
C
T
D
S
What is a Tangent in your own words? (no more than 160 characters)
Given an ellipse, and any point on it, we can draw a straight line through the point that will be tangent to this ellipse
Tangent to an ellipse
F1 F2
Step 1: Draw a line through the point T and through one of the two foci, say F1
How do we do it?
F1 F2
T
Step 2: Draw a circle about T that passes through F2, and let D denote the other end of that circle’s diameter
How? (continued)
F1 F2
T D
Step 3: Locate the midpoint M of the line-segment joining F2 and D
How? (continued)
F1 F2
T DM
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)
How? (continued)
F1 F2
T DM
tangent-line
Observe that line MT is the perpendicular bisector of segment DF2 (because ΔTDF2 will be an isosceles triangle)
Proof that it’s a tangent
F1 F2
T DM
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
Proof (continued)
F1 F2
T DM
tangent-line
S
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!
Why are these ideas relevant?
Do you understand what has been covered so far today?
1 2
0%
100%1. Yes2. No
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
Conic Sections
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.
Conic Sections
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.
Conic Sections
Conic Sections
In the picture in front of you (B) is a….
1. Circle2. Ellipse3. Parabola4. Hyperbola
1 2 3 4
86%
0%7%7%
In the picture in front of you (E) is a …
1. Circle2. Ellipse3. Parabola4. Hyperbola
1 2 3 4
0%
92%
0%8%
Quadrants of a Circle
How many Quadrants are there on a circle?
1 2 3 4
0%
29%
71%
0%
1. 22. 33. 44. 5
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.
Drawing an ellipse by the concentric circles method.
Draw any diagonal XX to the large circle through the center O, and find its intersections HH with the small circle.
Drawing an ellipse by the concentric circles method.
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.
Drawing an ellipse by the concentric circles method.
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.
Drawing an ellipse by the trammel method.
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.
Drawing an ellipse by the trammel method.
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.
Drawing an ellipse by the trammel method.
Using a French curve, complete the ellipse by drawing a smooth curve through the points.
Drawing an ellipse by the trammel method.
Drawing an ellipse by the trammel method.
Bisecting a line is?
1 2 3
0%
100%
0%
1. Splitting a line in 3rd’s
2. Splitting a line in 4ths
3. Splitting a line in Half
Which method of creating an ellipse uses a French curve?
1 2
46%
54%1. Trammel2. Concentric
Circles
Do you understand what was covered?
1 2
31%
69%1. Yes2. No
Tell me one thing you now get….
Tell me one thing you still don’t understand….
Tell me one thing you need more explanation….