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….