thespian : theater :: musician :

1. thespian : theater :: musician : • symphony • instrument • cd • movie

2. Things to Review….

3. C L Geometric Symbols Angle Triangle Radius Diameter Parallel Perpendicular Square Centerline R

4. What type of Bisect does this picture show? • With a compass • With a triangle

5. 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

6. Bisect a Line w/ a Triangle H D F C E A B G • Given line AB • Draw line CD from endpoint A • Draw line EF from endpoint B • Draw line GH through intersection

7. 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

8. Bisect an Angle • 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

9. Circumscribed is out side of circle • True • False

10. A R B R C D Construct an Arc Tangent to Two Lines at an Acute Angle • Given lines AB and CD • Construct parallel lines at distance R O • Construct the perpendiculars to locate points of tangency • With O as the point, construct the tangent arc using distance R

11. Construct an Arc Tangent to Two Lines at an Obtuse Angle A R C B R D • Given lines AB and CD • Construct parallel lines at distance R • Construct the perpendiculars to locate points of tangency O • With O as the point, construct the tangent arc using distance R

12. Construct an Arc Tangent to Two Lines at Right Angles A D R2 R2 R1 B C E • 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 O • With O as the point, strike arc R equal to given radius

13. Construct an Arc Tangent to a Line and an Arc C R1 R1 A B O D • 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 E T1 • Draw tangent arc R from T1 to T2with center E T2

14. Construct an Arc Tangent to Two Arcs A R1 R O B C R1 D S • Given arc AB with center O and arc CD with center S • Strike arcs R1 = radius R • Draw construction arcs parallel to given arcs, using centers O and S E T • Join E to O and E to S to get tangent points T T • Draw tangent arc R from T to T, with center E

15. Prism Right Rectangular Right Triangular Solids

16. Solids Cylinder Cone Sphere

17. Solids Pyramid Torus

18. Which solid is shown here as an orthographic? • Torus • Sphere • Cylinder • Pyramid

19. Position of Side Views An alternative postion for the side view is rotated and aligned with the top view.

20. First Angle Projection

21. Symbols for 1st & 3rd Angle Projection Third angle projection is used in the U.S., and Canada

22. In class we use…. • First Angle projection • Second Angle projection • Third Angle Projection

23. Summary • 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.

24. Day Two Review

25. D_A_T_N_ • R I F G • U Z D P • I F B H • E B H B

26. grape : raisin :: plum : • peach • fig • apricot • prune

27. Alexander : Macedonia :: Hannibal : • Carthage • Rome • Jerusalem • Babylon

28. 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.

29. Isometric Drawing is done at what angles? • 30/30/120 • 60/60/40 • 90/60/30 14 of 15

30. Unnatural Appearance ofOblique Drawing Oblique drawings of objects having a lot of depth can appear very unnatural due to the lack of foreshortening.

31. Perspective Drawings • 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.

32. 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.

33. What is the vanishing point? • Where all the lines converge together. • Where the earth ends. • Where the view point comes together.

34. Tangents to Curves A review of some ideas, That are both relevant to calculus and drafting.

35. Straightedge and Compass • The physical tools for drawing the figures are: • The unmarked ruler (i.e., a ‘straightedge’) • The compass (used for drawing of circles)

36. Lines and 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

37. The ‘perpendicular bisector’ • 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 Q M

38. Tangent-line to a Circle • Given a circle, and any point on it, we can draw a straight line through the point that will be tangent to this circle

39. How do we do it? • Step 1: Draw the line through C and T C T

40. How? (continued) • 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

41. How? (continued) • Step 3: Construct the straight line which is the perpendicular bisector of segment CD tangent-line C T D

42. Proof that it’s a tangent • 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

43. What is a Tangent in your own words? (no more than 160 characters) 15 14

44. Tangent to an ellipse • 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

45. How do we do it? • Step 1: Draw a line through the point T and through one of the two foci, say F1 T F1 F2

46. How? (continued) • Step 2: Draw a circle about T that passes through F2, and let D denote the other end of that circle’s diameter T D F1 F2

47. How? (continued) • Step 3: Locate the midpoint M of the line-segment joining F2 and D T D M F1 F2

48. How? (continued) • 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

49. Proof that it’s a tangent • 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

50. Proof (continued) • 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 )