Math 445. Assignment 8

Due Wed 2/25 at the beginning of lab.

This assignment is based on the Lenart Sphere handout that was also used in lab 7.

8.1 - Adventures 6 (10 points)

Answer 6.2

8.2 - Adventures 4 (15 points)

Give informal but clear and convincing explanations of WHY? You may find 6.2 helpful in some cases.

• (a) Adventure 4.1
• (b) Adventure 4.2
• (c) Adventure 4.3

Instructions for 8.3, 8.4, 8.5

Do each figure with the Lenart sphere (instructions for how and where to do this in class) and Sketchpad.

8.3 - Adventure 8.1 (10 points)

• Do the figure in both Sketchpad and on the sphere.
• State clearly the relation.
• Extra credit if you can explain why.

8.4 - Adventure 8.2 (10 points)

• Do the figure in both Sketchpad and on the sphere.
• State clearly the relation.
• Extra credit if you can explain why.

8.5 - Concurrence theorems for triangles (15 points)

• Pick out two of the concurrent theorems for triangles and investigate their analogs on the sphere by experiment with the Lenart sphere and Sketchpad.
• Extra credit for good explanations of why your observations are correct.

Reading:

Read the Lenart Sphere handout. It does not have many answers, but you can use it to structure your questions. There is a discussion of stereographic projection in Sved, chapter 2, pp. 29-32 (this has most of the diagram that explains how to construct the stereographic image of the antipodal point). Sved has a brief discussion of the properties of spherical geometry on pp. 73-74. Also, note that the formula for the excess of a spherical triangle in the problems in Sved, pp.78-79, #9. Finally, notice that the argument that the angular defect is additive (Sved problem #5, p. 77) does not depend on which geometry you are in.