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Reading for Weeks 5-6
Read Sved Chapters 4 thoroughly and enough of 5 to understand the P disk model.
Lab Assignment Due Monday, 2/7/00
This assignment consists of the Portfolio Figures assigned in Lab #5. They can be checked off in the lab or turned in.
Math 445. Assignment 6. Due Mon 2/7/00
Problems Part A:
Preparation and review for midterm. You will be asked some substantial question about Sved, Chapter 4. If you know how to do the problems at the end of the chapter, you will be well prepared. Thus you should write up and be prepared to present these problems and know the others.
- 6.1 (10 points) – (Hyperbolic parallel axiom implies triangle defect > 0) Sved, Chapter 4, page 77, #3
- 6.2 (10 points) - (Angle defect greater than 0 implies multiple parallels) Sved, Chapter 4, page 77, #5
- 6.3 (10 points) - (Saccheri quadrilaterals) Sved, Chapter 4, page 78, #6
Definition. If A and B are points in the Euclidean plane. The circles d which are orthogonal to all circles c passing through both A and B are called the Apollonian circles of A and B. (In other words the hyperbolic pencil defined by A and B is made up of the Apollonian circles.
- 6.4 (10 points) - (DWEG circles are Apollonian) In Lab 4, recall that a DWEG circle with center A through B was obtained by inverting B through the circles representing DWEG lines to get points B' and B''. Then the DWEG circle was constructed as the circle through B, B' and B''. Prove that this circle is actually the Apollonian circle of A and O through B.
- 6.5. (10 points) - (Points symmetric by two mirrors) Given two disjoint circles c and d, prove there is exactly one pair of points A and B with the following property: the inversion of A is B, both with mirror c and with mirror d. Hint: If the circles are concentric, this is much easier. Hint 2. Note the connection with the elliptic and hyperbolic pencils of circles defined by A and B.
- 6.6 (10 points) - (P-lines of symmetry)
(a) Given two P-points A and B. Prove that there is exactly one P-line m which P-reflects A to B. (P-reflection is just inversion if the P-line is an arc and reflection if the P-line is a segment.)
(b) Carry out a construction of m in a (random) example of points A and B.
Comment: This line of symmetry of AB is the P-perpendicular bisector of AB, but here it is defined by a symmetry and not by distance. A useful consequence of this is that any point A in the P-disk model can be reflected to the center point; this can simplify a figure.
Study Problems
If m is a P-line and A is a P-point, why is the inversion B of A in the support circle of m also a P-point? (In other words, why is B inside the P-disk and not outside?
- Explain what happens to the Thales figure in non-Euclidean geometry: given lines OA and OB, and points A' on OA and B' on OB with OA'/OA = OB'/PB, is the triangle OA'B' similar to OAB? Explain your answer.
- Write the correct formula for the inversion of point P(x,y) in the circle with center (0,0) and radius r.
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