Study Problems for Quiz 3
In the constructions below, also look for familiar orthogonal circle constructions
and pencils of circles. Be able to recognize and describe these connections.
Poincare model
Be able to carry out the constructions from
Lab 8, including the ones below.
- Given a P-line AB and a P-point C, construct the (convergent) parallels
to AB through C.
- Given a P-line AB and a P-point C, construct the perpendicular P-line to
AB through C.
- Given a P-line AB and a P-line CD that are ultraparallel, construct the
P-line perpendicular to both given P-lines.
- Given P-points A and B, construct the P-line that reflects A to (this is
the P-perpendicular bisector of AB).
- 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?
- Describe 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'/OB, is the triangle OA'B' similar to OAB? Explain your answer.
General Hyperbolic
- Define the defect of a triangle. How big can the defect be? How small?
- State and prove the additive property of defect.
- If a geometry contains a defective triangle, what does then there exist a defective right triangle?
- If a geometry contains a rectangle, why does it have an arbitrarily large rectangle?
- If a geometry contains both a defective triangle and a rectangle, explain how the addition of defect in this figure produces a contradiction.
