hyperbolic geometry

Question 1

Exercise: Circle inversion

Question 2

What is Euclid’s Fifth Postulate?

Answer 2

For every line (L) and point (P) that does not lie on L, there exists a unique line:
- through P
- and parallel to L

Question 3

In relation to the Poincaré Model of a Hyperbolic Geometry, lines of the geometry are of two types, what are those types?

Answer 3

1. diameters
2. parts of circles that are perpendicular to H (horizon circle)

Question 4

What is the rule for the Poincaré Model of a Hyperbolic Geometry?

Answer 4

Given the horizon circle (H), a circle with center O:
- points of the geometry are ALL THE POINTS THAT ARE STRICTLY INSIDE OF H
— does not include the points on the circle itself

Question 5

Exercise: Poincaré Model of Hyperbolic Geometry

Question 6

What are diameters?

Answer 6

All hyperbolic lines that pass through O
** when O = the center of a given circle

Question 7

In relation to Hyperbolic lines through A (a point other than the center), what is the center line for A?

Answer 7

The perp. bisector of the line Ainv(A)
- sometimes called LA

Question 8

Exercise: Hyperbolic Lines through A

Question 9

Exercise: Hyperbolic line passing through A and B

Question 10

Exercise: angle of 90 degrees; construct a hyperbolic line through A that is perpendicular to a given hyperbolic line L

Question 11

Exercise: given an angle; given an angle and a hyperbolic line passing through a point A, find a hyperbolic line that meets L at A with the angle

Question 12

Exercise: an equilateral triangle

Question 13

In a hyperbolic plane, what is the sum of the angles of a triangle?

Answer 13

LESS THAN 180 DEGREES