Given any two points A and B in the Euclidean Plane, is there always
one and only one line such that B is the reflection of A in that line?
Given any two points A and B in the Hyperbolic Plane, is there always
one and only one line such that B is the reflection of A in that line?
(Try this on the Non-Euclid Applet, if you're not sure of the answer!)
There are at least three statements in the proof of Proposition 3.1.13
for which the book gives no justification. Can you find them? Can you
justify any of them?
[Optional] I wish we had time to discuss this more fully, but we don't. So
do this problem for your own interest only.
In Definition 3.1.12, "initial" and "terminal" are not
defined. So perhaps a better way to define a directed angle would be as an
ordered pair of rays: (ray AB, ray AC). (Ordered pairs are defined precisely
in Set Theory). But, what about "clockwise" and
"counterclockwise"? The book gives no definition for these either.
Any suggestions? (The answer is tricky, and would probably surprise you!)
The book doesn't really define what a rotation is, either. It just
says you "revolve the plane" (and it doesn't even acknowledge that
it's only giving an informal definition). But with a bit of care we can give
a precise definition. Can you do this (assuming that Definition 3.1.12 has
already been fixed)?