Axiomatic Geometry - Mathematics 360 -
Spring 2009
For Homework 8
- There's something wrong with the definition of invariant on pages
40-41. Find it!
- For every pair of points z, w in the complex plane, define d(z,w) = |z -
w|. We proved in Problem 1 on page 46 that d is an invariant function for
Euclidean geometry. What is the "implicitly understood" set of figures on
which we're defining d? (That is, for this function d, what is
the "D" that's mentioned in the definition of "invariant function" on
p. 40-41?)
- For every pair of points z, w in the complex plane, define d(z,w) = |z +
w|. Answer each of the following questions. (Hint: first think pictorially,
then algebraically.)
- Is d an invariant function in Euclidean geometry? Prove your answer.
- Is d an invariant function in translational geometry? Prove your answer.
- Is d an invariant function in rotational geometry? Prove your answer.