Linear Algebra

290 Chapter 3. Maps Between Spaces
Frame 1: Frame 2: Frame 3:
−.2 radians −.4 radians −.6 radians
We could also make the cube appear to be coming closer to us by producing
film frames of it gradually enlarging.
Frame 1: Frame 2: Frame 3:
110 percent 120 percent 130 percent
In practice, computer graphics incorporate many interesting techniques from
linear algebra (see Exercise 4).
So the analysis above of distance-preserving maps is useful as well as interesting.
For instance, it shows that to include in graphics software all possible
rigid motions of the plane, we need only include a few cases. It is not possible
that we’ve somehow ovelooked some rigid motions.
A beautiful book that explores more in this area is [Weyl]. More on groups,
of transformations and otherwise, can be found in any book on Modern Algebra,
for instance [Birkhoff & MacLane]. More on Klein and the Erlanger Program is
in [Yaglom].
Exercises
1 Decide � if √each of these √ is an orthonormal matrix.
1/ 2 −1/ 2
(a)
−1/ √ 2 −1/ √ �
2
� √ √
1/ 3 −1/ 3
(b)
−1/ √ 3 −1/ √ �
3
� √ √ √
1/ 3 − 2/ 3
(c)
− √ 2/ √ 3 −1/ √ �
3
2 Write down the formula for each of these distance-preserving maps.
(a) the map that rotates π/6 radians, and then translates by �e2
(b) the map that reflects about the line y =2x
(c) the map that reflects about y = −2x andtranslatesover1andup1
3 (a) The proof that a map that is distance-preserving and sends the zero vector
to itself incidentally shows that such a map is one-to-one and onto (the
point in the domain determined by d0, d1, and d2 corresponds to the point
in the codomain determined by those three numbers). Therefore any distancepreserving
map has an inverse. Show that the inverse is also distance-preserving.
(b) Using the definitions given in this Topic, prove that congruence is an equivalence
relation between plane figures.