Transformation Geometry -- Math 331
March 3, 2004
Order 2 transformations
Definition. An affine transformation f of R^{n} has
order k if k > 0, the identity results from composing
it with itself k times, and k is the smallest positive integer
with that property.
When f has order k, clearly f is inverted by f^{k-1}.
In particular, an affine transformation of order 2 is a transformation
not the identity that is its own inverse.
-
Proposition 1.
An affine transformation f(x) = A x + b is of order
2 if and only if A^{2} = 1 and Ab = -b.
-
Proposition 2.
If f(x) = A x + b is of order 2, then A cannot be
the identity matrix.
-
Proposition 3.
If f(x) = A x + b is of order 2, then
det(A) = {+/-} 1.
-
Proposition 4.
When n = 2 if f is orientation-preserving and of order 2,
then f must be the half turn about some point.
-
Proposition 5.
When n = 2 if f(x) = A x + b is orientation-reversing and of
order 2, then
- a. The characteristic polynomial of
A must be t^{2} - 1.
- b. f must have a line of fixed points.
Comment on past exercises
- February 27, No. 1 Represent the two rotations as compositions
of reflections in lines through their centers where the line between the
two centers is used twice to obtain the required rotation as the composition
of reflection in the line through (1, 0) with elevation 5pi/8 followed
by reflection in the line through (0, 1) with elevation -pi/6.
The center of the required rotation is the point where these two lines meet,
which is
| ( | {3 - SQRT{3} |
| }/{4 - SQRT{2} |
| }, { SQRT{3} |
| + SQRT{2} |
| }/{4 - SQRT{2} |
| } | ) |
| .
|
- March 1, No. 1 The line segment from X to f(X) must
be perpendicular to the given line containing A and B, and the
midpoint of that segment must lie on the given line. Therefore
{X + f(X)}/{2} = s A + t B with s = 1 - t. Of course, t
depends on X, and may be determined by using the condition of
perpendicularity in the form
.
= 0.
One finds
Exercises due Friday, March 5
Let f be the affine transformation defined by
Show that f is orientation-reversing and of order 2.
Show that for any point x in R^{2} the vector from
x to f(x) is parallel to the vector (2, 1).
Find the line of fixed points of f.
Write proofs for as many as you can of the propostions above.
AUTHOR
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COMMENT