Transformation Geometry — Math 331

February 20, 2004

Discussion

• Theorem. Every orientation-preserving isometry of R^{2} with a fixed point is a rotation.

Proof. Let f be a given orientation-preserving isometry of R^{2} with fixed point c. Let τ be “translation by c”, i.e., τ(x) = x + c. Then the isometry g = τ^{−1} ∘ f ∘ τ has the property g(0) = 0. Since g is an affine map that fixes the origin, g must be a linear transformation of R^{2} that is distance-preserving. Therefore, g(x) = Ux for some 2 × 2 orthogonal matrix U. By an exercise in the previous assignment U must be one of the matrices formed using cos θ and sin θ for some value of θ, and since g is orientation-preserving, detU > 0 with the result that U must be the specific matrix

( \begin{matrix}cos θ & − sin θ \\ sin θ & cos θ \end{matrix} ) .

Therefore, g is the rotation about the origin through the angle θ, and f is the rotation about the point c through the angle θ.

• Theorem. Every orientation-reversing isometry of R^{2} with a given fixed point is the reflection in some line containing the fixed point.

Proof. The argument is very similar to the preceding argument except that the 2 × 2 orthogonal matrix U satisfies det(U) < 0 since the isometry is orientation-reversing, and, therefore,

U = ( \begin{matrix}cos θ & sin θ \\ sin θ & − cos θ \end{matrix} ) ,

which is the matrix of reflection in the line through the origin forming the angle θ/2 with the positive first coordinate axis.

• Proposition Every rotation of R^{2} is the composition of the reflections in two lines passing through its center.

Proof. For example, let σ_{1} be reflection in the horizontal line through the center and let σ_{2} be reflection in the line through the center forming angle θ/2 with the horizontal, where θ is the angle of rotation about the center. Then σ_{2} ∘ σ_{1} is the given rotation.

Exercises due Monday, February 23

1. Prove: If an isometry f of the plane is a rotation about the point p, then for every point x in the plane p must lie on the perpendicular bisector of the line segment from x to f(x).

2. Show that every translation of R^{2} is the composition of the reflections in two parallel lines that are perpendicular to the direction of translation.

3. Show that the composition of a rotation with the reflection in a line through the center of the rotation is another such reflection.

4. Let A, B, C, and P be four points in a plane, no three of which are collinear. Let P A meet BC at D, P B meet CA at E, and P C meet AB at F. Prove that D, E, and F are not collinear.