Wednesday, 11/16

How isometries compose I

Line reflection R = R_m

For any P, let P' = R_m (P). Then if P is on m, P = P'; if not, m is the perpendicular bisector of PP'.
R_m R_m = identity. In other words, R_m^-1 = R_m.

Composition of two line reflections T = R_nR_m

See Lab 8.
S = R_mR_n is the inverse of T, since ST = (R_mR_n)(R_nR_m) = R_m(R_nR_n)R_m = R_mR_m = identity, by the associative law.

If m and n are parallel, then T is a translation

The translation vector has length twice the distance from m to n.
The direction of the vector is from m towards n.
Specifically, if M is a point on m and M' is the reflection of M in n, then the translation takes M to M', so the translation vector is MM'.
The translation S = R_mR_n is the inverse of T with translation vector -MM'.
If p and q are any lines parallel to m and n so that the distance from p to q equals the distance from m to n and the direction from p to q is the same as the direction from m to n, then the translation R_qR_p = R_nR_m.

If m and n intersect at a point O, then T is a rotation.

The angle of rotation is twice the (signed, or directed) angle from m to n.
Specifically, if M is a point on m and M' is the reflection of M in n, then the rotation takes M to M', so the angle is MOM'.
The rotation S = R_mR_n is the inverse of T with rotation angle –MOM'.
If p and q are any other lines through O so that the (signed, or directed) angle from p to q equals the angle from m to n, then the rotation R_qR_p = R_nR_m.

Half Turns (Point Reflections) are special rotations

See Lab 8 and Brown 2.4
A point reflection H_A with center A is rotation by 180 degrees. Thus H_A = R_nR_m for any perpendicular lines through A, since if the angle between the lines = 90 degrees, the rotation angle = 2*90 = 180 degrees.
If m and n are perpendicular at A, then H_A = R_mR_n = R_nR_m . This is true since one rotation is by angle +2*90 = +180 and the other is by -2*90 = -180. But + and - 180 degrees are the same angle.
The composition of two point reflections, T = H_B H_A is translation by vector 2AB.
If vector AB = vector CD, then the translation H_B H_A = H_D H_C. Unless the points are collinear, this means that ABDC is a parallelogram.
The composition of 3 point reflections, T = H_C H_B H_A is another half-turn H_E . This is the same as saying H_B H_A = H_C H_E, or ABCE is a parallelogram

Translation Composition