How isometries compose II

Mirror (or Factor) Adjustment -- A major idea

The theorems about double lines reflections (and double point reflections) have more power than might be noticed for the first time. For example, if one wants to factor a rotation with center A and write it as a double line reflection, one of the two mirror lines can be taken as any line through A. Then the other one is determined. This means that if we are given a double reflection such as this, the lines can be moved so that one of the lines is in an advantageous position for canceling or other simplification.

Mirror Adjustment: Statements

• 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 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.
• 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.

Mirror Adjustment Consequences

Constructions:

1. Given 3 concurrent lines, a, b, c , find the fourth line d whose reflection R_d = R_c R_b R_a, the triple reflection.
2. Given 3 parallel lines, a, b, c , find the fourth line d whose reflection R_d = R_c R_b R_a, the triple reflection.
3. Compose two rotations with centers at A and B and angles a and b.
4. Special related cases: compose two translations or a translation and a rotation

Glide reflections and triple reflection in triangle sides

Show that composition of a HT and a line reflection is a glide reflection.

• Mirror adjustment proof
• Proof with triangles  – several cases
• Proof with coordinates

Use mirror adjustment to turn any triple line reflection into the composition of a half turn and a line reflection.

Theorem:  Any triple line reflection is a glide reflection.

Construction:  Given a triangle ABC, construct the invariant line and glide vector of the triple line reflection R_ABR_CAR_BC.

Odds and Evens and orientation

For any isometry T that is the composition of four line reflections, T is also the composition of two line reflections.

• Thus if T is the composition of n line reflections, for n > 3, then T is also the composition of n – 2 line reflections.

• Continuing this process of reducing the number of reflections by 2 eventually shows that T is either the product of 2 or 3 line reflections, depending on whether T was the product of an even or an odd number of reflections.

An isometry cannot be both the product of an even number and an odd number of line reflections.

• If R_a R_b R_c = R_d R_e, then R_a R_b R_c R_e R_d = identity.  But the 5 reflections on the left can be reduced to 3, so this says that a triple reflection = identity.  But we know triple reflections are either line reflections or glide reflections, and neither is the identity, since most points are not fixed.

Call an isometry Odd (or orientation reversing) if it is the produce of an odd number of line reflections and an Even (or orientation preserving) if it is a product of an even number of line reflections.

• The consquences of the results above is that every isometry of the plane is Odd or Even but cannot be both.
• Every Odd is the composition of 3 line reflections (or one) and thus either a line reflection or a glide reflection.
• Every Even is the composition of 2 line reflections and thus is either a translation or a rotation (or the identity).

Theorem:  All isometries of the plane are either the identity, a translation, a rotation, a line reflection, or a glide reflection.

Composition:

• The composition of 2 Odds is Even. For example the product of a line reflection and a glide reflection is Even, and so is either a translation or a rotation. (It can't be the identity since the line reflection is not the inverse of the glide reflection.)
• The composition of 2 Evens is Even. For example, the product of two rotations is a rotation or a translation.
• A mixed product of an Odd and an Even is Odd. For example, the composition of a rotation and a line reflection is a line reflection or a glide reflection.