Reflections, rotations and glide reflections

Try

Reflect a point first across y=0 and then across y=3. Its y-coordinate changes from y to −y to 6+y, so the composition is translation upward by six. Reverse the order and the translation reverses. A composition of reflections need not itself be a reflection.

This lesson classifies plane isometries by their algebraic form and fixed sets. It also proves a bounded construction: every plane isometry is a composition of at most three line reflections.

Predict

Predict the result when the reflecting lines intersect instead of remaining parallel. Reflection reverses angular direction about its line; performing two reflections adds twice the signed angle between the lines. The composition is a rotation about their intersection.

Predict whether every orientation-reversing isometry has a fixed line. A reflection followed by a nonzero translation along that line has no fixed points. It is a glide reflection.

Experiment

Track a point symbolically through each composition.

Reflecting lines, in order	Coordinate effect	Result
y=0 then y=3	y→−y→y+6	Translation
x-axis then y-axis	(x,y)→(x,−y)→(−x,−y)	Half-turn
x-axis then line angle θ	angle φ→−φ→2θ+φ	Rotation by 2θ

The angular calculation uses coordinates centered at the lines’ intersection. It is not valid unchanged for two displaced lines. Choose a numerical point and verify the first two rows, then explain why checking one point alone does not prove equality of maps.

Understand

For a unit normal vector n, reflection in n·x=c is R(x)=x−2(n·x−c)n. The tangent component stays fixed and the normal displacement reverses. Its linear part Q=I−2nnᵀ is symmetric and satisfies Q²=I, hence QᵀQ=I. The formula is therefore an isometry.

Two reflections with the same normal and constants a then b give x↦x+2(b−a)n. Intersecting-line reflections give the rotation already calculated. Thus translations and rotations are products of two reflections.

For an orientation-reversing orthogonal matrix, choose orthonormal coordinates with Q=diag(1,−1). An affine isometry then has form F(x,y)=(x+a,−y+b). Shift the y-origin to b/2 to obtain (x+a,−y). If a=0 this is a reflection; if a≠0 it is a glide reflection. The fixed-point equation x=x+a explains the distinction exactly.

Apply

Every plane isometry now falls into one of five cases: identity, translation, rotation, reflection, or glide reflection. Identity uses zero reflections; a reflection uses one; translations and rotations use two. A glide reflection is a reflection in its axis composed with a translation along that axis, so three suffice.

Repair “Three reflections always produce a reflection.” Their composition reverses orientation, but can be a glide reflection. Determinant only separates orientation types; it does not settle the fixed set.

For optional depth, the three-reflection bound is not claimed minimal in every case. Classifying the fixed set and orientation tells when fewer suffice. A nontrivial glide cannot be one reflection because it has no fixed points.

Continue

For a proposed transformation, record its determinant sign, fixed set, and reflection factorization. These are independent checks of the classification. Continue to similarities and invariants to see which conclusions survive when uniform scaling is allowed.