What are facts about the line of reflection?

Answer 1

Given a line L and a point P. Reflection P' of P in L is the point such that PP' is perpendicular to L, and PM = MP', where M is the point of intersection of PP' and L. In other words, P' is located on the other side of L, but at the same distance from L as P. P' is said to be a mirror or symmetric image of P in L. The line L is called the axis of symmetry or axis of reflection.

P' exists for any P. Let's write SL(P) = P'. If P' = SL(P), then P is the reflection in L of P':P = SL(P'). So that repeated reflection does noting: it does not move a point. Formally,

SL2 = I,

Where I is the identity transform. SL(P) = P, iff P lies on L.

The reflection transform SL applies to arbitrary shapes point-by-point. Each point of a given shape S is reflected in L, and the collection of these reflections is the symmetric image of S: SL(S). To determine SL(S) when S is a polygon, suffice it to reflect its vertices. This is exactly what has been done in the applet below.

On the other hand, if S' is known to be a mirror image of S, then any pair of points P and P' not fixed by the reflection (P ≠ P'), the axes of reflection is uniquely determined as the perpendicular bisector of PP'.

In the applet, you can create polygons with a desired number of vertices, drag the vertices one at a time, or drag the polygon as a whole. Axes of reflection can also be dragged. They rotate if dragged near the applet's border, or translate if dragged nearer their midpoint.