Plane Mirrors — Question 10

Question 10

A sixty-degree mirror wedge generates a finite orbit of image points

Two ideal plane mirrors meet at 60 degrees. A point object lies at polar angle 20 degrees between them, at radius r from their intersection. In the cross section, reflections in the two mirror lines map an angle theta to -theta and to 120 degrees minus theta.

1. Generate all distinct angular positions in the orbit under repeated reflections.

2. Count the images excluding the object, and identify why no new positions arise after three ordered pairs of reflections.

Question 10 - Solution

A sixty-degree mirror wedge generates a finite orbit of image points

(a) Method and calculation

$$S_{60}S_0(\theta)=120^\circ+\theta\pmod{360^\circ}.$$$$\boxed{\theta\in\{20^\circ,100^\circ,140^\circ,220^\circ,260^\circ,340^\circ\}}.$$

(b) Interpretation and further calculation

$$\boxed{N_{\mathrm{images}}=6-1=5}.$$$$(S_{60}S_0)^3(\theta)=\theta+360^\circ\equiv\boxed{\theta}.$$

Physical check

All orbit points remain at radius r; the 20-degree object is not on a mirror or a symmetry axis that would merge positions.