Plane Mirrors — Question 7

Question 7

Two parallel reflections translate an image by twice the mirror separation

Two ideal plane mirrors have coordinates x=0 and x=L. An object lies at x=a with zero less than a less than L. Reflection maps are $S_0(x)=-x$ and $S_L(x)=2L-x$.

1. Find the image coordinate after reflection first at x=0 and then at x=L, and after the reverse order.

2. Find the coordinate after N repetitions of the first ordered pair and its displacement from the original object.

Question 7 - Solution

Two parallel reflections translate an image by twice the mirror separation

(a) Method and calculation

$$\boxed{S_L[S_0(a)]=2L+a}.$$$$\boxed{S_0[S_L(a)]=a-2L}.$$

(b) Interpretation and further calculation

$$(S_LS_0)^N(a)=\boxed{a+2NL}.$$$$\boxed{\Delta x_N=2NL}.$$

Physical check

The two ordered compositions are opposite translations; image order matters even though each individual reflection is its own inverse.