Reflection as an Example of the Shortest Path for Light

  There's a simpler version of the Steiner Tree Problem and that is Hero's problem of finding the shortest path for a reflected ray of light. Again, for the general problem, we have the "gradient" equal to the sum of two unit vectors pointing to the unknown point, x. An additional complication is the constraint of the motion of x along a line so that dx=îdx'.

A solution for the reduced normal equations verifies that the angle of incidence equals the angle of reflection.

Reflecting the second point above the line illustrates Euclid's Prop. XX in his Elements Bk 1 asserting the sum of any two sides of a triangle is greater than the third or, equivalently, a straight line is the shortest distance between two points.

One can see that the triangles in the two problems are similar and the math works out the same.