Examples

Archimedes' Trammel
The Archimedes' Trammel mechanism generates an ellipse. Here we use a point A as the ellipse's center, and put a point E on the same horizontal line as A. The points C and E are constrained to run along the ellipse's axes (the green lines), while the distance between them is set to a-b. When we trace the locus of the point D at distance b from E along the same line, we get an ellipse with semimajor axis a and semiminor axis b.

Billiards
In American billiards, you need to bounce your cue ball off two cushions before hitting another ball. How long is the path taken by the cue ball, A? We can create the correct path by reflecting the target ball B in the two cushions (IJ and JG), drawing a line between the cue ball and the twice-reflected target B", then reflecting this line back. The path length is the same as the distance between the cue ball and the twice-reflected target.