An “elliptical trainer” is an exercise machine, intended to give its user the same physical experience that he could get from climbing up stairs or just going for a walk. Here’s my “artist’s impression” of one of these things:
The basic idea is that the athlete (blue in this picture) steps up and down on two boards (only one shown here) which are attached to a wheel behind him (that can be slowed down with a brake to make him exert himself more) and to vertical bars that he holds on to, which swivel fore and back around a pivot marked “T” in this sketch.
All this time, the athlete’s foot remains on a spot “F” on the moving board, and apparently this point F moves along an elliptical curve (green in the sketch above) — hence the name of this machine.
This contraption was recently the subject of a limerick in the OEDILF, and in a private discussion with my fellow OEDILF author PGS (Holger Martin), the question arose whether the green curve in this sketch really is an ellipse, or only an egg-shaped non-circular shape. In other words: is the elliptical trainer really “elliptical”?
Obviously, this is just a simple question about planar geometry involving a circle, a circular arc and some straight lines, so I thought I’d be able to determine the green shape quite easily. It’s turned out to be a real pest though. I hope I can get it worked out later this week.