The elliptical trainer

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.

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