Edit: I noticed a small mistake in the formulas below. I’ll try to get it fixed this week.

I visited a railroad museum today, and I saw a display showing how the piston is linked to the train wheel. For fun and learning I wanted to model the basic mathematics of how the linkage moves with the wheel and the piston, without looking up the answer on the Internet. That part seemed very simple:

Since l and p are fixed length, it was a matter of simple trigonometry, as seen above. Then I threw the math into a simple Racket program to simulate the movement. That part not hard, but it took an hour or two to add enough lines and circles to make the graphic look half-way decent. Here is a video recording of it running (about 10 seconds):

One interesting part of the math is the connection point of l and p (see the diagram above). Until you get very long lengths of l, you get something close to the cosine function but not quite the same.

Nice graphic! Very interesting math problem!

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I just noticed a typo in the trig diagram so I fixed that.

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