How much to turn to make a circle?

Question

It's weird that I never thought about this before and it looks so simple. You are swimming in a lake (or maybe on a boat) and you have a compass or something with which you can measure the angle of your turns. How can you make a circle of radius $r$?

I just realize that we undergo a motion that's both rotational and transnational. Seems more like a physics question now.

Answer 1

This is a pure mathematical question, and the answer is simple: with just a compass you can not even make sure to draw a circle. At best, a convex spiral (not necessarily closed).

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If you add the condition that you are able to swim in a perfectly regular way (?), you can indeed follow a circle. Of unknown radius. (More precisely, with an angular deviation always proportional to the swimmed distance.) But then a compass is of no use.

Answer 2

Assuming that you can measure velocity: just keep the angular deviation per time unit by the value: $$ \omega=\frac vr, $$ where the angle is assumed to be measured in radians.