Consider a large number of points distributed on the circumference of a circle with radius r. If I rotate each point with a randomly chosen Euler angle around a randomly chosen coordinate inside this circle, then the result of this transformation may not be distributed within the circle and it may exceed the circle's dimension. I am interested in the calculation (and drawing) of the largest possible shape (or dimension) that this transformation can make.
2026-03-29 13:21:17.1774790477
Determining the rotation shape
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The maximal shape is a circle with thrice the radius of the original. You can obtain it by rotating every point on the rim around another point on the rim directly opposite (i.e. antipodal) to the first one, with an angle of 180°. The distance between these two points is the diameter of the circle, i.e. twice the radius, and it will be pointing away from the center of the circle. Since the center of rotation is one radius away, you obtain a total of $3r$ distance from the center.