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Mapping the unit interval onto circles maintaining length

Move the value of between values of 0 and using the slider and watch P trace out a path along the polar graph . P also lies on the circle with center and radius . Furthermore, the length of the arc from O to P always has length 1.
Prove the relationships described above. You want to consider using the law of sines on the isosceles triangle OPC where C is the center of the circle. The center C will change as changes.