Unwrapping a circle dynamically using its involutes:

One can now construct a curved line that interpolates between the two endpoints of circle and line:

The involutes in blue of the red circle are given parametrically on the right-hand side by

$[ \sin t - (t-a) \cos t, \cos t + (t-a) \sin t ]$

and on the left-hand side by reflecting the x-coordinate:

$[ -\sin t + (t-a) \cos t, \cos t + (t-a) \sin t ]$

The portions of the involutes begin with $$t = a$$ on the circle and all end at $$y = -1$$ when $$t = \pi$$. The moving purple line connects points along the involutes that are linearly interpolated between the endpoints.

The configuration explicitly sets minima and maxima to keep the plot centered during the evolution. It includes the setting for equal aspect to ensure the appearance of the circle.

Complete code for this example:

Examples Page