Circle Involute Unwinding

The involute of a circle is the path traced by a point on a taut string as it is unwound from a circle. Unlike a cycloid, which is formed by rolling a circle along a line, an involute relies on keeping a tangent string straight as it detaches from the circumference.

Experiment

Explore the unwinding of the string interactively.

1. Open the Circle Involute Unwinding module.
2. Set the Circle radius to 1.
3. Increase the Unwind angle from 0 to 2. Notice how the length of the unwound tangent string equals the angle t (since r=1), and the total arc length of the involute grows as (1/2)t^2.
4. Observe the curvature at t=0; it is undefined because the tracing point starts at a cusp with zero speed.