Circle Involute Unwinding

Worked Example: Finding the Involute

Consider a circle of radius a = 1 and an unwind angle t = 1 radian.

1. The contact point of the string with the circle is (a cos t, a sin t) = (cos(1), sin(1)) ≈ (0.540, 0.841).
2. The length of the unwound tangent string is the arc length of the circle that has been unwound, which is a · t = 1 · 1 = 1.
3. The parametric equations for the involute are x = a(cos t + t sin t) and y = a(sin t - t cos t).
4. Plugging in a = 1, t = 1, we get x = cos(1) + sin(1) ≈ 1.38177 and y = sin(1) - cos(1) ≈ 0.30116.
5. The speed of the point is a · t = 1. The total arc length traced by the end of the string from t=0 is (1/2)at^2 = 0.5.
6. The curvature κ for t > 0 is 1/(a · t) = 1.0.

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.

Limitations

This visualization covers the involute up to small values of t. For large t, the involute spirals infinitely outward, which requires scaling adjustments to display on a finite screen. The curvature at t=0 is mathematically unbounded (singular) because the initial speed is zero; the model displays this as "Unavailable" instead of an arbitrary infinite number.

References

For more details on the involute of a circle and its properties, see the following resource:

• The Involute of a Circle - UNT Digital Library. This text discusses the geometric and kinematic properties of the unwinding string model.