Circle Involute

First studied by Huygens when he was considering clocks without pendula for use on ships at sea. He used the circle involute in his first pendulum clock in an attempt to force the pendulum to swing in the path of a Cycloid.

For a Circle with $a=1$, the parametric equations of the circle and their derivatives are given by

$$x=\cos t\qquad x'=-\sin t\qquad x''=-\cos t$$ (1)

$$y=\sin t\qquad y'=\cos t\qquad y''=-\sin t.$$ (2)

The Tangent Vector is

$$\hat{\bf T}=\left[{\matrix{-\sin t\cr \cos t\cr}}\right]$$ (3)

and the Arc Length along the circle is

$$s=\int \sqrt{x'^2+y'^2}\,dt = \int dt=t,$$ (4)

so the involute is given by

$${\bf r}_i={\bf r}-s\hat{\bf T} = \left[{\matrix{\cos t\cr \s... ...= \left[{\matrix{\cos t+t\sin t\cr \sin t-t\cos t\cr}}\right],$$ (5)

or

$$x = a(\cos t+t\sin t)$$ (6)

$$y = a(\sin t-t\cos t).$$ (7)

The Arc Length, Curvature, and Tangential Angle are

$$s = \int ds=\int\sqrt{x'^2+y'^2}\,dt = {\textstyle{1\over 2}}at^2$$ (8)

$$\kappa = {1\over at}$$ (9)

$$\phi = t.$$ (10)

The Cesàro Equation is

$$\kappa={1\over\sqrt{as}}.$$ (11)

See also Circle, Circle Evolute, Ellipse Involute, Involute

References

Gray, A. Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, p. 83, 1993.

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 190-191, 1972.

MacTutor History of Mathematics Archive. ``Involute of a Circle.'' http://www-groups.dcs.st-and.ac.uk/~history/Curves/Involute.html.