Involute

Attach a string to a point on a curve. Extend the string so that it is tangent to the curve at the point of attachment. Then wind the string up, keeping it always taut. The Locus of points traced out by the end of the string is the involute of the original curve, and the original curve is called the Evolute of its involute. Although a curve has a unique Evolute, it has infinitely many involutes corresponding to different choices of initial point. An involute can also be thought of as any curve Orthogonal to all the Tangents to a given curve.

The equation of the involute is

$${\bf r}_{\rm i}={\bf r}-s\hat{\bf T},$$ (1)

where $\hat {\bf T}$ is the Tangent Vector

$$\hat{\bf T}={{d{\bf r}\over dt}\over \left\vert{d{\bf r}\over dt}\right\vert}$$ (2)

and $s$ is the Arc Length

$$s =\int ds = \int{ds\over dt}\, dt = \int {\sqrt{ds^2}\over dt}\,dt= \int \sqrt{f'^2+g'^2}\,dt.$$ (3)

This can be written for a parametrically represented function $(f(t),g(t))$ as

$$x(t) = f-{s f'\over\sqrt{f'^2+g'^2}}$$ (4)

$$y(t) = g-{s g'\over\sqrt{f'^2+g'^2}}.$$ (5)

Curve	Involute
Astroid	Astroid 1/2 as large
Cardioid	Cardioid 3 times as large
Catenary	Tractrix
Circle Catacaustic for a point source	Limaçon
Circle	Circle Involute (a Spiral)
Cycloid	equal Cycloid
Deltoid	Deltoid 1/3 as large
Ellipse	Ellipse Involute
Epicycloid	reduced Epicycloid
Hypocycloid	similar Hypocycloid
Logarithmic Spiral	equal Logarithmic Spiral
Neile's Parabola	Parabola
Nephroid	Cayley's Sextic
Nephroid	Nephroid 2 times as large

See also Evolute, Humbert's Theorem

References

Cundy, H. and Rollett, A. ``Roulettes and Involutes.'' §2.6 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 46-55, 1989.

Dixon, R. ``String Drawings.'' Ch. 2 in Mathographics. New York: Dover, pp. 75-78, 1991.

Gray, A. ``Involutes.'' §5.4 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 81-85, 1993.

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 40-42 and 202, 1972.

Lee, X. ``Involute.'' http://www.best.com/~xah/SpecialPlaneCurves_dir/Involute_dir/involute.html.

Lockwood, E. H. ``Evolutes and Involutes.'' Ch. 21 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 166-171, 1967.

Pappas, T. ``The Involute.'' The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 187, 1989.

Yates, R. C. ``Involutes.'' A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 135-137, 1952.