An evolute is the locus of centers of curvature (the envelope) of a plane curve's normals. The original curve is then said to be the involute
of its evolute. Given a plane curve represented parametrically by 
, the equation of the evolute is given by
where 
are the coordinates of the running
point, 
is the radius
of curvature
 |
(3) |
and 
is the angle between the unit tangent
vector
![T^^=(x^')/(|x^'|)=1/(sqrt(f^('2)+g^('2)))[f^'; g^']](https://txtdot.deep-swarm.xyz/proxy/img?url=https%3A%2F%2Fmathworld.wolfram.com%2Fimages%2Fequations%2FEvolute%2FNumberedEquation2.svg) |
(4) |
and the x-axis,
Combining gives
The definition of the evolute of a curve is independent of parameterization for any differentiable function (Gray 1997). If 
is the evolute of a curve 
, then 
is said to be the involute of 
. The centers of the osculating
circles to a curve form the evolute to that curve (Gray 1997, p. 111).
The following table lists the evolutes of some common curves, some of which are illustrated above.
See also
Envelope, Involute, Osculating Circle, Roulette
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References
Cayley, A. "On Evolutes of Parallel Curves." Quart. J. Pure Appl. Math. 11, 183-199, 1871.Dixon, R. "String Drawings." Ch. 2 in Mathographics. New York: Dover, pp. 75-78, 1991.Gray, A. "Evolutes." ยง5.1 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 98-103, 1997.Jeffrey, H. M. "On the Evolutes of Cubic Curves." Quart. J. Pure Appl. Math. 11, 78-81 and 145-155, 1871.Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 40 and 202, 1972.Lockwood, E. H. "Evolutes and Involutes." Ch. 21 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 166-171, 1967.Yates, R. C. "Evolutes." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 86-92, 1952.
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Cite this as:
Weisstein, Eric W. "Evolute." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Evolute.html