Evolute
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 (f(t),g(t)), the equation of the evolute is given by
where (x,y) are the coordinates of the running point, R is the radius of curvature
and tau is the angle between the unit tangent vector
![画像: T^^=(x^')/(|x^'|)=1/(sqrt(f^('2)+g^('2)))[f^'; g^']](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/Evolute/NumberedEquation2.svg){kind=link}
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 E is the evolute of a curve I, then I is said to be the involute of E. 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, RouletteExplore with Wolfram|Alpha
More things to try:
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." §5.1 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.Referenced on Wolfram|Alpha
EvoluteCite this as:
Weisstein, Eric W. "Evolute." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Evolute.html