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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

$\displaystyle x$ $\textstyle =$ $\displaystyle f-R\sin\tau$ (1)
$\displaystyle y$ $\textstyle =$ $\displaystyle g+R\cos\tau,$ (2)

where $(x,y)$ are the coordinates of the running point, $R$ is the Radius of Curvature
R={(f'^2+g'^2)^{3/2}\over f'g''-f''g'},
\end{displaymath} (3)

and $\tau$ is the angle between the unit Tangent Vector
\hat{\bf T} = {{\bf x}'\over \vert{\bf x}'\vert} = {1\over \sqrt{f'^2+g'^2}} {\left[{\matrix{f'\cr g'\cr}}\right]}
\end{displaymath} (4)

and the x-Axis,
$\displaystyle \cos\tau$ $\textstyle =$ $\displaystyle \hat {\bf T}\cdot\hat{\bf x}$ (5)
$\displaystyle \sin\tau$ $\textstyle =$ $\displaystyle \hat {\bf T}\cdot\hat{\bf y}.$ (6)

Combining gives
$\displaystyle x$ $\textstyle =$ $\displaystyle f-{(f'^2+g'^2)g'\over f'g''-f''g'}$ (7)
$\displaystyle y$ $\textstyle =$ $\displaystyle g+{(f'^2+g'^2)g'\over f'g''-f''g'}.$ (8)

The definition of the evolute of a curve is independent of parameterization for any differentiable function (Gray 1993). 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 1993, p. 90).

The following table lists the evolutes of some common curves.

Curve Evolute
Astroid Astroid 2 times as large
Cardioid Cardioid 1/3 as large
Cayley's Sextic Nephroid
Circle point (0, 0)
Cycloid equal Cycloid
Deltoid Deltoid 3 times as large
Ellipse Lamé Curve
Epicycloid enlarged Epicycloid
Hypocycloid similar Hypocycloid
Limaçon Circle Catacaustic for a point source
Logarithmic Spiral equal Logarithmic Spiral
Nephroid Nephroid 1/2 as large
Parabola Neile's Parabola
Tractrix Catenary

See also Involute, Osculating Circle


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. Boca Raton, FL: CRC Press, pp. 76-80, 1993.

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.

Lee, X. ``Evolute.''

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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© 1996-9 Eric W. Weisstein