The pedal of a curve with respect to a point is the locus of the foot of the perpendicular from to the tangent to the curve. More precisely, given a curve , the pedal curve of with respect to a fixed point (called the pedal point) is the locus of the point of intersection of the perpendicular from to a tangent to . The parametric equations for a curve relative to the pedal point are given by

			(1)
			(2)

If a curve is the pedal curve of a curve , then is the negative pedal curve of (Lawrence 1972, pp. 47-48).

When a closed curve rolls on a straight line, the area between the line and roulette after a complete revolution by any point on the curve is twice the area of the pedal curve (taken with respect to the generating point) of the rolling curve.

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Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagination. New York: Chelsea, p. 25, 1999.

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 46-49 and 204, 1972.

Lockwood, E. H. "Pedal Curves." Ch. 18 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 152-155, 1967.

Porteous, I. R. Geometric Differentiation for the Intelligence of Curves and Surfaces. Cambridge, England: Cambridge University Press, 1994.

Trott, M. The Mathematica GuideBook for Graphics. New York: Springer-Verlag, p. 19, 2004. http://www.mathematicaguidebooks.org/.

Ueda, K. In Mathematical Methods for Curves and Surfaces (Ed. T. Lyche and L. L. Shumaker). Nashville, TN: Vanderbilt University Press, 2001.

Yates, R. C. "Pedal Curves." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 160-165, 1952.

Zwikker, C. The Advanced Geometry of Plane Curves and Their Applications. New York: Dover, pp. 150-158, 1963.

CITE THIS AS:

Weisstein, Eric W. "Pedal Curve." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PedalCurve.html