Pedal Curve
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
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.
SEE ALSO: Contrapedal Curve,
Negative Pedal Curve,
Pedal
Point
REFERENCES:
Ameseder, A. "Ueber Fusspunktcurven der Kegelschnitte." Archiv Math.
u. Phys. 64, 143-144, 1879.
Ameseder, A. "Zur Theorie der Fusspunktencurven der Kegelschnitte." Archiv
Math. u. Phys. 64, 145-163, 1879.
Gray, A. "Pedal Curves." §5.8 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed.
Boca Raton, FL: CRC Press, pp. 117-125, 1997.
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.
Referenced on Wolfram|Alpha:
Pedal Curve
CITE THIS AS:
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