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Pedal Curve

Given a curve $C$, the pedal curve of $C$ with respect to a fixed point $O$ (the Pedal Point) is the locus of the point $P$ of intersection of the Perpendicular from $O$ to a Tangent to $C$. The parametric equations for a curve $(f(t),g(t))$ relative to the Pedal Point $(x_0,y_0)$ are

$\displaystyle x$ $\textstyle =$ $\displaystyle {x_0f'^2+f g'^2+(y_0-g)f'g'\over f'^2+g'^2}$  
$\displaystyle y$ $\textstyle =$ $\displaystyle {g f'^2+y_0g'^2+(x_0-f)f'g'\over f'^2+g'2^2}.$  

Curve Pedal Point Pedal Curve
Astroid center Quadrifolium
Cardioid cusp Cayley's Sextic
Central Conic Focus Circle
Circle any point Limaçon
Circle on Circumference Cardioid
Circle Involute center of Circle Archimedean Spiral
Cissoid of Diocles Focus Cardioid
Deltoid center Trifolium
Deltoid cusp simple Folium
Deltoid on curve unsymmetric double folium
Deltoid Vertex double folium
Epicycloid center Rose
Hypocycloid center Rose
Line any point point
Logarithmic Spiral pole Logarithmic Spiral
Parabola Focus Line
Parabola foot of Directrix Right Strophoid
Parabola on Directrix Strophoid
Parabola reflection of Focus by Directrix Maclaurin Trisectrix
Parabola Vertex Cissoid of Diocles
Sinusoidal Spiral pole Sinusoidal Spiral
Tschirnhausen Cubic Focus of Pedal Parabola

See also Negative Pedal Curve


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

Lee, X. ``Pedal.''

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

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

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© 1996-9 Eric W. Weisstein