10182

Pedal Curves of Conics

The pedal curve of a curve with respect to a point (the pedal point) is the locus of the foot of the perpendicular from to the tangent line of the curve .
Drag the locator (red pedal point) to change the pedal curve of a given conic.
Choose an item from the popup menu to change the type of conic and drag the slider to change its shape.

DETAILS

Generally, the pedal curve of a conic is a bicircular quartic (a fourth-order curve with double absolute points) with a real double point at the pedal point . The point is a node, a cusp, or an isolated double point depending on whether it is outside, on, or inside the conic , respectively.
In special cases, this quartic splits into a pair of lines and a curve of the lower order.
• If is a parabola, splits into the line at infinity and a circular cubic.
• If is an ellipse or a hyperbola and is one of its foci, is a circle. More precisely, the pedal curve splits into the pair of isotropic lines through and a circle.
• If is a parabola and is its focus, is a line. More precisely, the pedal curve splits into the line at infinity, the pair of isotropic lines through , and the tangent line to the parabola at its vertex.
References
[1] A. Gray, Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed., Boca Raton: CRC Press LLC, 1998.
[2] G. Salmon, A Treatise on the Higher Plane Curves, New York: Chelsea Publishing Company (reprint), 1960.

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