Pedal Curves of Conics
![](/img/demonstrations-branding.png)
Requires a Wolfram Notebook System
Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.
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
.
Contributed by: Sonja Gorjanc (September 2012)
GEFFA Summer School 2012 - Rijeka
Open content licensed under CC BY-NC-SA
Snapshots
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.
Permanent Citation