Pedal Curve

For a plane curve C and a given fixed point P, the pedal curve of C is the locus of points X so that PX is perpendicular to a tangent to the curve passing through X. The point P is called the pedal point. The pedal curve is the first in a series of curves C₁, C₂, C₃, etc., where C₁ is the pedal of C, C₂ is the pedal of C₁, and so on. In this scheme, C₁ is known as the first positive pedal of C, C₂ is the to second positive pedal of C, and so on. Going the other direction, C is the first negative pedal of C₁, the second negative pedal of C₂, etc.

More precisely, at any point R on C, let T be the tangent line at R. There is then a unique point X on T which is either P (in case P lies on T) or forms with P a line perpendicular to T. The pedal curve is the set of such points X, called the foot of the perpendicular to T from P, as R ranges over points C.

Similarly, there is a unique point Y on the line normal to C at R so that PY is perpendicular to the normal, so PXRY is a (possibly degenerate) rectangle. The locus of points Y is called the contrapedal curve.

The orthotomic of a curve is its pedal magnified by a factor of 2 so that the center of similarity is P. This is locus of the reflection of P through the tangent line T.