Pedal curve
The pedal curve results from the orthogonal projection of a fixed point on the tangent lines of a given curve. More precisely, for a plane curve C and a given fixed pedal point P, the pedal curve of C is the locus of points X so that the line PX is perpendicular to a tangent T to the curve passing through the point X. Conversely, at any point R on the curve C, let T be the tangent line at that point R; then there is a unique point X on the tangent T which forms with the pedal point P a line perpendicular to the tangent T (for the special case when the fixed point P lies on the tangent T, the points X and P coincide) – the pedal curve is the set of such points X, called the foot of the perpendicular to the tangent T from the fixed point P, as the variable point R ranges over the curve C.
Complementing the pedal curve, 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.
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 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.
Take P to be the origin. For a curve given by the equation F(x, y)=0, if the equation of the tangent line at R=(x0, y0) is written in the form
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