Given a curve C1 and a
(pedal) point O, construct
for each tangent l of C1 a point P, for which OP is perpendicular to the tangent. The
collection of points P forms a curve C2, the (positive)
pedal of C1 (with respect to the pedal point).
When C1 is given by (x, y) = (f(t), g(t)), and we translate C1 in such a way that the
pedal point is the origin, then C2 has the form:
Two curves are invariant for making a pedal:
The pedal of the parabola is the curve
given by the equation:
Some other pedals are:
The reverse operation of making a pedal is to construct from each point P of C2 a line l
that is perpendicular to OP. The lines l together form an envelope of the curve C1. Now we
call C1 the negative pedal 1)
of C2. When C1 is
a pedal of C2, then C2 is the negative pedal of C1.
Because of this definition, the curve is in fact also an orthocaustic:
the orthocaustic of a curve C1 (with respect to a point O) is the envelope of
the perpendiculars of P on OP (P on C1).
Instead of tangents to a curve we can consider normals to that curve. This pedal curve is
called the normal pedal curve.
MacLaurin was the first author to investigate pedal curves (1718).
1) In French:
antipodaire. In German: Gegenfusspunktskurve.