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).
notes 1) In French:
antipodaire. In German: Gegenfusspunktskurve.
