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The pedal of a curve C with respect to a point O is the locus of the foot of the perpendicular from P to
the tangent to the curve. More precisely, given a curve C, the pedal curve P of C with respect to a fixed point
O (called the pedal point) is the locus of the point P of intersection of the perpendicular from O to a tangent to C. The parametric equations for a curve
relative to the pedal point
are
given by
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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 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.
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