Pedal Curves: Calculus in Motion

Pedal curves are an important class of derived curves (new curves which are obtained from given, known curve or curves). Given a curve C on a plane and a point O, one can consider a variable point P moving along the curve C. At each position of the moving point P, we have a tangent line to the curve C at P, which we denote by T_P. If we drop a perpendicular from O onto the tangent line T_P, suppose the foot of the perpendicular is M. Now, as P moves along the curve C, the tangent line T_P keeps changing, and so does the foot M of the perpendicular dropped from O onto T_P. The locus of the feet M of the perpendicular from O onto the tangent line T_P, as P varies along the curve C, is called the Pedal Curve of C with respect to the point O.

**Figure:** C is the given curve and O is the specified point from which perpendiculars are dropped onto ***T_P*** , which is the tangent line to the curve C at the point P on the curve. The foot of the perpendicular is M, as P moves along C, the locus of M is called the pedal curve of C with respect to the specified point O (*The picture shows two different positions of the foot of the perpendicular viz. M and M’, corresponding to two different positions of the moving point along the curve viz. P and P’*)

As a familiar example, one can check that the pedal curve of the parabola 4y=x² with respect to its vertex O (0,0) turns out to be the famous Cissoid of Diocles with parametric coordinates as follows: