Orthogonal Trajectories/Examples/Cardioids

Theorem
Consider the one-parameter family of curves of cardioids given in polar form as:
 * $(1): \quad r = c \paren {1 + \cos \theta}$

Its family of orthogonal trajectories is given by the equation:
 * $r = c \paren {1 - \cos \theta}$


 * CardioidsOrthogonalTrajectories.png

Proof
We use the technique of formation of ordinary differential equation by elimination.

Differentiating $(1)$ $r$ gives:

Thus from Orthogonal Trajectories of One-Parameter Family of Curves, the family of orthogonal trajectories is given by:
 * $r \dfrac {\d \theta} {\d r} = \dfrac {\sin \theta} {1 + \cos \theta}$

So:

Hence the result.