Orthogonal Trajectories/Cardioids

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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

Differentiating $(1)$ with respect to $r$ gives:

\((2):\quad\) \(\displaystyle \frac {\d r} {\d \theta}\) \(=\) \(\displaystyle - c \sin \theta\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \frac {\d r} {\d \theta}\) \(=\) \(\displaystyle - \frac {r \sin \theta} {1 + \cos \theta}\) eliminating $c$ between $(1)$ and $(2)$
\(\displaystyle \leadsto \ \ \) \(\displaystyle r \frac {\d \theta} {\d r}\) \(=\) \(\displaystyle - \frac {1 + \cos \theta} {\sin \theta}\)


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:

\(\displaystyle r \dfrac {\d \theta} {\d r}\) \(=\) \(\displaystyle \frac {\sin \theta} {1 + \cos \theta}\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \int \frac {\d r} r\) \(=\) \(\displaystyle \int \frac {1 + \cos \theta} {\sin \theta} \rd \theta\) Separation of Variables
\(\displaystyle \) \(=\) \(\displaystyle \int \paren {\csc \theta + \cot \theta} \rd \theta\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \ln r\) \(=\) \(\displaystyle \ln \size {\csc \theta - \cot \theta} + \ln \size {\sin \theta} + c\)
\(\displaystyle \) \(=\) \(\displaystyle \ln \size {\paren {\csc \theta - \cot \theta} \sin \theta} + c\)
\(\displaystyle \) \(=\) \(\displaystyle \ln \size {1 - \cos \theta} + c\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle r\) \(=\) \(\displaystyle c \paren {1 - \cos \theta}\)

Hence the result.

$\blacksquare$


Sources