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:

$r = c \paren {1 - \cos \theta}$

Proof

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

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

\(\text {(2)}: \quad\)

\(\ds \frac {\d r} {\d \theta}\)

\(=\)

\(\ds - c \sin \theta\)

\(\ds \leadsto \ \ \)

\(\ds \frac {\d r} {\d \theta}\)

\(=\)

\(\ds -\frac {r \sin \theta} {1 + \cos \theta}\)

eliminating $c$ between $(1)$ and $(2)$

\(\ds \leadsto \ \ \)

\(\ds r \frac {\d \theta} {\d r}\)

\(=\)

\(\ds -\frac {1 + \cos \theta} {\sin \theta}\)

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:

\(\ds r \dfrac {\d \theta} {\d r}\)

\(=\)

\(\ds \frac {\sin \theta} {1 + \cos \theta}\)

\(\ds \leadsto \ \ \)

\(\ds \int \frac {\d r} r\)

\(=\)

\(\ds \int \frac {1 + \cos \theta} {\sin \theta} \rd \theta\)

Solution to Separable Differential Equation

\(\ds \)

\(=\)

\(\ds \int \paren {\csc \theta + \cot \theta} \rd \theta\)

\(\ds \leadsto \ \ \)

\(\ds \ln r\)

\(=\)

\(\ds \ln \size {\csc \theta - \cot \theta} + \ln \size {\sin \theta} + c\)

\(\ds \)

\(=\)

\(\ds \ln \size {\paren {\csc \theta - \cot \theta} \sin \theta} + c\)

\(\ds \)

\(=\)

\(\ds \ln \size {1 - \cos \theta} + c\)

\(\ds \leadsto \ \ \)

\(\ds r\)

\(=\)

\(\ds c \paren {1 - \cos \theta}\)

Hence the result.

$\blacksquare$

Sources

• 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $1$: The Nature of Differential Equations: $\S 3$: Families of Curves. Orthogonal Trajectories: Problem $1 \ \text c$