Cardioid

In geometry, the cardioid is an epicycloid which has one and only one cusp. That is, a cardioid is a curve that can be produced as a locus — by tracing the path of a chosen point of a circle which rolls without slipping around another circle which is fixed but which has the same radius as the rolling circle.

The cardioid is also a special type of limaçon: it is the limaçon with one cusp.

The name comes from the heart shape of the curve. Compared to the ♥ symbol, though, it doesn't have the sharp point at the bottom.

The cardioid is an inversion of a parabola.

Equations

Since the cardioid is an epicycloid with one cusp, its parametric equations are

$x(\theta )=\cos \theta +{1 \over 2}\cos 2\theta ,\qquad \qquad (1)$

$y(\theta )=\sin \theta +{1 \over 2}\sin 2\theta .\qquad \qquad (2)$

The same shape can be defined in polar coordinates by the equation

$\rho (\theta )=1+\cos \theta .\$

Proof

Equations (1) and (2) define a cardioid whose cuspidal point is (-1/2, 0). To convert to polar, the cusp is preferably at the origin, so add 1/2 to the abscissa:

$x(\theta )={1 \over 2}+\cos \theta +{1 \over 2}\cos 2\theta ,$

$y(\theta )=\sin \theta +{1 \over 2}\sin 2\theta .$

The polar radius $\rho (\theta )$ is given by

$\rho (\theta )={\sqrt {x^{2}(\theta )+y^{2}(\theta )}}$

$={\sqrt {\left({1 \over 2}+\cos \theta +{1 \over 2}\cos 2\theta \right)^{2}+\left(\sin \theta +{1 \over 2}\sin 2\theta \right)^{2}}}.$

Expand,

$\rho ={\sqrt {{1 \over 4}+\cos ^{2}\theta +{1 \over 4}\cos ^{2}2\theta +\cos \theta +{1 \over 2}\cos 2\theta +\cos \theta \cos 2\theta +\sin ^{2}\theta +{1 \over 4}\sin ^{2}2\theta +\sin \theta \sin 2\theta }}.$

Simplify by noticing that

$\cos ^{2}\theta +\sin ^{2}\theta =1,\qquad \qquad {\mbox{(trig. ident.)}}$

${1 \over 4}\cos ^{2}2\theta +{1 \over 4}\sin ^{2}2\theta ={1 \over 4},\qquad \qquad {\mbox{(variation of the above)}}$

$\cos \theta \cos 2\theta +\sin \theta \sin 2\theta =\cos(\theta -2\theta )=\cos -\theta =\cos \theta .\$

Thus,

$\rho ={\sqrt {{1 \over 4}+1+{1 \over 4}+2\cos \theta +{1 \over 2}\cos 2\theta }}$

$={\sqrt {{3 \over 2}+{4 \over 2}\cos \theta +{1 \over 2}\cos 2\theta }}$

$={\sqrt {3+4\cos \theta +\cos 2\theta \over 2}}.$

Then, since

$\cos 2\theta =\cos ^{2}\theta -\sin ^{2}\theta =2\cos ^{2}\theta -1,\qquad \qquad {\mbox{(trigonometric identity)}}$

it follows that

$\rho ={\sqrt {3+4\cos \theta +2\cos ^{2}\theta -1 \over 2}}={\sqrt {2+4\cos \theta +2\cos ^{2}\theta \over 2}},$

$\rho ={\sqrt {1+2\cos \theta +\cos ^{2}\theta }}=1+\cos \theta ,$

quod erat demonstrandum.

Graphs

Four graphs of cardioids oriented in the four cardinal directions, with their respective polar equations.