Cardioid Guide, Meaning , Facts, Information and Description

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 (Greek kardioeides = kardia:heart + eidos:shape). Compared to the ♥ symbol, though, it doesn't have the sharp point at the bottom.

The cardioid is an inverse transform of a parabola.

The large, central, black figure in a Mandelbrot set is a cardioid. This cardioid is surrounded by a fractal arrangement of circles.

Table of contents

1 Equations 1.1 Proof 2 Graphs

Equations

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

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

Proof

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

The polar radius is given by

Expand,

Simplify by noticing that

Thus,

Then, since

it follows that

quod erat demonstrandum.

Graphs

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

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