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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 limašon: 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 large, central, black figure in a Mandelbrot set is a cardioid. This cardioid is surrounded by a fractal arrangement of circles.
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
The polar radius ρ(θ) is given by
Simplify by noticing that
it follows that
- Four graphs of cardioids oriented in the four cardinal directions, with their respective polar equations.
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