Cardioid

The red curve is a cardioid.

In geometry, the cardioid is an epicycloid with one cusp. That is, a cardioid is a curve that can be produced as the path (or locus) of a point on the circumference of a circle as that circle rolls around another fixed circle with the same radius.

The cardioid is also a special type of limaçon: it is the limaçon with one cusp. The cusp is formed when the ratio of a to b in the equation is equal to one.

The name comes from the heart shape of the curve (Greek kardioeides = kardia:heart + eidos:shape). Compared to the heart symbol (♥), though, a cardioid only has one sharp point (or cusp). It is rather shaped more like the outline of the cross section of a plum.

The cardioid is an inverse transform of a parabola.

The large central figure in the Mandelbrot set is a cardioid.

Caustics can take the shape of cardioids. The caustic seen at the bottom of a coffee cup, for instance, may be a cardioid. The specific curve depends on the angle the light source makes relative to the bottom of the cup. The shape can be a nephroid, which looks quite similar.

Contents 1 Equations 2 Graphs 3 Area 4 See also 5 References

Equations

Since the cardioid is an epicycloid with one cusp, in cartesian coordinates it has parametric equations

$x(t)\; =\; 2r\; \backslash left(\; \backslash cos\; t\; -\; \{1\; \backslash over\; 2\}\; \backslash cos\; 2\; t\; \backslash right),$

$y(t)\; =\; 2r\; \backslash left(\; \backslash sin\; t\; -\; \{1\; \backslash over\; 2\}\; \backslash sin\; 2\; t\; \backslash right)$

where r is the radius of the circles which generate the curve, and the fixed circle is centered at the origin. The cusp is at (r,0).

The polar equation

$\backslash rho(\backslash theta)\; =\; 2r(1\; -\; \backslash cos\; \backslash theta).\; \backslash$

yields a cardioid with the same shape. It is the same curve as the cardioid given above, shifted to the left r units, so the cusp is at the origin.

For a proof, see cardioid proofs.

Graphs

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

Area

The area of a cardioid with polar equation

$\backslash rho(\backslash theta)\; =\; a(1\; -\; \backslash cos\; \backslash theta)$

is

$A\; =\; \{3\backslash over\; 2\}\; \backslash pi\; a^2$.

See proof.

See also

• Wittgenstein\'s rod
• microphone - for a discussion of cardioid microphones
• Loop antenna
• Radio direction finder
• Radio direction finding
• Yagi antenna

References

• Hearty Munching on Cardioids at cut-the-knot
• Xah Lee, Cardioid (1998) (This site provides a number of alternative constructions).
• Jan Wassenaar, Cardioid, (2005)

This article is licensed under the GNU Free Documentation License. It uses material from Wikipedia