Cardioid

A cardioid (from the Greek καρδία "heart") is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. It can also be defined as an epicycloid having a single cusp. It is also a type of sinusoidal spiral, and an inverse curve of the parabola with the focus as the center of inversion.

The name was coined by de Castillon in 1741 but had been the subject of study decades beforehand. Named for its heart-like form, it is shaped more like the outline of the cross section of a round apple without the stalk.

A cardioid microphone exhibits an acoustic pickup pattern that, when graphed in two dimensions, resembles a cardioid, (any 2d plane containing the 3d straight line of the microphone body.) In three dimensions, the cardioid is shaped like an apple centred around the microphone which is the "stalk" of the apple.

Based on the rolling circle description, with the fixed circle having the origin as its center, and both circles having radius a, the cardioid is given by the following parametric equations:

In the complex plane this becomes

Here a is the radius of the circles which generate the curve, and the fixed circle is centered at the origin. The point generating the curve touches the fixed circle at (a, 0), the cusp. The parameter t can be eliminated giving

or, in rectangular coordinates,

These equations can be simplified somewhat by shifting the fixed circle to the right a units and choosing the point on the rolling circle so that it touches the fixed circle at the origin; this changes the orientation of the curve so that the cusp is on the left. The parametric equations are then:

or, in the complex plane,

With the substitution u=tan t/2,

giving a rational parameterization:

or

The parametrization can also be written

and in this form it is apparent that the equation for this cardioid may be written in polar coordinates as

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