Cardioid

The cardioid is a plane curve with a heart-like shape. It can be defined as the locus of a point on a circle as it rolls around another circle of the same radius. It is a special case of an epicycloid with one cusp.

In polar form, it is written as r = a(1 − cos θ) (the orientation can be reversed with r = a(1 + cos θ)). In Cartesian coordinates it satisfies (x^2 + y^2 − a x)^2 = a^2(x^2 + y^2). The area enclosed by the cardioid r = a(1 − cos θ) is A = (3/2)πa^2. The curve is symmetric about the axis of symmetry and has a single cusp at the origin.

A convenient parametric representation is x = a(2 cos t − cos 2t), y = a(2 sin t − sin 2t), with t in [0, 2π]. This form highlights its generation as an epicycloid: a circle of radius a rolling around another circle of radius a, with a point on the rolling circle tracing the cardioid. The cusp is at the origin, and the curve closes after one complete revolution.

Applications of the cardioid appear in acoustics and engineering, notably in cardioid microphone and speaker pickup patterns and in antenna radiation patterns for directional sensitivity or emission. It also appears in computer graphics and design as a simple, well-understood heart-shaped outline. The term cardioid reflects its heart-like appearance; it has been studied as a classical plane curve in mathematics since the early modern period.