caridoid

Caridoid, often intended to refer to the cardioid, is a heart-shaped plane curve that belongs to the family of epicycloids. The term cardioid derives from the Greek kardia, meaning heart. In some contexts, the word is misspelled as caridoid.

In polar form, a cardioid is described by r = a(1 − cos θ), yielding a cusp at the origin and symmetry about the x-axis. The orientation can be flipped with r = a(1 + cos θ). Its Cartesian equation is (x^2 + y^2 − a x)^2 = a^2 (x^2 + y^2). A common parametric representation is x = a(2 cos t − cos 2t), y = a(2 sin t − sin 2t).

Key properties include a single cusp, smooth interior otherwise, and symmetry about the x-axis. The curve is closed and simple. Its area is A = (3/2)πa^2, and its arc length is L = 8a. The curve can be generated by a circle of radius a rolling around another circle of the same radius; a point on the circumference traces the cardioid.

Variations of the cardioid arise from changing orientation or radius ratios, leading to r = a(1 ± cos θ) and related forms. In applications, cardioid shapes appear in acoustics and antenna theory, where cardioid patterns describe directional sensitivity or radiation. The cardioid also arises in optics and caustics and serves as a common example in studies of plane curves and geometric loci.