Cardioidal

Cardioidal is an adjective relating to a cardioid, a heart-shaped plane curve. The cardioid is a classical geometric form obtained by tracing a point on the circumference of a circle as it rolls around another circle of equal radius. It can also be described as a special limaçon with eccentricity one.

In polar coordinates, a cardioid is commonly given by r = a(1 − cos θ) or r = a(1 + cos θ), with a > 0, yielding two oppositely oriented copies. In Cartesian form this curve satisfies (x^2 + y^2 − a x)^2 = a^2 (x^2 + y^2). A standard parametric representation is x = a(2 cos t − cos 2t), y = a(2 sin t − sin 2t).

The cardioid has a cusp at the point where r = 0 (for r = a(1 − cos θ) at θ = 0) and is symmetric about the x-axis. It attains a maximum radius of 2a at θ = π. Its area is A = (3/2)πa^2, and its arc length is L = 8a. As a member of the family of limaçons, the cardioid is the e = 1 case, making it a classical example in the study of plane curves.

Cardioidal curves appear in various contexts, including optical caustics formed by circular mirrors and in acoustics and antenna design where cardioid radiation or reception patterns are desirable. The term cardioidal is used to describe curves of this class or other cardioid-like shapes. See also cardioid, limaçon, polar coordinates.