cardioidi

A cardioid is a heart-shaped plane curve that is used in mathematics and applied fields. In the standard form it is a special type of epicycloid: a curve generated when a circle of radius r rolls around another circle of the same radius. The resulting curve has a single cusp and a smooth, rounded exterior.

In polar coordinates, a common representation is r = a(1 ± cos θ), where a > 0. The plus sign yields a cardioid facing left or right depending on orientation, and the minus sign yields the opposite orientation. In Cartesian coordinates, the cardioid with cusp at the origin satisfies (x^2 + y^2 − a x)^2 = a^2(x^2 + y^2). The curve is symmetric about the x-axis and has its maximum radius at θ = π, equal to 2a.

Key geometric properties include its single cusp, the heart-like shape, and a finite area given by A = (3/2)πa^2 for the standard form r = a(1 − cos θ). The arc length of this cardioid is L = 8a, a result that can be obtained from the polar arc-length integral.

Variants of the cardioid arise from rotation and reflection, yielding different orientations of the cusp and axis. It can also be described as an epicycloid generated by a circle of radius a rolling around an identical fixed circle, or as the envelope of circles with center on a cardioid.

Applications and occurrences span acoustics and signal processing, where cardioid patterns describe directional sensitivity in microphones and antennas. In mathematical contexts, cardioids serve as examples of cusped curves with explicit polar and Cartesian equations and well-defined area and length.