Kardioidset

Kardioidset is a term found in some geometric and computational contexts to denote a region associated with a cardioid curve. It is not a standardized object in mathematics, and uses of the term can vary between sources. In the most common interpretation, a kardioidset refers to the filled interior of a cardioid described by a polar equation, together with its boundary.

The typical definition uses polar coordinates with a > 0. The cardioid boundary is given by r = a(1 − cos θ). The kardioidset then consists of all points with 0 ≤ r ≤ a(1 − cos θ) for 0 ≤ θ < 2π. The boundary curve forms a heart-shaped locus with a cusp at the origin, while the interior is a simply connected region.

Key properties of this kardioidset include its area and perimeter. The area of the cardioid interior defined by r = a(1 − cos θ) is (3/2)πa^2. The boundary length (arc length) of the cardioid is 16a. In Cartesian form, the boundary can be parameterized as x = a(1 − cos θ)cos θ and y = a(1 − cos θ)sin θ.

Variants and usage of the term can extend beyond the strict interior of the standard cardioid. Some sources describe kardioidsets as any region bounded by cardioid-shaped curves, or as masks and regions produced by cardioid-based mappings in computer graphics and signal processing. Because the term is not universally standardized, definitions may differ across disciplines.

See also: cardioid, polar coordinates, planar curves, image masks.