radiiand

Radiiand is a geometric concept describing a planar region defined by a central point and a radial distance function. In this framework, a radiiand is the set of points whose distance from the center does not exceed a prescribed function of angle, making the boundary a polar curve r = f(φ).

Formally, choose a center as the origin and an angular interval [α, β]. A radiiand is the region R = { (r, φ) | φ ∈ [α, β], 0 ≤ r ≤ f(φ) }, where f is a nonnegative, typically continuous function. If α and β cover 0 to 2π and f is continuous, R is a closed, star-shaped region with respect to the center. The boundary of a radiiand is the polar curve r = f(φ). The area of a radiiand is given by A = (1/2) ∫α to β [f(φ)]^2 dφ.

Examples of radiiands include circles (f(φ) constant), limaçons (f(φ) = a + b cos φ), and many rose or irregular shapes produced by other angular functions. The radiiand framework unifies radial shapes under a single polar-description approach, facilitating analysis and rendering.

Applications of radiiand concepts appear in computer graphics, geographic information systems, and robotics, where shapes are naturally described by distance from a center rather than by Cartesian coordinates. In practice, radiiands serve as convenient models for radial features, collision detection, and area estimation when the boundary can be expressed as a function of angle. The term radiiand is used variably, and some authors refer to the same objects as polar regions or radial domains.