Angulaarset

Angulaarset is a theoretical construct in geometry used to describe sets of points defined by angular constraints relative to a fixed origin. In standard terms, given a fixed origin O and a specification of angular intervals I, a corresponding angulaarset A comprises points P whose polar coordinates (r, θ) satisfy θ ∈ I and 0 ≤ r ≤ R(θ), where R is a radial bound function defined for θ in I. Note that angulaarset is not a widely established term in mainstream geometry; the concept is used here to illustrate how angular constraints can carve out planar regions.

Construction and variants: The angular specification I may be a single interval or a union of disjoint intervals, producing a range of sector-like regions. The boundary of A is defined by the function R(θ); if R is constant, A is a union of circular sectors, while a variable R(θ) yields a boundary that follows the curve r = R(θ). By combining multiple angular intervals and radial bounds, diverse, wedge-shaped regions can be created.

Properties: Angulaarsets are generally non-convex unless I is a single interval with a monotone radial bound. They are closed under finite union and intersection when angular intervals are combined coherently, and under positive radial scaling (multiplying R by a positive factor expands or contracts A about the origin).

Applications: The concept serves as an instructional tool in computational geometry, robotics planning, and data visualization, where angular constraints model directional reach, sensor coverage, or visibility fields.

See also: angular sector, polar coordinates, sector, wedge.

References: As a theoretical construct, formal treatments are limited; angulaarset appears mainly in didactic contexts and exploratory discussions.