CircleAreaField

CircleAreaField is a mathematical construct used in computational geometry and related fields to describe a scalar field derived from the geometry of a planar region. Given a compact region R in the plane with boundary ∂R, for every point p in R the function r(p) is defined as the Euclidean distance from p to ∂R. The CircleAreaField at p is defined as F(p) = π [r(p)]^2. Equivalently, F(p) equals the area of the largest circle centered at p that lies entirely within R.

The domain of F is R, and F vanishes on the boundary since r(p) = 0 there. The maximum value occurs at points farthest from ∂R, where r(p) equals the inradius of R; the corresponding maximum area is π ρ^2. The field thus encodes the shape of R through the distance-to-boundary function.

Properties and relationships: F is continuous and 1-Lipschitz; away from sharp corners its gradient exists almost everywhere. It is closely related to the Euclidean distance transform and to the medial axis of R. For simple regions such as rectangles or disks, closed-form expressions may be available or easily computed. Variants include using the signed distance to the boundary or restricting the domain to a subregion.

Computations and applications: F can be computed by standard distance-transform algorithms on grids or meshes. Applications include morphological erosion and buffering in image processing, collision detection and shading in computer graphics, and region-based analysis in geographic information systems.

See also: distance transform, signed distance function, medial axis, morphological erosion.