onsurface

Onsurface is an informal term used in geometry, computer graphics, and related fields to indicate that a point lies on the surface of a geometric object. A surface is typically a two-dimensional manifold embedded in three-dimensional space, though surfaces may be algebraic, parametric, or polygonal representations.

Common representations of surfaces include implicit, parametric, and polygonal forms. An implicit surface is given by a scalar function F(x, y, z) = 0, where a point p = (x, y, z) is on the surface if F(p) = 0. A parametric surface uses a mapping X(u, v) = (x(u, v), y(u, v), z(u, v)); a point is on the surface if there exist parameters (u, v) such that X(u, v) = p. In practice, many surfaces are approximated by triangulated meshes, which consist of polygonal facets whose union approximates the underlying surface.

Determining whether a point is on the surface involves tolerance handling due to numerical computation. For implicit surfaces, one checks |F(p)| ≤ ε. For parametric surfaces, one may compute the closest point on the surface to p and verify that the distance is within a chosen tolerance; projection methods may also be used. Robust on-surface tests are important in rendering, collision detection, ray tracing, mesh processing, and surface fitting.

In applications, the concept aids tasks such as intersection tests, surface registration, and geometric queries. The term is often used informally in software APIs and algorithm descriptions to denote a predicate or check that a point lies on the given surface. See also implicit surface, parametric surface, and distance field.