Esurface

Esurface is a term used in computer graphics and differential geometry to denote a surface defined by an explicit parametric representation, as opposed to implicit surfaces defined by F(x,y,z) = 0. An esurface is typically described by a mapping S: D → R^3, where D is a two-dimensional parameter domain and S(u,v) = (x(u,v), y(u,v), z(u,v)). This explicit form supports straightforward sampling for rendering, ray tracing, and numerical analysis, and makes it easy to compute derivatives for shading and curvature.

Common realizations of esurfaces include Bezier patches, B-spline surfaces, NURBS, and other tensor-product or subdivision-based patches. They are often represented by a control grid and a corresponding parameterization, with continuity classes such as C^0, C^1, C^2 governing smoothness across patch boundaries. The surface can be edited by moving control points, changing knot vectors, or refining the parameter grid via subdivision.

Construction methods include lofting between boundary curves, interpolation of a point grid, or generating surfaces from a mesh of curves. Esurfaces are widely used in computer-aided design, computer graphics, and geometric modeling due to their intuitive control and exact evaluation.

Notes: The term esurface is not universally standardized; in many contexts explicit parametric surfaces are simply called parametric surfaces, rather than “esurfaces.” See also: parametric surface, Bezier surface, B-spline surface, NURBS.