pointsrepresented

Pointsrepresented is a term used in mathematics, computer graphics, and geometric data structures to denote the set of points that a particular object, surface, or dataset stands for or stores. In a broad sense, it refers to the collection of coordinates that describe a shape or space.

Definition and scope: Pointsrepresented can be finite, such as the vertex coordinates of a polygon mesh, or infinite, as in the image of a parametric surface map. For a discrete mesh, pointsrepresented often corresponds to the mesh vertices. For a parametric surface S(u,v), the pointsrepresented are the points in space obtained by evaluating S over its parameter domain, producing a continuum of points.

Coordinate representations: Points can be expressed in various coordinate systems, with Cartesian (x,y,z) being common in three dimensions. Other systems, such as homogeneous coordinates or polar representations, are used for specific computations. The chosen representation affects storage, precision, and algorithms used to manipulate the point set.

Implicit and explicit forms: In implicit representations, a point lies in the represented set if it satisfies an equation F(x)=0. In explicit or parametric forms, pointsrepresented are the outputs of a mapping from parameters to space. Both forms aim to capture the same underlying geometry, but differ in how the points are generated and stored.

Applications: The concept appears in rendering, collision detection, sampling, and processing of point clouds. It is central to workflows that convert continuous surfaces into discrete samples, as well as to operations that query, transform, or analyze a set of spatial points.

See also: point set, point cloud, mesh, parametric surface, implicit surface, sampling.