Barycentrictype

Barycentrictype is a term used in geometry and computer graphics to denote a data representation and associated operations for barycentric coordinates with respect to a simplex, such as a triangle in two dimensions or a tetrahedron in three dimensions. A barycentrictype encapsulates the weights assigned to the simplex vertices and provides methods for computing, normalizing, and applying these weights to interpolate attributes or evaluate linear functions.

In a typical implementation, a barycentrictype stores a set of barycentric weights w0, w1, w2 (and w3 in three-dimensional cases) that sum to one. It offers constructors from Cartesian coordinates, from known vertex weights, and functions to convert to and from Cartesian coordinates. It also supports operations such as affine combination of vertex attributes, interpolation of vertex colors or textures, and checking point containment within the simplex by testing nonnegative weights.

Applications include surface shading in computer graphics, texture mapping, finite element analysis, and geometric search algorithms that require robust point-in-simplex tests. Barycentrictype can be extended to higher dimensions and to generalized barycentric coordinates where weights may be negative or satisfy different constraints, enabling interpolation outside a simplex or on non-convex domains.

Relation to theory: It is closely related to barycentric coordinates, affine geometry, and mesh processing. While barycentric coordinates describe how a point decomposes relative to a simplex, barycentrictype focuses on a practical data representation and a set of accompanying operations.

See also: barycentric coordinates, generalized barycentric coordinates, affine transformation, interpolation, finite element method, mesh processing.