Baricèntrica

Baricèntrica refers to barycentric coordinates, a coordinate system in geometry. In barycentric coordinates, a point is expressed as a weighted average of a set of other points, called the reference points or vertices. For example, in a triangle with vertices A, B, and C, a point P can be represented as P = uA + vB + wC, where u, v, and w are the barycentric coordinates. The sum of these coordinates (u + v + w) is typically normalized to 1.

The name "barycentric" comes from the Greek words "barys" (heavy) and "kentron" (center), reflecting the concept's origin in the physics of centers of mass. If the barycentric coordinates represent masses placed at the vertices, the point P is the center of mass of this system. This property makes barycentric coordinates useful in physics and engineering for calculating centers of mass and related quantities.

In geometry, barycentric coordinates offer an alternative to Cartesian coordinates. They are particularly useful for operations that involve affine transformations, such as interpolation, and for describing points inside or on the boundary of a simplex (a triangle in 2D, a tetrahedron in 3D, etc.). For a point P to be inside a triangle with vertices A, B, and C, its barycentric coordinates (u, v, w) must all be non-negative (u >= 0, v >= 0, w >= 0) and sum to 1 (u + v + w = 1). This property makes them invaluable in computer graphics for interpolating vertex attributes like color or texture coordinates across polygons.