Barycentriska

Barycentriska refers to barycentric coordinates, a coordinate system used to describe a point within a simplex in terms of its relationship to the vertices of that simplex. A simplex is a generalization of a triangle or tetrahedron to any number of dimensions. For a triangle, the vertices are A, B, and C. Any point P inside or on the boundary of the triangle can be represented as a weighted average of these vertices.

The barycentric coordinates of a point P with respect to a simplex with vertices V0, V1, ..., Vn are a set of n+1 numbers, denoted as (λ0, λ1, ..., λn), such that P = λ0*V0 + λ1*V1 + ... + λn*Vn. A crucial property is that the sum of these coordinates is always equal to one: λ0 + λ1 + ... + λn = 1.

If all the barycentric coordinates are non-negative (λi ≥ 0 for all i), then the point P lies within the simplex. If any coordinate is negative, the point lies outside the simplex. The coordinates have a geometric interpretation: they represent the relative "mass" that needs to be placed at each vertex to balance the system at the point P. For a triangle, if a mass is placed at each vertex, the barycentric coordinate corresponding to a vertex is proportional to the area of the subtriangle formed by the opposite side and the point P, divided by the total area of the triangle.

Barycentric coordinates are widely used in various fields, including computer graphics for interpolation (e.g., color or texture coordinates), finite element analysis for discretizing domains, and computational geometry. Their independence from a specific coordinate system (like Cartesian) makes them useful for geometric transformations and invariant properties.