Barycentric coordinate system

In geometry, the barycentric coordinate system is a coordinate system in which the location of a point of a simplex (a triangle, tetrahedron, etc.) is specified as the center of mass, or barycenter, of usually unequal masses placed at its vertices. Coordinates also extend outside the simplex, where one or more coordinates become negative. The system was introduced in 1827 by August Ferdinand Möbius. In geometry, the barycentric coordinate system is a coordinate system in which the location of a point of a simplex (a triangle, tetrahedron, etc.) is specified as the center of mass, or barycenter, of usually unequal masses placed at its vertices. Coordinates also extend outside the simplex, where one or more coordinates become negative. The system was introduced in 1827 by August Ferdinand Möbius. Let x 1 , … , x n {displaystyle mathbf {x} _{1},ldots ,mathbf {x} _{n}} be the vertices of a simplex in an affine space A. If, for some point p {displaystyle mathbf {p} } in A, and at least one of a 1 , … , a n {displaystyle a_{1},ldots ,a_{n}} does not vanishthen we say that the coefficients ( a 1 , … , a n {displaystyle a_{1},ldots ,a_{n}} ) are barycentric coordinates of p {displaystyle mathbf {p} } with respect to x 1 , … , x n {displaystyle mathbf {x} _{1},ldots ,mathbf {x} _{n}} . The vertices themselves have the coordinates x 1 = ( 1 , 0 , 0 , … , 0 ) , x 2 = ( 0 , 1 , 0 , … , 0 ) , … , x n = ( 0 , 0 , 0 , … , 1 ) {displaystyle mathbf {x} _{1}=(1,0,0,ldots ,0),mathbf {x} _{2}=(0,1,0,ldots ,0),ldots ,mathbf {x} _{n}=(0,0,0,ldots ,1)} . Barycentric coordinates are not unique: for any b not equal to zero, ( b a 1 , … , b a n {displaystyle ba_{1},ldots ,ba_{n}} ) are also barycentric coordinates of p. When the coordinates are not negative, the point p {displaystyle mathbf {p} } lies in the convex hull of x 1 , … , x n {displaystyle mathbf {x} _{1},ldots ,mathbf {x} _{n}} , that is, in the simplex which has those points as its vertices. Barycentric coordinates, as defined above, are a form of homogeneous coordinates: indeed, the 'usual' homogeneous coordinates are the barycentric coordinates defined in the extended affine n-space on the simplex whose vertices are the points at infinity on the coordinate axes, plus the origin. Sometimes values of coordinates are restricted with a condition which makes them unique; then, they are affine coordinates. The classical terminology in this case is that of absolute barycentric coordinates. In the context of a triangle, barycentric coordinates are also known as area coordinates or areal coordinates, because the coordinates of P with respect to triangle ABC are equivalent to the (signed) ratios of the areas of PBC, PCA and PAB to the area of the reference triangle ABC. Areal and trilinear coordinates are used for similar purposes in geometry. Barycentric or areal coordinates are extremely useful in engineering applications involving triangular subdomains. These make analytic integrals often easier to evaluate, and Gaussian quadrature tables are often presented in terms of area coordinates. Consider a triangle T {displaystyle T} defined by its three vertices, r 1 {displaystyle mathbf {r} _{1}} , r 2 {displaystyle mathbf {r} _{2}} and r 3 {displaystyle mathbf {r} _{3}} . Each point r {displaystyle mathbf {r} } located inside this triangle can be written as a unique convex combination of the three vertices. In other words, for each r {displaystyle mathbf {r} } there is a unique sequence of three numbers, λ 1 , λ 2 , λ 3 ≥ 0 {displaystyle lambda _{1},lambda _{2},lambda _{3}geq 0} such that λ 1 + λ 2 + λ 3 = 1 {displaystyle lambda _{1}+lambda _{2}+lambda _{3}=1} and