barycentric coordinates <coordinate geometry>
point in a n-dimension coordinate system using `n+1` numbers, \([u_0, u_1, u_2, u_3, ... , u_n] \backepsilon [[0 \leq u_i \leq 1] \wedge \sum u_i = 1.0]\), in which the location of a point of an n-simplex (of any dimension) is specified by a weighted center of mass of equal masses placed at its vertices using vector algebra of the `RR^n` used in the coordinate reference system
Note to entry: Even though there are `n+1` coordinates in a barycentric coordinate system, the topological dimension is n, since the restriction (sums to 1.0) loses 1 degree of freedom (once you have n ordinates, the remaining one is determined such as `u_n = 1.0 − sum_(i=0)^(n-1) u_i`. he coordinates for the simplex are all non-negative, but the system can be extended outside of the simples by using negative numbers. If the ordinates are all positive, then the point is inside (interior to) the n-simplex. If one of them is 1.0 and the other 0, this is a corner of the simplex. If one of them is zero and the others still each greater than or equal to zero, the point is on the n-1-simplex opposite the vertex zeroed out. If any are negative, the point is outside of the simplex. The coordinates are dependent on the underlying coordinate reference system of the source data.
ORIGIN: ISO/TC 211 Glossary of Terms - English (last updated: 2020-06-02)