Conversion of Cartesian to Barycentric coordinates.
Given the Cartesian coordinates of one or more points, compute the barycentric coordinates of these points with respect to a simplex.
cart2bary(X, P)
X |
Reference simplex in N dimensions represented by a N+1-by-N matrix |
P |
M-by-N matrix in which each row is the Cartesian coordinates of a point. |
Given a reference simplex in N dimensions represented by a N+1-by-N matrix an arbitrary point P in Cartesian coordinates, represented by a 1-by-N row vector, can be written as
P = β X
where β is an N+1 vector of the barycentric coordinates. A criterion on β is that
∑_iβ_i = 1
Now partition the simplex into its first N rows X_N and its N+1th row X_{N+1}. Partition the barycentric coordinates into the first N columns β_N and the N+1th column β_{N+1}. This allows us to write
P_{N+1} - X_{N+1} = β_N X_N + β_{N+1} X_{N+1} - X_{N+1}
which can be written
P_{N+1} - X_{N+1} = β_N(X_N - 1_N X_{N+1})
where 1_N is an N-by-1 matrix of ones. We can then solve for β_N:
β_N = (P_{N+1} - X_{N+1})(X_N - 1_N X_{N+1})^{-1}
and compute
β_{N+1} = 1 - ∑_{i=1}^Nβ_i
This can be generalised for multiple values of P, one per row.
M-by-N+1 matrix in which each row is the
barycentric coordinates of corresponding row of P. If the
simplex is degenerate a warning is issued and the function returns
NULL.
Based on the Octave function by David Bateman.
David Sterratt
## Define simplex in 2D (i.e. a triangle)
X <- rbind(c(0, 0),
c(0, 1),
c(1, 0))
## Cartesian cooridinates of points
P <- rbind(c(0.5, 0.5),
c(0.1, 0.8))
## Plot triangle and points
trimesh(rbind(1:3), X)
text(X[,1], X[,2], 1:3) # Label vertices
points(P)
cart2bary(X, P)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.