Conversion of Cartesian to Barycentric coordinates.

Description

Given the Cartesian coordinates of one or more points, compute the barycentric coordinates of these points with respect to a simplex.

Usage

cart2bary(X, P)

Arguments

Details

Given a reference simplex in N dimensions represented by a N+1-by-N matrix an arbitrary point mathbf{P} in Cartesian coordinates, represented by a 1-by-N row vector, can be written as

mathbf{P} = mathbf{β}mathbf{X}

where mathbf{β} is a N+1 vector of the barycentric coordinates. A criterion on mathbf{β} is that

∑_iβ_i = 1

Now partition the simplex into its first N rows mathbf{X}_N and its N+1th row mathbf{X}_{N+1}. Partition the barycentric coordinates into the first N columns mathbf{β}_N and the N+1th column β_{N+1}. This allows us to write

mathbf{P - X}_{N+1} = mathbf{β}_Nmathbf{X}_N + mathbf{β}_{N+1}mathbf{X}_{N+1} - mathbf{X}_{N+1}

which can be written

mathbf{P - X}_{N+1} = mathbf{β}_N(mathbf{X}_N - mathbf{1}mathbf{X}_{N+1})

where mathbf{1} is a N-by-1 matrix of ones. We can then solve for mathbf{β}_N:

mathbf{β}_N = (mathbf{P - X}_{N+1})(mathbf{X}_N - mathbf{1}mathbf{X}_{N+1})^{-1}

and compute

β_{N+1} = 1 - ∑_{i=1}^Nβ_i

This can be generalised for multiple values of mathbf{P}, one per row.

Value

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.

Note

Based on the Octave function by David Bateman.

Author(s)

David Sterratt

See Also

bary2cart

Examples

## 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)

Results