cart2bary {geometry}

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

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.

Details

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 = \beta X

where \beta is an N+1 vector of the barycentric coordinates. A criterion on \beta is that

\sum_i\beta_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 \beta_N and the N+1th column \beta_{N+1}. This allows us to write

P_{N+1} - X_{N+1} = \beta_N X_N + \beta_{N+1} X_{N+1} - X_{N+1}

which can be written

P_{N+1} - X_{N+1} = \beta_N(X_N - 1_N X_{N+1})

where 1_N is an N-by-1 matrix of ones. We can then solve for \beta_N:

\beta_N = (P_{N+1} - X_{N+1})(X_N - 1_N X_{N+1})^{-1}

and compute

\beta_{N+1} = 1 - \sum_{i=1}^N\beta_i

This can be generalised for multiple values of 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 coordinates 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)