cart2bary: Conversion of Cartesian to Barycentric coordinates.

In geometry: Mesh Generation and Surface Tessellation

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)

geometry documentation built on Sept. 11, 2024, 9 p.m.