R: Conversion of Cartesian to Barycentric coordinates.

cart2bary {geometry} | R Documentation

## 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 = 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.

### 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)

[Package geometry version 0.4.0 Index]