cart2bary: 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)

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.

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.

Author

David Sterratt

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 $N+1$th row $X_{N+1}$. Partition the barycentric coordinates into the first $N$ columns $\beta_N$ and the $N+1$th 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.

Examples

Run this code

## 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)

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