Barycentric coordinates of points within a triangle

The weights that say how much of each corner a point is made of. They sum to one, and they are all non-negative exactly when the point is inside the triangle – which makes them a point-in-triangle test and an interpolation rule at the same time.

Usage

bary_weights(triangle, xy)

Arguments

triangle

three corners, as a 3 by 2 matrix

xy

points to place within it, a two column matrix

Value

A matrix with one row per point and three columns, one weight per corner of triangle.

Details

This is the whole of what geometry::tsearch(bary = TRUE) computes in C, written out so it can be read. For a triangle with corners \((x_1,y_1)\), \((x_2,y_2)\), \((x_3,y_3)\) and a point \((x,y)\), solve for the weights by Cramer's rule on

$$w_1 (x_1,y_1) + w_2 (x_2,y_2) + w_3 (x_3,y_3) = (x,y), \quad w_1 + w_2 + w_3 = 1$$

Estimating a value is then one line: the weighted sum of the corner values.

See also

find_triangle(), which uses this to locate points, and grid_barycentric(), which uses it to interpolate.

Examples

tri <- cbind(c(0, 1, 0), c(0, 0, 1))
## the corners themselves
bary_weights(tri, tri)
#>      [,1] [,2] [,3]
#> [1,]    1    0    0
#> [2,]    0    1    0
#> [3,]    0    0    1

## the centroid is one third of each
bary_weights(tri, cbind(1/3, 1/3))
#>           [,1]      [,2]      [,3]
#> [1,] 0.3333333 0.3333333 0.3333333

## outside the triangle, a weight goes negative
bary_weights(tri, cbind(1, 1))
#>      [,1] [,2] [,3]
#> [1,]   -1    1    1