barycoords {tigers}

Computes Barycentric Coordinates

Description

The barycentric coordinates of a point inside a polygon are weighted coordinates of the vertices of this polygon. The algorithm implemented in this function works for any concave or convex polygon (Hormann and Floater, 2006).

Usage

barycoords(XY, point)

Arguments

XY

A two-column matrix giving the coordinates of a polygon.

point

a vector with two values giving the coordinates of a point.

Details

If the polygon is a triangle, the trilinear2Cartesian can be used instead.

The polygon must be open (see is.open), and can be either in clockwise or in counterclockwise order (see is.clockwise).

For the moment, the function is not vectorized with respect to point, so it must be called for each point separately (see examples). This is likely to change in the future.

Value

a numeric vector giving the barycentric coordinates of the point (second argument). The length of the returned vector is equal to the number of vertices in the polygon (first argument).

Author(s)

Emmanuel Paradis

References

Hormann, K. and Floater, M. S. (2006) Mean value coordinates for arbitrary planar polygons. ACM Transactions on Graphics 25, 1424–1441. <doi:10.1145/1183287.1183295>

See Also

trilinear2Cartesian

Examples

## a square:
xy <- cbind(c(0, 1, 1, 0), c(0, 0, 1, 1))

## a small function to get the coordinates directly:
f <- function(Pxy) barycoords(xy, Pxy)
## the CMYK scale:
F <- col2rgb(c("cyan", "magenta", "yellow", "black"))

n <- 1e5L
## random points in the square
Pxys <- matrix(runif(2 * n), n, 2)
system.time(res <- t(apply(Pxys, 1, f))) # < 1 sec
colnames(res) <- as.character(1:4)

## all rows should (approximately) sum to one:
all.equal(rowSums(res), rep(1, n), tol = 1e-15)

## transform the barycentric coordinates into colours:
COLS <- t(F %*% t(res)) / 255
rgbCOLS <- apply(COLS, 1, function(x) do.call(rgb, as.list(x)))
## add transparency:
rgbCOLS <- paste0(rgbCOLS, "33")
## plot the results:
plot(0:1, 0:1, "n", asp = 1, ann = FALSE, axes = FALSE)
points(Pxys, pch = ".", col = rgbCOLS, cex = 20)
## the visual effect is nicer with n <- 1e6L above and cex = 7
## in the last command


## the example below follows the same logic than the previous one

## an 8-vertex polygon:
xy <- cbind(c(0, 0.5, 1, 3, 1, 0.5, 0, -2),
            c(0, -2, 0, 0.5, 1, 3, 1, 0.5))

## random points in the square and in the 4 triangles:
Pxys <- rbind(matrix(runif(2 * n), n, 2),
              rpit(n, xy[1:3, ]),
	      rpit(n, xy[3:5, ]),
	      rpit(n, xy[5:7, ]),
	      rpit(n, xy[c(7:8, 1), ]))

system.time(res <- t(apply(Pxys, 1, f))) # < 5 sec

colnames(res) <- as.character(1:8)
F <- col2rgb(c("black", "red", "orange", "green",
               "yellow", "blue", "purple", "white"))
## F <- col2rgb(rainbow(8)) # alternative
COLS <- t(F %*% t(res)) / 255.001
rgbCOLS <- apply(COLS, 1, function(x) do.call(rgb, as.list(x)))
rgbCOLS <- paste0(rgbCOLS, "33") # add transparency
plot(xy, , "n", asp = 1, ann = FALSE, axes = FALSE)
points(Pxys, pch = ".", col = rgbCOLS, cex = 5)