Triangulate a two-dimensional polygon.

This algorithm decomposes a general polygon into simple polygons and uses the “ear-clipping” algorithm to triangulate it. Polygons with holes are supported.

triangulate(x, y = NULL, z = NULL, random = TRUE, plot = FALSE, partial = NA)
x, y, z

Coordinates of a two-dimensional polygon in a format supported by xyz.coords. See Details for how z is handled.


Whether to use a random or deterministic triangulation.


Whether to plot the triangulation; mainly for debugging purposes.


If the triangulation fails, should partial results be returned?


Normally triangulate looks only at the x and y coordinates. However, if one of those is constant, it is replaced with the z coordinate if present.

The algorithm works as follows. First, it breaks the polygon into pieces separated by NA values in x or y. Each of these pieces should be a simple, non-self-intersecting polygon, separate from the other pieces. (Though some minor exceptions to this rule may work, none are guaranteed). The nesting of these pieces is determined.

The “outer” polygon(s) are then merged with the polygons that they immediately contain, and each of these pieces is triangulated using the ear-clipping algorithm.

Finally, all the triangulated pieces are put together into one result.


A three-by-n array giving the indices of the vertices of each triangle. (No vertices are added; only the original vertices are used in the triangulation.)

The array has an integer vector attribute "nextvert" with one entry per vertex, giving the index of the next vertex to proceed counter-clockwise around outer polygon boundaries, clockwise around inner boundaries.


Not all inputs will succeed, even when a triangulation is possible. Generally using random = TRUE will find a successful triangulation if one exists, but it may occasionally take more than one try.


See the Wikipedia article “polygon triangulation” for a description of the ear-clipping algorithm.

See Also

extrude3d for a solid extrusion of a polygon, polygon3d for a flat display; both use triangulate.

  • triangulate
theta <- seq(0, 2*pi, len = 25)[-25]
theta <- c(theta, NA, theta, NA, theta, NA, theta, NA, theta)
r <- c(rep(1.5, 24), NA, rep(0.5, 24), NA, rep(0.5, 24), NA, rep(0.3, 24), NA, rep(0.1, 24))
dx <- c(rep(0, 24), NA, rep(0.6, 24), NA, rep(-0.6, 24), NA, rep(-0.6, 24), NA, rep(-0.6, 24))
x <- r*cos(theta) + dx
y <- r*sin(theta)
plot(x, y, type = "n")
polygon(x, y)
triangulate(x, y, plot = TRUE)
polygon3d(x, y, x - y, col = "red")
# }
Documentation reproduced from package rgl, version 0.100.50, License: GPL

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