Delaunay triangulation in N dimensions

The Delaunay triangulation is a tessellation of the convex hull of the points such that no \(N\)-sphere defined by the \(N\)- triangles contains any other points from the set.

math, dplot, graphs
delaunayn(p, options = NULL, output.options = NULL, full = FALSE)

An \(M\)-by-\(N\) matrix whose rows represent \(M\) points in \(N\)-dimensional space.


String containing extra control options for the underlying Qhull command; see the Qhull documentation (../doc/qhull/html/qdelaun.html) for the available options.

The Qbb option is always passed to Qhull. The default options are Qcc Qc Qt Qz for \(N<4\) and Qcc Qc Qt Qx for \(N>=4\). If neither of the QJ or Qt options are supplied, the Qt option is passed to Qhull. The Qt option ensures all Delaunay regions are simplical (e.g., triangles in 2D). See ../doc/qhull/html/qdelaun.html for more details. Contrary to the Qhull documentation, no degenerate (zero area) regions are returned with the Qt option since the R function removes them from the triangulation.

If options is specified, the default options are overridden. It is recommended to use output.options for options controlling the outputs.


String containing Qhull options to control output. Currently Fn (neighbours) and Fa (areas) are supported. Causes an object of return value for details. If output.options is TRUE, select all supported options.


Deprecated and will be removed in a future release. Adds options Fa and Fn.


If neither of the Qhull options Fn or Fa are specified, return the Delaunay triangulation as a matrix with \(M\) rows and \(N+1\) columns in which each row contains a set of indices to the input points p. Thus each row describes a simplex of dimension \(N\), e.g. a triangle in 2D or a tetrahedron in 3D.

If the output.options argument contains Fn or Fa, return a list with class delaunayn comprising the named elements:


The Delaunay triangulation described above


If Fa is specified, an \(M\)-dimensional vector containing the generalise area of each simplex (e.g. in 2D the areas of triangles; in 3D the volumes of tetrahedra). See ../doc/qhull/html/qh-optf.html#Fa.


If Fn is specified, a list of neighbours of each simplex. See ../doc/qhull/html/qh-optf.html#Fn


This function interfaces the Qhull library and is a port from Octave ( to R. Qhull computes convex hulls, Delaunay triangulations, halfspace intersections about a point, Voronoi diagrams, furthest-site Delaunay triangulations, and furthest-site Voronoi diagrams. It runs in 2D, 3D, 4D, and higher dimensions. It implements the Quickhull algorithm for computing the convex hull. Qhull handles round-off errors from floating point arithmetic. It computes volumes, surface areas, and approximations to the convex hull. See the Qhull documentation included in this distribution (the doc directory ../doc/qhull/index.html).

Qhull does not support constrained Delaunay triangulations, triangulation of non-convex surfaces, mesh generation of non-convex objects, or medium-sized inputs in 9D and higher. A rudimentary algorithm for mesh generation in non-convex regions using Delaunay triangulation is implemented in distmesh2d (currently only 2D).


Barber, C.B., Dobkin, D.P., and Huhdanpaa, H.T., “The Quickhull algorithm for convex hulls,” ACM Trans. on Mathematical Software, Dec 1996.

See Also

tri.mesh, convhulln, surf.tri, distmesh2d

  • delaunayn
# example delaunayn
d <- c(-1,1)
pc <- as.matrix(rbind(expand.grid(d,d,d),0))
tc <- delaunayn(pc)

# example tetramesh
# }
tetramesh(tc,pc, alpha=0.9)
# }
tc1 <- delaunayn(pc, output.options="Fa")
## sum of generalised areas is total volume of cube

# }
Documentation reproduced from package geometry, version 0.4.1, License: GPL (>= 3)

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