DT = delaunay(P) creates a 2-D or 3-D Delaunay triangulation from the points in a matrix P. The output DT is a three-column (for two dimensions) or four-column (for three dimensions) matrix where each row contains the row indices of the input points that make up a triangle or tetrahedron in the triangulation.
For more functionality with Delaunay triangulations, consider creating a delaunayTriangulation object and using object functions to compute geometric quantities and query properties of the triangulation.
Delaunay triangulation example (red-yellow points are the data points from which the triangulation was generated):EXAMPLECommands used with the North Carolina dataset to create the above figure (subset shown in figure).g.region n=220750 s=219950 w=638300 e=639000 -pv.delaunay input=elev_lid792_randpts output=elev_lid792_randpts_delaunayREFERENCESLeonid Guibas and Jorge Stolfi, (1985). Primitives for the Manipulation of General Subdivisions and the Computation ofVoronoi Diagrams, ACM Transactions on Graphics, Vol 4, No. 2, April 1985, Pages 74-123SEE ALSOv.voronoi, v.hullAUTHORSMartin Pavlovsky, Google Summer of Code 2008, StudentPaul Kelly, MentorBased on "dct" by Geoff Leach, Department of Computer Science, RMIT.SOURCE CODE Available at: v.delaunay source code (history)
The furthest-site Delaunay triangulation corresponds to the upper facets of the Delaunay construction.Its vertices are theextreme points of the input sites.It is the dual of the furthest-site Voronoi diagram. Example: rbox 10 D2 | qdelaunay Qu Qt s i TO result Compute the 2-d, furthest-site Delaunay triangulation of 10 random points. Triangulate the output. Write a summary to the console and the regions to 'result'. Example: rbox 10 D2 | qdelaunay Qu QJ s i TO result Compute the 2-d, furthest-site Delaunay triangulation of 10 random points. Joggle the input to guarantee triangular output. Write a summary to the console and the regions to 'result'. Example: rbox r y c G1 D2 | qdelaunay Qu s Fv TO result Compute the 2-d, furthest-site Delaunay triangulation of a triangle inside a square. Write a summary to the console and unoriented regions to 'result'. Merge regions for cocircular input sites (e.g., the square). The square is the only furthest-site Delaunay region.As with the Delaunay triangulation, Qhull computes thefurthest-site Delaunay triangulation by lifting the input sites to aparaboloid. The lower facets correspond to the Delaunaytriangulation while the upper facets correspond to thefurthest-site triangulation. Neither triangulation includes"vertical" facets (i.e., facets whose last hyperplanecoefficient is nearly zero). Vertical facets correspond to inputsites that are coplanar to the convex hull of the input. Anexample is points on the boundary of a lattice.
By default, qdelaunay merges cocircular and cospherical regions.For example, the furthest-site Delaunay triangulation of a square inside a diamond('rbox D2 c d G4 | qdelaunay Qu') consists of one region (the diamond).If you use 'Qt' (triangulated output),all furthest-site Delaunay regions will be simplicial (e.g., triangles in 2-d).Some regions may bedegenerate and have zero area.If you use 'QJ' (joggled input), all furthest-siteDelaunay regionswill be simplicial (e.g., triangles in 2-d). Joggled inputis less accurate than triangulated output ('Qt'). See Merged facets or joggled input.
Use I/O redirection (e.g., qdelaunay Qu < data.txt), a pipe (e.g., rbox 10 | qdelaunay Qu),or the 'TI' option (e.g., qdelaunay Qu TI data.txt).For example, this is a square containing four random points.Its furthest-site Delaunaytriangulation contains one square.rbox c 4 D2 > data2 RBOX c 4 D28-0.4999921736307369 -0.36846221179558170.2556053225468894 -0.04134986786297510.0327672376602583 -0.2810408135699488-0.452955383763607 0.17886471718444 -0.5 -0.5 -0.5 0.5 0.5 -0.5 0.5 0.5qdelaunay Qu i < dataFurthest-site Delaunay triangulation by the convex hull of 8 points in 3-d: Number of input sites: 8 Number of Delaunay regions: 1 Number of non-simplicial Delaunay regions: 1Statistics for: RBOX c 4 D2 | QDELAUNAY s Qu i Number of points processed: 8 Number of hyperplanes created: 20 Number of facets in hull: 11 Number of distance tests for qhull: 34 Number of merged facets: 1 Number of distance tests for merging: 107 CPU seconds to compute hull (after input): 0.0217 6 4 5»furthest-site qdelaunayoutputsThese options control the output of furthest-site Delaunay triangulations:
The furthest-site Delaunay triangulation does notrecord coincident input sites. Use qdelaunay instead.qdelaunay Qu does not work for purely cocircularor cospherical points (e.g., rbox c | qdelaunay Qu). Instead,use qdelaunay Qz -- when all points are vertices of the convexhull of the input sites, the Delaunay triangulation is the sameas the furthest-site Delaunay triangulation.A non-simplicial, furthest-site Delaunay region indicates nearly cocircular orcospherical input sites. To avoid non-simplicial regions triangulatethe output ('Qt') or jogglethe input ('QJ'). Joggled inputis less accurate than triangulated output.You may also triangulatenon-simplicial regions with option 'Ft'. It addsthe centrum to non-simplicial regions. Alternatively, use an exact arithmetic code.
Furthest-site Delaunay triangulations do not include facets that arecoplanar with the convex hull of the input sites. A facet iscoplanar if the last coefficient of its normal isnearly zero (see qh_ZEROdelaunay).»furthest-site qdelaunay conventionsThe following terminology is used for furthest-site Delaunaytriangulations in Qhull. The underlying structure is the upperfacets of a convex hull in one higher dimension. See convex hull conventions, Delaunay conventions,and Qhull's data structures 2b1af7f3a8