delaunayTriangulation

Delaunay triangulation in 2-D and 3-D

Description

Use the delaunayTriangulation object to create a 2-D or 3-D Delaunay triangulation from a set of points. For 2-D data, you can also specify edge constraints.

You can perform a variety of topological and geometric queries on a delaunayTriangulation, including any triangulation query. For example, locate a facet that contains a specific point, find the vertices of the convex hull, or compute the Voronoi Diagram.

Creation

To create a delaunayTriangulation object, use the delaunayTriangulation function with input arguments that define the triangulation's points and constrained edges.

Syntax

DT = delaunayTriangulation(P)

DT = delaunayTriangulation(P,C)

DT = delaunayTriangulation(x,y)

DT = delaunayTriangulation(x,y,C)

DT = delaunayTriangulation(x,y,z)

DT = delaunayTriangulation()

Description

example

DT = delaunayTriangulation(P) creates a Delaunay triangulation from the points in P. The matrix P has 2 or 3 columns, depending on whether your points are in 2-D or 3-D space.

DT = delaunayTriangulation(P,C) specifies the edge constraints in the matrix C for the 2-D points in P. Each row of C defines the start and end vertex IDs of a constrained edge. Vertex IDs are the row numbers of the corresponding vertices in the property DT.Points.

DT = delaunayTriangulation(x,y) creates a 2-D Delaunay triangulation from the point coordinates in the column vectors x and y.

DT = delaunayTriangulation(x,y,C) specifies the edge constraints in a matrix C.

example

DT = delaunayTriangulation(x,y,z) creates a 3-D Delaunay triangulation from the point coordinates in the column vectors x, y, and z.

DT = delaunayTriangulation() creates an empty Delaunay triangulation.

Input Arguments

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`P` — Points
matrix

Points, specified as a matrix whose columns are the x-coordinates, y-coordinates, and (possibly) z-coordinates of the triangulation points. The row numbers of P are the vertex IDs in the triangulation.

`x` — x-coordinates
column vector

x-coordinates of triangulation points, specified as a column vector.

`y` — y-coordinates
column vector

y-coordinates of triangulation points, specified as a column vector.

`z` — z-coordinates
column vector

z-coordinates of triangulation points, specified as a column vector.

`C` — Vertex IDs of constrained edges
2-column matrix

Vertex IDs of constrained edges, specified as a 2-column matrix. Each row of C corresponds to a constrained edge and contains two IDs:

C(j,1) is the ID of the vertex at the start of an edge.
C(j,2) is the ID of the vertex at end of the edge.

You can specify edge constraints for 2-D triangulations only.

Properties

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`Points` — Triangulation points
matrix

Points in the triangulation, represented as a matrix with the following characteristics:

Each row in DT.Points contains the coordinates of a vertex.
Each row number of DT.Points is a vertex ID.

`ConnectivityList` — Triangulation connectivity list
matrix

Triangulation connectivity list, represented as a matrix with the following characteristics:

Each element in DT.ConnectivityList is a vertex ID.
Each row represents a triangle or tetrahedron in the triangulation.
Each row number of DT.ConnectivityList is a triangle or tetrahedron ID.

`Constraints` — Constrained edges
2-column matrix of vertex IDs

Constrained edges, represented as a 2-column matrix of vertex IDs. Each row of DT.Constraints corresponds to a constrained edge and contains two IDs:

DT.Constraints(j,1) is the ID of the vertex at the start of an edge.
DT.Constraints(j,2) is the ID of the vertex at end of the edge.

DT.Constraints is an empty matrix when the triangulation has no constrained edges.

Object Functions

`convexHull`	Convex hull of Delaunay triangulation
`isInterior`	Query interior points of Delaunay triangulation
`voronoiDiagram`	Voronoi diagram of Delaunay triangulation
`barycentricToCartesian`	Convert coordinates from barycentric to Cartesian
`cartesianToBarycentric`	Convert coordinates from Cartesian to barycentric
`circumcenter`	Circumcenter of triangle or tetrahedron
`edgeAttachments`	Triangles or tetrahedra attached to specified edge
`edges`	Triangulation edges
`faceNormal`	Triangulation unit normal vectors
`featureEdges`	Handle sharp edges of triangulation
`freeBoundary`	Free boundary facets
`incenter`	Incenter of triangulation elements
`isConnected`	Test if two vertices are connected by an edge
`nearestNeighbor`	Closest vertex
`neighbors`	Triangle or tetrahedron neighbors
`pointLocation`	Triangle or tetrahedron enclosing point
`size`	Size of triangulation connectivity list
`vertexAttachments`	Triangles or tetrahedra attached to vertex
`vertexNormal`	Triangulation vertex normal

Examples

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2-D Delaunay Triangulation

Open Live Script

Create a 2-D delaunayTriangulation object for 30 random points.

rng default;
P = rand([30 2]);
DT = delaunayTriangulation(P)

DT = 
  delaunayTriangulation with properties:

              Points: [30x2 double]
    ConnectivityList: [48x3 double]
         Constraints: []

Compute the center points of each triangle, and plot the triangulation with the center points.

IC = incenter(DT);
triplot(DT)
hold on
plot(IC(:,1),IC(:,2),'*r')

3-D Delaunay Triangulation

Open Live Script

Create a 3-D delaunayTriangulation object for 30 random points.

rng default;
x = rand([30 1]);
y = rand([30 1]);
z = rand([30 1]);
DT = delaunayTriangulation(x,y,z)

DT = 
  delaunayTriangulation with properties:

              Points: [30x3 double]
    ConnectivityList: [102x4 double]
         Constraints: []

Plot the triangulation.

tetramesh(DT,'FaceAlpha',0.3);

Compute and plot the convex hull of the triangulation.

[K,v] = convexHull(DT);
trisurf(K,DT.Points(:,1),DT.Points(:,2),DT.Points(:,3))

More About

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