Documentation

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.

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

expand all

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

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

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

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

expand all

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.

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.

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

collapse all

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

P = gallery('uniformdata',[30 2],0);
DT = delaunayTriangulation(P)
DT =
delaunayTriangulation with properties:

Points: [30x2 double]
ConnectivityList: [50x3 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') Create a 3-D delaunayTriangulation object for 30 random points.

x = gallery('uniformdata',[30 1],0);
y = gallery('uniformdata',[30 1],1);
z = gallery('uniformdata',[30 1],2);
DT = delaunayTriangulation(x,y,z)
DT =
delaunayTriangulation with properties:

Points: [30x3 double]
ConnectivityList: [111x4 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)) 