# DelaunayMesh

DelaunayMesh[{p1,p2,}]

gives a MeshRegion representing the Delaunay mesh from the points p1, p2, .

# Details and Options

• DelaunayMesh is also known as Delaunay triangulation and Delaunay tetrahedralization.
• A Delaunay mesh consists of intervals (in 1D), triangles (in 2D), tetrahedra (in 3D), and -dimensional simplices (in D).
• A Delaunay mesh has simplex cells defined by points, such that the circumsphere for the same points contains no other points from the original points pi.
• The Delaunay mesh gives a triangulation where the minimum interior angle is maximized.
• DelaunayMesh takes the same options as MeshRegion.

# Examples

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## Basic Examples(4)

A 1D Delaunay mesh:

A 2D Delaunay mesh from a list of points:

A 3D Delaunay mesh from a list of points:

Delaunay mesh from points corresponding to minimal vectors of the hexagonal close packing lattice:

## Scope(3)

Create a 1D Delaunay mesh from a set of points:

Basic properties:

Delaunay meshes are full dimensional:

Delaunay meshes are bounded:

Find its measure and centroid:

Find nearest distance and nearest point:

Create a 2D Delaunay mesh from a set of points:

Basic properties:

Delaunay meshes are full dimensional:

Delaunay meshes are bounded:

Find its area and centroid:

Test for point membership or distance to the closest point in the region:

Create a 3D Delaunay mesh from a set of points:

Basic properties:

Delaunay meshes are full dimensional:

Delaunay meshes are bounded:

Find its area and centroid:

Test for point membership or distance to the closest point in the region:

## Options(11)

### MeshCellHighlight(2)

MeshCellHighlight allows you to specify highlighting for parts of a DelaunayMesh:

Individual cells can be highlighted using their cell index:

Or by the cell itself:

### MeshCellLabel(2)

MeshCellLabel can be used to label parts of a DelaunayMesh:

Individual cells can be labeled using their cell index:

Or by the cell itself:

### MeshCellMarker(1)

MeshCellMarker can be used to assign values to parts of a DelaunayMesh:

Use MeshCellLabel to show the markers:

### MeshCellShapeFunction(2)

MeshCellShapeFunction allows you to specify functions for parts of a DelaunayMesh:

Individual cells can be drawn using their cell index:

Or by the cell itself:

### MeshCellStyle(2)

MeshCellStyle allows you to specify styling for parts of a DelaunayMesh:

Individual cells can be highlighted using their cell index:

Or by the cell itself:

### PlotTheme(2)

Use a theme with grid lines and a legend:

Use a theme to draw a wireframe:

## Applications(5)

Generate lattice points of a 3D lattice basis:

Construct and visualize the mesh region:

Construct a 3D region from a point set:

Compare original region to Delaunay mesh:

Visualize the piecewise constant interpolation of city elevations in Colorado:

Voronoi mesh from city coordinates:

Create a function to map a given coordinate pair to the nearest known elevation:

Function to rescale elevation values to , suitable for color functions:

Piecewise constant contour plot of city elevations:

A similar plot can also be achieved with ListContourPlot:

Solve a PDE over a region defined by point set:

Create a mesh from selected points on a raster:

Initial locator points:

Function to convert a raster and a mesh region to polygons:

Function to create an overlay mesh:

Click the image to add and remove draggable vertices:

## Properties & Relations(7)

The output of DelaunayMesh is always a full-dimensional MeshRegion:

DelaunayMesh consists of intervals in 1D:

Triangles in 2D:

Tetrahedra in 3D:

The circumcircle for each triangle in a DelaunayMesh contains no other point:

Find circumcircles for all triangles:

Plot the circumcircles as disks:

The circumsphere for each tetrahedron in a DelaunayMesh contains no other point:

Find circumspheres for all tetrahedra:

Plot the circumspheres:

ConvexHullMesh is effectively the BoundaryMesh of a DelaunayMesh:

Use TriangulateMesh to retriangulate a region:

VoronoiMesh is the dual of the DelaunayMesh:

Each Voronoi cell has a single point from the original point set:

Wolfram Research (2014), DelaunayMesh, Wolfram Language function, https://reference.wolfram.com/language/ref/DelaunayMesh.html (updated 2015).

#### Text

Wolfram Research (2014), DelaunayMesh, Wolfram Language function, https://reference.wolfram.com/language/ref/DelaunayMesh.html (updated 2015).

#### CMS

Wolfram Language. 2014. "DelaunayMesh." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2015. https://reference.wolfram.com/language/ref/DelaunayMesh.html.

#### APA

Wolfram Language. (2014). DelaunayMesh. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/DelaunayMesh.html

#### BibTeX

@misc{reference.wolfram_2022_delaunaymesh, author="Wolfram Research", title="{DelaunayMesh}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/DelaunayMesh.html}", note=[Accessed: 03-June-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2022_delaunaymesh, organization={Wolfram Research}, title={DelaunayMesh}, year={2015}, url={https://reference.wolfram.com/language/ref/DelaunayMesh.html}, note=[Accessed: 03-June-2023 ]}