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VoronoiMesh—Wolfram Language Documentation

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• See Also
  • DelaunayMesh
  • ConvexHullMesh
  • MeshRegion
  • RegionNearest
• Related Guides
  • Mesh-Based Geometric Regions
  • Geometric Computation
  • Spatial Point Collections

VoronoiMesh[{p₁,…,p_n},{{x_min,x_max},…}]

clips the mesh to the bounds .

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

Each point corresponds to a Voronoi cell, which is an interval in the 1D case:

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

Each point corresponds to a Voronoi cell:

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

Basic properties:

Voronoi meshes are full-dimensional:

Voronoi meshes are bounded by their clipping values:

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

Basic properties:

Voronoi meshes are full-dimensional:

Voronoi meshes are bounded by their clipping values:

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

Individual cells can be highlighted using their cell index:

Or by the cell itself:

MeshCellLabel can be used to label parts of a VoronoiMesh:

Individual cells can be labeled using their cell index:

Or by the cell itself:

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

Individual cells can be highlighted using their cell index:

Or by the cell itself:

Use a theme with grid lines and a legend:

Use a theme to draw a wireframe:

Create an interactive Voronoi mesh with draggable points. Use Click on the Voronoi mesh to add and remove draggable points:

Voronoi meshes for simple point configurations including a grid:

A circle of points:

Create an interactive map of the closest large city in Italy. Start by getting names and coordinate data for large cities in Italy:

Nearest function for labeling region cells:

Generate a Voronoi mesh from city coordinates:

Create a tooltip map from the Voronoi mesh:

Mouse over the map to get the name of the closest large city:

Given a 2D lattice basis:

Generate lattice points:

Generate and visualize the Voronoi cells, also known as Brillouin zones:

Create a polygonal mosaic from an image:

Find the edges of the image:

Create a Voronoi mesh from the edge positions:

Recursively apply VoronoiMesh to the mean vertex positions of the precursive Voronoi cells to create a more uniform mesh:

Color each polygon with the image color coinciding with its mean vertex positions:

Create a jigsaw puzzle from a random Voronoi mesh:

Extract lines from the mesh:

Define a function that converts a line to an interlocking edge:

Replace lines of sufficient length with an interlocking edge:

Visualize the result:

Visualize a piecewise constant interpolation of a function over a set of random point samples:

Voronoi mesh from , sample coordinates:

Function to rescale values to :

Piecewise constant contour plot of sample data:

A similar plot can also be achieved with ListContourPlot:

Plan a path for a point robot through a random set of point obstacles by following Voronoi edges:

Generate Voronoi edges around the obstacles:

Create an undirected Graph from the Voronoi edges with their lengths as edge weights:

Use Nearest function to find the closest Voronoi vertices to the start and end points:

Drag starting or ending points to explore different paths (red):

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

Each point of the original data is contained in exactly one cell in the Voronoi mesh:

VoronoiMesh is the dual of the DelaunayMesh:

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

The Voronoi cell for p_i is given by :

Generate the conditions for each cell:

DistanceTransform for black points on a white background will be similar to VoronoiMesh:

2D Voronoi diagrams can be rendered in Graphics3D using Cone primitives:

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