# Voronoi diagram#

This module provides the class VoronoiDiagram for computing the Voronoi diagram of a finite list of points in $$\RR^d$$.

class sage.geometry.voronoi_diagram.VoronoiDiagram(points)#

Bases: SageObject

Base class for the Voronoi diagram.

Compute the Voronoi diagram of a list of points.

INPUT:

OUTPUT:

An instance of the VoronoiDiagram class.

EXAMPLES:

Get the Voronoi diagram for some points in $$\RR^3$$:

sage: V = VoronoiDiagram([[1, 3, .3], [2, -2, 1], [-1, 2, -.1]]); V
The Voronoi diagram of 3 points of dimension 3 in the Real Double Field

sage: VoronoiDiagram([])
The empty Voronoi diagram.


Get the Voronoi diagram of a regular pentagon in AA^2. All cells meet at the origin:

sage: DV = VoronoiDiagram([[AA(c) for c in v]                                   # needs sage.rings.number_field
....:                      for v in polytopes.regular_polygon(5).vertices_list()]); DV
The Voronoi diagram of 5 points of dimension 2 in the Algebraic Real Field
sage: all(P.contains([0, 0]) for P in DV.regions().values())                    # needs sage.rings.number_field
True
sage: any(P.interior_contains([0, 0]) for P in DV.regions().values())           # needs sage.rings.number_field
False


If the vertices are not converted to AA before, the method throws an error:

sage: polytopes.dodecahedron().vertices_list()[0][0].parent()                   # needs sage.rings.number_field
Number Field in sqrt5 with defining polynomial x^2 - 5 with sqrt5 = 2.236067977499790?
sage: VoronoiDiagram(polytopes.dodecahedron().vertices_list())                  # needs sage.rings.number_field
Traceback (most recent call last):
...
NotImplementedError: Base ring of the Voronoi diagram must be
one of QQ, RDF, AA.


ALGORITHM:

We use hyperplanes tangent to the paraboloid one dimension higher to get a convex polyhedron and then project back to one dimension lower.

Todo

• The dual construction: Delaunay triangulation

• improve 2d-plotting

• implement 3d-plotting

• more general constructions, like Voroi diagrams with weights (power diagrams)

REFERENCES:

AUTHORS:

• Moritz Firsching (2012-09-21)

ambient_dim()#

Return the ambient dimension of the points.

EXAMPLES:

sage: V = VoronoiDiagram([[.5, 3], [2, 5], [4, 5], [4, -1]])
sage: V.ambient_dim()
2
sage: V = VoronoiDiagram([[1, 2, 3, 4, 5, 6]]); V.ambient_dim()
6

base_ring()#

Return the base_ring of the regions of the Voronoi diagram.

EXAMPLES:

sage: V = VoronoiDiagram([[1, 3, 1], [2, -2, 1], [-1, 2, 1/2]]); V.base_ring()
Rational Field
sage: V = VoronoiDiagram([[1, 3.14], [2, -2/3], [-1, 22]]); V.base_ring()
Real Double Field
sage: V = VoronoiDiagram([[1, 3], [2, 4]]); V.base_ring()
Rational Field

plot(cell_colors=None, **kwds)#

Return a graphical representation for 2-dimensional Voronoi diagrams.

INPUT:

• cell_colors – (default: None) provide the colors for the cells, either as dictionary. Randomly colored cells are provided with None.

• **kwds – optional keyword parameters, passed on as arguments for plot().

OUTPUT:

A graphics object.

EXAMPLES:

sage: # needs sage.plot
sage: P = [[0.671, 0.650], [0.258, 0.767], [0.562, 0.406],
....:      [0.254, 0.709], [0.493, 0.879]]
sage: V = VoronoiDiagram(P); S=V.plot()
sage: show(S, xmin=0, xmax=1, ymin=0, ymax=1, aspect_ratio=1, axes=false)
sage: S = V.plot(cell_colors={0: 'red', 1: 'blue', 2: 'green',
....:                         3: 'white', 4: 'yellow'})
sage: show(S, xmin=0, xmax=1, ymin=0, ymax=1, aspect_ratio=1, axes=false)
sage: S = V.plot(cell_colors=['red', 'blue', 'red', 'white', 'white'])
sage: show(S, xmin=0, xmax=1, ymin=0, ymax=1, aspect_ratio=1, axes=false)
sage: S = V.plot(cell_colors='something else')
Traceback (most recent call last):
...
AssertionError: 'cell_colors' must be a list or a dictionary


Trying to plot a Voronoi diagram of dimension other than 2 gives an error:

sage: VoronoiDiagram([[1, 2, 3], [6, 5, 4]]).plot()                         # needs sage.plot
Traceback (most recent call last):
...
NotImplementedError: Plotting of 3-dimensional Voronoi diagrams not
implemented

points()#

Return the input points (as a PointConfiguration).

EXAMPLES:

sage: V = VoronoiDiagram([[.5, 3], [2, 5], [4, 5], [4, -1]]); V.points()
A point configuration in affine 2-space over Real Field
with 53 bits of precision consisting of 4 points.
The triangulations of this point configuration are
assumed to be connected, not necessarily fine,
not necessarily regular.

regions()#

Return the Voronoi regions of the Voronoi diagram as a dictionary of polyhedra.

EXAMPLES:

sage: V = VoronoiDiagram([[1, 3, .3], [2, -2, 1], [-1, 2, -.1]])
sage: P = V.points()
sage: V.regions() == {P[0]: Polyhedron(base_ring=RDF, lines=[(-RDF(0.375), RDF(0.13888888890000001), RDF(1.5277777779999999))],
....:                                                 rays=[(RDF(9), -RDF(1), -RDF(20)), (RDF(4.5), RDF(1), -RDF(25))],
....:                                                 vertices=[(-RDF(1.1074999999999999), RDF(1.149444444), RDF(9.0138888890000004))]),
....:                 P[1]: Polyhedron(base_ring=RDF, lines=[(-RDF(0.375), RDF(0.13888888890000001), RDF(1.5277777779999999))],
....:                                                 rays=[(RDF(9), -RDF(1), -RDF(20)), (-RDF(2.25), -RDF(1), RDF(2.5))],
....:                                                  vertices=[(-RDF(1.1074999999999999), RDF(1.149444444), RDF(9.0138888890000004))]),
....:                 P[2]: Polyhedron(base_ring=RDF, lines=[(-RDF(0.375), RDF(0.13888888890000001), RDF(1.5277777779999999))],
....:                                                 rays=[(RDF(4.5), RDF(1), -RDF(25)), (-RDF(2.25), -RDF(1), RDF(2.5))],
....:                                                 vertices=[(-RDF(1.1074999999999999), RDF(1.149444444), RDF(9.0138888890000004))])}
True