Base class for polyhedra: Methods for triangulation and volume computation#

class sage.geometry.polyhedron.base7.Polyhedron_base7(parent, Vrep, Hrep, Vrep_minimal=None, Hrep_minimal=None, pref_rep=None, mutable=False, **kwds)#

Bases: Polyhedron_base6

Methods related to triangulation and volume.

centroid(engine='auto', **kwds)#

Return the center of the mass of the polytope.

The mass is taken with respect to the induced Lebesgue measure, see volume().

If the polyhedron is not compact, a NotImplementedError is raised.


  • engine – either ‘auto’ (default), ‘internal’, ‘TOPCOM’, or ‘normaliz’. The ‘internal’ and ‘TOPCOM’ instruct this package to always use its own triangulation algorithms or TOPCOM’s algorithms, respectively. By default (‘auto’), TOPCOM is used if it is available and internal routines otherwise.

  • **kwds – keyword arguments that are passed to the triangulation engine (see triangulate()).

OUTPUT: The centroid as vector.


We triangulate the polytope and find the barycenter of the simplices. We add the individual barycenters weighted by the fraction of the total mass.


sage: P = polytopes.hypercube(2).pyramid()
sage: P.centroid()
(1/4, 0, 0)

sage: P = polytopes.associahedron(['A', 2])                                 # needs sage.combinat
sage: P.centroid()                                                          # needs sage.combinat
(2/21, 2/21)

sage: P = polytopes.permutahedron(4, backend='normaliz')    # optional - pynormaliz
sage: P.centroid()                                          # optional - pynormaliz
(5/2, 5/2, 5/2, 5/2)

The method is not implemented for unbounded polyhedra:

sage: P = Polyhedron(vertices=[(0, 0)], rays=[(1, 0), (0, 1)])
sage: P.centroid()
Traceback (most recent call last):
NotImplementedError: the polyhedron is not compact

The centroid of an empty polyhedron is not defined:

sage: Polyhedron().centroid()
Traceback (most recent call last):
ZeroDivisionError: rational division by zero
integrate(function, measure='ambient', **kwds)#

Return the integral of function over this polytope.


  • self – Polyhedron

  • function – a multivariate polynomial or a valid LattE description string for polynomials

  • measure – string, the measure to use

    Allowed values are:

    • ambient (default): Lebesgue measure of ambient space,

    • induced: Lebesgue measure of the affine hull,

    • induced_nonnormalized: Lebesgue measure of the affine hull without the normalization by \(\sqrt{\det(A^\top A)}\) (with \(A\) being the affine transformation matrix; see affine_hull()).

  • **kwds – additional keyword arguments that are passed to the engine


The integral of the polynomial over the polytope


The polytope triangulation algorithm is used. This function depends on LattE (i.e., the latte_int optional package).


sage: P = polytopes.cube()
sage: x, y, z = polygens(QQ, 'x, y, z')
sage: P.integrate(x^2*y^2*z^2)                              # optional - latte_int

If the polyhedron has floating point coordinates, an inexact result can be obtained if we transform to rational coordinates:

sage: P = 1.4142*polytopes.cube()
sage: P_QQ = Polyhedron(vertices=[[QQ(vi) for vi in v] for v in P.vertex_generator()])
sage: RDF(P_QQ.integrate(x^2*y^2*z^2))                      # optional - latte_int

Integral over a non full-dimensional polytope:

sage: x, y = polygens(QQ, 'x, y')
sage: P = Polyhedron(vertices=[[0,0], [1,1]])
sage: P.integrate(x*y)
sage: ixy = P.integrate(x*y, measure='induced'); ixy        # optional - latte_int
sage: ixy.parent()                                          # optional - latte_int
Algebraic Real Field

Convert to a symbolic expression:

sage: ixy.radical_expression()                              # optional - latte_int

Another non full-dimensional polytope integration:

sage: R.<x, y, z> = QQ[]
sage: P = polytopes.simplex(2)
sage: V = AA(P.volume(measure='induced'))                                   # needs sage.rings.number_field
sage: V.radical_expression()                                                # needs sage.rings.number_field sage.symbolic
sage: P.integrate(R(1), measure='induced') == V             # optional - latte_int, needs sage.rings.number_field sage.symbolic

Computing the mass center:

sage: (P.integrate(x, measure='induced')                    # optional - latte_int, needs sage.rings.number_field sage.symbolic
....:     / V).radical_expression()
sage: (P.integrate(y, measure='induced')                    # optional - latte_int, needs sage.rings.number_field sage.symbolic
....:     / V).radical_expression()
sage: (P.integrate(z, measure='induced')                    # optional - latte_int, needs sage.rings.number_field sage.symbolic
....:     / V).radical_expression()
triangulate(engine='auto', connected=True, fine=False, regular=None, star=None)#

Return a triangulation of the polytope.


  • engine – either ‘auto’ (default), ‘internal’, ‘TOPCOM’, or ‘normaliz’. The ‘internal’ and ‘TOPCOM’ instruct this package to always use its own triangulation algorithms or TOPCOM’s algorithms, respectively. By default (‘auto’), TOPCOM is used if it is available and internal routines otherwise.

The remaining keyword parameters are passed through to the PointConfiguration constructor:

  • connected – boolean (default: True). Whether the triangulations should be connected to the regular triangulations via bistellar flips. These are much easier to compute than all triangulations.

  • fine – boolean (default: False). Whether the triangulations must be fine, that is, make use of all points of the configuration.

  • regular – boolean or None (default: None). Whether the triangulations must be regular. A regular triangulation is one that is induced by a piecewise-linear convex support function. In other words, the shadows of the faces of a polyhedron in one higher dimension.

    • True: Only regular triangulations.

    • False: Only non-regular triangulations.

    • None (default): Both kinds of triangulation.

  • star – either None (default) or a point. Whether the triangulations must be star. A triangulation is star if all maximal simplices contain a common point. The central point can be specified by its index (an integer) in the given points or by its coordinates (anything iterable.)


A triangulation of the convex hull of the vertices as a Triangulation. The indices in the triangulation correspond to the Vrepresentation() objects.


sage: cube = polytopes.hypercube(3)
sage: triangulation = cube.triangulate(
....:    engine='internal')  # to make doctest independent of TOPCOM
sage: triangulation
(<0,1,2,7>, <0,1,5,7>, <0,2,3,7>, <0,3,4,7>, <0,4,5,7>, <1,5,6,7>)
sage: simplex_indices = triangulation[0]; simplex_indices
(0, 1, 2, 7)
sage: simplex_vertices = [cube.Vrepresentation(i) for i in simplex_indices]
sage: simplex_vertices
[A vertex at (1, -1, -1),
 A vertex at (1, 1, -1),
 A vertex at (1, 1, 1),
 A vertex at (-1, 1, 1)]
sage: Polyhedron(simplex_vertices)
A 3-dimensional polyhedron in ZZ^3 defined as the convex hull of 4 vertices

It is possible to use 'normaliz' as an engine. For this, the polyhedron should have the backend set to normaliz:

sage: P = Polyhedron(vertices=[[0,0,1], [1,0,1],            # optional - pynormaliz
....:                          [0,1,1], [1,1,1]],
....:                backend='normaliz')
sage: P.triangulate(engine='normaliz')                      # optional - pynormaliz
(<0,1,2>, <1,2,3>)

sage: P = Polyhedron(vertices=[[0,0,1], [1,0,1],
....:                          [0,1,1], [1,1,1]])
sage: P.triangulate(engine='normaliz')
Traceback (most recent call last):
TypeError: the polyhedron's backend should be 'normaliz'

The normaliz engine can triangulate pointed cones:

sage: # optional - pynormaliz
sage: C1 = Polyhedron(rays=[[0,0,1], [1,0,1],
....:                       [0,1,1], [1,1,1]],
....:                 backend='normaliz')
sage: C1.triangulate(engine='normaliz')
(<0,1,2>, <1,2,3>)
sage: C2 = Polyhedron(rays=[[1,0,1], [0,0,1],
....:                       [0,1,1], [1,1,10/9]],
....:                 backend='normaliz')
sage: C2.triangulate(engine='normaliz')
(<0,1,2>, <1,2,3>)

They can also be affine cones:

sage: K = Polyhedron(vertices=[[1,1,1]],                    # optional - pynormaliz
....:                rays=[[1,0,0], [0,1,0], [1,1,-1], [1,1,1]],
....:                backend='normaliz')
sage: K.triangulate(engine='normaliz')                      # optional - pynormaliz
(<0,1,2>, <0,1,3>)
volume(measure='ambient', engine='auto', **kwds)#

Return the volume of the polytope.


  • measure – string. The measure to use. Allowed values are:

    • ambient (default): Lebesgue measure of ambient space (volume)

    • induced: Lebesgue measure of the affine hull (relative volume)

    • induced_rational: Scaling of the Lebesgue measure for rational polytopes, such that the unit hypercube has volume 1

    • induced_lattice: Scaling of the Lebesgue measure, such that the volume of the hypercube is factorial(n)

  • engine – string. The backend to use. Allowed values are:

    • 'auto' (default): choose engine according to measure

    • 'internal': see triangulate()

    • 'TOPCOM': see triangulate()

    • 'lrs': use David Avis’s lrs program (optional)

    • 'latte': use LattE integrale program (optional)

    • 'normaliz': use Normaliz program (optional)

  • **kwds – keyword arguments that are passed to the triangulation engine


The volume of the polytope


sage: polytopes.hypercube(3).volume()
sage: (polytopes.hypercube(3)*2).volume()
sage: polytopes.twenty_four_cell().volume()

Volume of the same polytopes, using the optional package lrslib (which requires a rational polytope):

sage: I3 = polytopes.hypercube(3)
sage: I3.volume(engine='lrs')                               # optional - lrslib
sage: C24 = polytopes.twenty_four_cell()
sage: C24.volume(engine='lrs')                              # optional - lrslib

If the base ring is exact, the answer is exact:

sage: P5 = polytopes.regular_polygon(5)                                     # needs sage.rings.number_field
sage: P5.volume()                                                           # needs sage.rings.number_field

sage: polytopes.icosahedron().volume()                                      # needs sage.rings.number_field
5/12*sqrt5 + 5/4
sage: numerical_approx(_)  # abs tol 1e9                                    # needs sage.rings.number_field

When considering lower-dimensional polytopes, we can ask for the ambient (full-dimensional), the induced measure (of the affine hull) or, in the case of lattice polytopes, for the induced rational measure. This is controlled by the parameter measure. Different engines may have different ideas on the definition of volume of a lower-dimensional object:

sage: P = Polyhedron([[0, 0], [1, 1]])
sage: P.volume()
sage: P.volume(measure='induced')                                           # needs sage.rings.number_field
sage: P.volume(measure='induced_rational')                  # optional - latte_int

sage: # needs sage.rings.number_field
sage: S = polytopes.regular_polygon(6); S
A 2-dimensional polyhedron in AA^2 defined as the convex hull of 6 vertices
sage: edge = S.faces(1)[4].as_polyhedron()
sage: edge.vertices()
(A vertex at (0.866025403784439?, 1/2), A vertex at (0, 1))
sage: edge.volume()
sage: edge.volume(measure='induced')

sage: # optional - pynormaliz
sage: P = Polyhedron(backend='normaliz',
....:                vertices=[[1,0,0], [0,0,1],
....:                          [-1,1,1], [-1,2,0]])
sage: P.volume()
sage: P.volume(measure='induced')                                           # needs sage.rings.number_field
sage: P.volume(measure='induced', engine='normaliz')
sage: P.volume(measure='induced_rational')                  # optional - latte_int
sage: P.volume(measure='induced_rational',
....:          engine='normaliz')
sage: P.volume(measure='induced_lattice')

The same polytope without normaliz backend:

sage: P = Polyhedron(vertices=[[1,0,0], [0,0,1], [-1,1,1], [-1,2,0]])
sage: P.volume(measure='induced_lattice', engine='latte')   # optional - latte_int

sage: # needs sage.groups sage.rings.number_field
sage: Dexact = polytopes.dodecahedron()
sage: F0 = Dexact.faces(2)[0].as_polyhedron()
sage: v = F0.volume(measure='induced', engine='internal'); v
sage: F4 = Dexact.faces(2)[4].as_polyhedron()
sage: v = F4.volume(measure='induced', engine='internal'); v
sage: RDF(v)    # abs tol 1e-9

sage: # needs sage.groups
sage: Dinexact = polytopes.dodecahedron(exact=False)
sage: F2 = Dinexact.faces(2)[2].as_polyhedron()
sage: w = F2.volume(measure='induced', engine='internal')
sage: RDF(w)    # abs tol 1e-9

sage: all(polytopes.simplex(d).volume(measure='induced')                    # needs sage.rings.number_field sage.symbolic
....:        == sqrt(d+1)/factorial(d)
....:     for d in range(1,5))

sage: I = Polyhedron([[-3, 0], [0, 9]])
sage: I.volume(measure='induced')                                           # needs sage.rings.number_field
sage: I.volume(measure='induced_rational')                  # optional - latte_int

sage: T = Polyhedron([[3, 0, 0], [0, 4, 0], [0, 0, 5]])
sage: T.volume(measure='induced')                                           # needs sage.rings.number_field
sage: T.volume(measure='induced_rational')                  # optional - latte_int

sage: Q = Polyhedron(vertices=[(0, 0, 1, 1), (0, 1, 1, 0), (1, 1, 0, 0)])
sage: Q.volume(measure='induced')
sage: Q.volume(measure='induced_rational')                  # optional - latte_int

The volume of a full-dimensional unbounded polyhedron is infinity:

sage: P = Polyhedron(vertices=[[1, 0], [0, 1]], rays=[[1, 1]])
sage: P.volume()

The volume of a non full-dimensional unbounded polyhedron depends on the measure used:

sage: P = Polyhedron(ieqs = [[1,1,1], [-1,-1,-1], [3,1,0]]); P
A 1-dimensional polyhedron in QQ^2 defined as the convex hull of 1 vertex and 1 ray
sage: P.volume()
sage: P.volume(measure='induced')
sage: P.volume(measure='ambient')
sage: P.volume(measure='induced_rational')                  # optional - pynormaliz
sage: P.volume(measure='induced_rational',engine='latte')

The volume in \(0\)-dimensional space is taken by counting measure:

sage: P = Polyhedron(vertices=[[]]); P
A 0-dimensional polyhedron in ZZ^0 defined as the convex hull of 1 vertex
sage: P.volume()
sage: P = Polyhedron(vertices=[]); P
The empty polyhedron in ZZ^0
sage: P.volume()