Triangulations of a point configuration

A point configuration is a finite set of points in Euclidean space or, more generally, in projective space. A triangulation is a simplicial decomposition of the convex hull of a given point configuration such that all vertices of the simplices end up lying on points of the configuration. That is, there are no new vertices apart from the initial points.

Note that points that are not vertices of the convex hull need not be used in the triangulation. A triangulation that does make use of all points of the configuration is called fine, and you can restrict yourself to such triangulations if you want. See PointConfiguration and restrict_to_fine_triangulations() for more details.

Finding a single triangulation and listing all connected triangulations is implemented natively in this package. However, for more advanced options [TOPCOM] needs to be installed. It is available as an optional package for Sage, and you can install it with the command:

sage: install_package('TOPCOM')     # not tested

Note

TOPCOM and the internal algorithms tend to enumerate triangulations in a different order. This is why we always explicitly specify the engine as engine='TOPCOM' or engine='internal' in the doctests. In your own applications, you do not need to specify the engine. By default, TOPCOM is used if it is available and the internal algorithms are used otherwise.

EXAMPLES:

First, we select the internal implementation for enumerating triangulations:

sage: PointConfiguration.set_engine('internal')   # to make doctests independent of TOPCOM

A 2-dimensional point configuration:

sage: p = PointConfiguration([[0,0],[0,1],[1,0],[1,1],[-1,-1]])
sage: p
A point configuration in QQ^2 consisting of 5 points. The
triangulations of this point configuration are assumed to
be connected, not necessarily fine, not necessarily regular.
sage: t = p.triangulate()  # a single triangulation
sage: t
(<1,3,4>, <2,3,4>)
sage: len(t)
2
sage: t[0]
(1, 3, 4)
sage: t[1]
(2, 3, 4)
sage: list(t)
[(1, 3, 4), (2, 3, 4)]
sage: t.plot(axes=False)
sage: list( p.triangulations() )
[(<1,3,4>, <2,3,4>),
 (<0,1,3>, <0,1,4>, <0,2,3>, <0,2,4>),
 (<1,2,3>, <1,2,4>),
 (<0,1,2>, <0,1,4>, <0,2,4>, <1,2,3>)]
sage: p_fine = p.restrict_to_fine_triangulations()
sage: p_fine
A point configuration in QQ^2 consisting of 5 points. The
triangulations of this point configuration are assumed to
be connected, fine, not necessarily regular.
sage: list( p_fine.triangulations() )
[(<0,1,3>, <0,1,4>, <0,2,3>, <0,2,4>),
 (<0,1,2>, <0,1,4>, <0,2,4>, <1,2,3>)]

A 3-dimensional point configuration:

sage: p = [[0,-1,-1],[0,0,1],[0,1,0], [1,-1,-1],[1,0,1],[1,1,0]]
sage: points = PointConfiguration(p)
sage: triang = points.triangulate()
sage: triang.plot(axes=False)

The standard example of a non-regular triangulation:

sage: p = PointConfiguration([[-1,-5/9],[0,10/9],[1,-5/9],[-2,-10/9],[0,20/9],[2,-10/9]])
sage: regular = p.restrict_to_regular_triangulations(True).triangulations_list()      # optional - TOPCOM
sage: nonregular = p.restrict_to_regular_triangulations(False).triangulations_list()  # optional - TOPCOM
sage: len(regular)     # optional - TOPCOM
16
sage: len(nonregular)  # optional - TOPCOM
2
sage: nonregular[0].plot(aspect_ratio=1, axes=False)   # optional - TOPCOM

Note that the points need not be in general position. That is, the points may lie in a hyperplane and the linear dependencies will be removed before passing the data to TOPCOM which cannot handle it:

sage: points = [[0,0,0,1],[0,3,0,1],[3,0,0,1],[0,0,1,1],[0,3,1,1],[3,0,1,1],[1,1,2,1]]
sage: points = [ p+[1,2,3] for p in points ]
sage: pc = PointConfiguration(points)
sage: pc.ambient_dim()
7
sage: pc.dim()
3
sage: pc.triangulate()
(<0,1,2,6>, <0,1,3,6>, <0,2,3,6>, <1,2,4,6>, <1,3,4,6>, <2,3,5,6>, <2,4,5,6>)
sage: _ in pc.triangulations()
True
sage: len( pc.triangulations_list() )
26

REFERENCES:

[TOPCOM]J. Rambau, TOPCOM <http://www.rambau.wm.uni-bayreuth.de/TOPCOM/>.
[GKZ](1, 2) Gel’fand, I. M.; Kapranov, M. M.; and Zelevinsky, A. V. “Discriminants, Resultants and Multidimensional Determinants” Birkhauser 1994.
[PUNTOS]Jesus A. De Loera http://www.math.ucdavis.edu/~deloera/RECENT_WORK/puntos2000

AUTHORS:

  • Volker Braun: initial version, 2010
  • Josh Whitney: added functionality for computing volumes and secondary polytopes of PointConfigurations
  • Marshall Hampton: improved documentation and doctest coverage
  • Volker Braun: rewrite using Parent/Element and catgories. Added a Point class. More doctests. Less zombies.
  • Volker Braun: Cythonized parts of it, added a C++ implementation of the bistellar flip algorithm to enumerate all connected triangulations.
  • Volker Braun 2011: switched the triangulate() method to the placing triangulation (faster).
class sage.geometry.triangulation.point_configuration.PointConfiguration(points, connected, fine, regular, star, defined_affine)

Bases: sage.structure.unique_representation.UniqueRepresentation, sage.geometry.triangulation.base.PointConfiguration_base

A collection of points in Euclidean (or projective) space.

This is the parent class for the triangulations of the point configuration. There are a few options to specifically select what kind of triangulations are admissible.

INPUT:

The constructor accepts the following arguments:

  • points – the points. Technically, any iterable of iterables will do. In particular, a PointConfiguration can be passed.

  • projective – boolean (default: False). Whether the point coordinates should be interpreted as projective (True) or affine (False) coordinates. If necessary, points are projectivized by setting the last homogeneous coordinate to one and/or affine patches are chosen internally.

  • 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 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.)

EXAMPLES:

sage: p = PointConfiguration([[0,0],[0,1],[1,0],[1,1],[-1,-1]])
sage: p
A point configuration in QQ^2 consisting of 5 points. The
triangulations of this point configuration are assumed to
be connected, not necessarily fine, not necessarily regular.
sage: p.triangulate()  # a single triangulation
(<1,3,4>, <2,3,4>)
Element

alias of Triangulation

Gale_transform(points=None)

Return the Gale transform of self.

INPUT:

  • points – a tuple of points or point indices or None (default). A subset of points for which to compute the Gale transform. By default, all points are used.

OUTPUT:

A matrix over base_ring().

EXAMPLES:

sage: pc = PointConfiguration([(0,0),(1,0),(2,1),(1,1),(0,1)])
sage: pc.Gale_transform()
[ 1 -1  0  1 -1]
[ 0  0  1 -2  1]

sage: pc.Gale_transform((0,1,3,4))
[ 1 -1  1 -1]

sage: points = (pc.point(0), pc.point(1), pc.point(3), pc.point(4))
sage: pc.Gale_transform(points)
[ 1 -1  1 -1]
an_element()

Synonymous for triangulate().

TESTS:

sage: p = PointConfiguration([[0, 1], [0, 0], [1, 0], [1,1]])
sage: p.an_element()
(<0,1,3>, <1,2,3>)
bistellar_flips()

Return the bistellar flips.

OUTPUT:

The bistellar flips as a tuple. Each flip is a pair \((T_+,T_-)\) where \(T_+\) and \(T_-\) are partial triangulations of the point configuration.

EXAMPLES:

sage: pc = PointConfiguration([(0,0),(1,0),(0,1),(1,1)])
sage: pc.bistellar_flips()
(((<0,1,3>, <0,2,3>), (<0,1,2>, <1,2,3>)),)
sage: Tpos, Tneg = pc.bistellar_flips()[0]
sage: Tpos.plot(axes=False)
sage: Tneg.plot(axes=False)

The 3d analog:

sage: pc = PointConfiguration([(0,0,0),(0,2,0),(0,0,2),(-1,0,0),(1,1,1)])
sage: pc.bistellar_flips()
(((<0,1,2,3>, <0,1,2,4>), (<0,1,3,4>, <0,2,3,4>, <1,2,3,4>)),)

A 2d flip on the base of the pyramid over a square:

sage: pc = PointConfiguration([(0,0,0),(0,2,0),(0,0,2),(0,2,2),(1,1,1)])
sage: pc.bistellar_flips()
(((<0,1,3>, <0,2,3>), (<0,1,2>, <1,2,3>)),)
sage: Tpos, Tneg = pc.bistellar_flips()[0]
sage: Tpos.plot(axes=False)
circuits()

Return the circuits of the point configuration.

Roughly, a circuit is a minimal linearly dependent subset of the points. That is, a circuit is a partition

\[\{ 0, 1, \dots, n-1 \} = C_+ \cup C_0 \cup C_-\]

such that there is an (unique up to an overall normalization) affine relation

\[\sum_{i\in C_+} \alpha_i \vec{p}_i = \sum_{j\in C_-} \alpha_j \vec{p}_j\]

with all positive (or all negative) coefficients, where \(\vec{p}_i=(p_1,\dots,p_k,1)\) are the projective coordinates of the \(i\)-th point.

OUTPUT:

The list of (unsigned) circuits as triples \((C_+, C_0, C_-)\). The swapped circuit \((C_-, C_0, C_+)\) is not returned separately.

EXAMPLES:

sage: p = PointConfiguration([(0,0),(+1,0),(-1,0),(0,+1),(0,-1)])
sage: p.circuits()
(((0,), (1, 2), (3, 4)), ((0,), (3, 4), (1, 2)), ((1, 2), (0,), (3, 4)))

TESTS:

sage: U=matrix([
...      [ 0, 0, 0, 0, 0, 2, 4,-1, 1, 1, 0, 0, 1, 0],
...      [ 0, 0, 0, 1, 0, 0,-1, 0, 0, 0, 0, 0, 0, 0],
...      [ 0, 2, 0, 0, 0, 0,-1, 0, 1, 0, 1, 0, 0, 1],
...      [ 0, 1, 1, 0, 0, 1, 0,-2, 1, 0, 0,-1, 1, 1],
...      [ 0, 0, 0, 0, 1, 0,-1, 0, 0, 0, 0, 0, 0, 0]
...   ])
sage: p = PointConfiguration(U.columns())
sage: len( p.circuits() )    # long time
218
circuits_support()

A generator for the supports of the circuits of the point configuration.

See circuits() for details.

OUTPUT:

A generator for the supports \(C_-\cup C_+\) (returned as a Python tuple) for all circuits of the point configuration.

EXAMPLES:

sage: p = PointConfiguration([(0,0),(+1,0),(-1,0),(0,+1),(0,-1)])
sage: list( p.circuits_support() )
[(0, 3, 4), (0, 1, 2), (1, 2, 3, 4)]
contained_simplex(large=True, initial_point=None)

Return a simplex contained in the point configuration.

INPUT:

  • large – boolean. Whether to attempt to return a large simplex.
  • initial_point – a Point or None (default). A specific point to start with when picking the simplex vertices.

OUTPUT:

A tuple of points that span a simplex of dimension dim(). If large==True, the simplex is constructed by sucessively picking the farthest point. This will ensure that the simplex is not unneccessarily small, but will in general not return a maximal simplex.

EXAMPLES:

sage: pc = PointConfiguration([(0,0),(1,0),(2,1),(1,1),(0,1)])
sage: pc.contained_simplex()
(P(0, 1), P(2, 1), P(1, 0))
sage: pc.contained_simplex(large=False)
(P(0, 1), P(1, 1), P(1, 0))
sage: pc.contained_simplex(initial_point=pc.point(0))
(P(0, 0), P(1, 1), P(1, 0))

sage: pc = PointConfiguration([[0,0],[0,1],[1,0],[1,1],[-1,-1]])
sage: pc.contained_simplex()
(P(-1, -1), P(1, 1), P(0, 1))

TESTS:

sage: pc = PointConfiguration([[0,0],[0,1],[1,0]])
sage: pc.contained_simplex()
(P(1, 0), P(0, 1), P(0, 0))
sage: pc = PointConfiguration([[0,0],[0,1]])
sage: pc.contained_simplex()
(P(0, 1), P(0, 0))
sage: pc = PointConfiguration([[0,0]])
sage: pc.contained_simplex()
(P(0, 0),)
sage: pc = PointConfiguration([])
sage: pc.contained_simplex()
()
convex_hull()

Return the convex hull of the point configuration.

EXAMPLES:

sage: p = PointConfiguration([[0,0],[0,1],[1,0],[1,1],[-1,-1]])
sage: p.convex_hull()
A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 4 vertices
distance(x, y)

Returns the distance between two points.

INPUT:

  • x, y – two points of the point configuration.

OUTPUT:

The distance between x and y, measured either with distance_affine() or distance_FS() depending on whether the point configuration is defined by affine or projective points. These are related, but not equal to the usual flat and Fubini-Study distance.

EXAMPLES:

sage: pc = PointConfiguration([(0,0),(1,0),(2,1),(1,2),(0,1)])
sage: [ pc.distance(pc.point(0), p) for p in pc.points() ]
[0, 1, 5, 5, 1]

sage: pc = PointConfiguration([(0,0,1),(1,0,1),(2,1,1),(1,2,1),(0,1,1)], projective=True)
sage: [ pc.distance(pc.point(0), p) for p in pc.points() ]
[0, 1/2, 5/6, 5/6, 1/2]
distance_FS(x, y)

Returns the distance between two points.

The distance function used in this method is \(1-\cos d_{FS}(x,y)^2\), where \(d_{FS}\) is the Fubini-Study distance of projective points. Recall the Fubini-Studi distance function

\[d_{FS}(x,y) = \arccos \sqrt{ \frac{(x\cdot y)^2}{|x|^2 |y|^2} }\]

INPUT:

  • x, y – two points of the point configuration.

OUTPUT:

The distance \(1-\cos d_{FS}(x,y)^2\). Note that this distance lies in the same field as the entries of x, y. That is, the distance of rational points will be rational and so on.

EXAMPLES:

sage: pc = PointConfiguration([(0,0),(1,0),(2,1),(1,2),(0,1)])
sage: [ pc.distance_FS(pc.point(0), p) for p in pc.points() ]
[0, 1/2, 5/6, 5/6, 1/2]
distance_affine(x, y)

Returns the distance between two points.

The distance function used in this method is \(d_{aff}(x,y)^2\), the square of the usual affine distance function

\[d_{aff}(x,y) = |x-y|\]

INPUT:

  • x, y – two points of the point configuration.

OUTPUT:

The metric distance-square \(d_{aff}(x,y)^2\). Note that this distance lies in the same field as the entries of x, y. That is, the distance of rational points will be rational and so on.

EXAMPLES:

sage: pc = PointConfiguration([(0,0),(1,0),(2,1),(1,2),(0,1)])
sage: [ pc.distance_affine(pc.point(0), p) for p in pc.points() ]
[0, 1, 5, 5, 1]
exclude_points(point_idx_list)

Return a new point configuration with the given points removed.

INPUT:

  • point_idx_list – a list of integers. The indices of points to exclude.

OUTPUT:

A new PointConfiguration with the given points removed.

EXAMPLES:

sage: p = PointConfiguration([[-1,0], [0,0], [1,-1], [1,0], [1,1]]);
sage: list(p)
[P(-1, 0), P(0, 0), P(1, -1), P(1, 0), P(1, 1)]
sage: q = p.exclude_points([3])
sage: list(q)
[P(-1, 0), P(0, 0), P(1, -1), P(1, 1)]
sage: p.exclude_points( p.face_interior(codim=1) ).points()
(P(-1, 0), P(0, 0), P(1, -1), P(1, 1))
face_codimension(point)

Return the smallest \(d\in\mathbb{Z}\) such that point is contained in the interior of a codimension-\(d\) face.

EXAMPLES:

sage: triangle = PointConfiguration([[0,0], [1,-1], [1,0], [1,1]]);
sage: triangle.point(2)
P(1, 0)
sage: triangle.face_codimension(2)
1
sage: triangle.face_codimension( [1,0] )
1

This also works for degenerate cases like the tip of the pyramid over a square (which saturates four inequalities):

sage: pyramid = PointConfiguration([[1,0,0],[0,1,1],[0,1,-1],[0,-1,-1],[0,-1,1]])
sage: pyramid.face_codimension(0)
3
face_interior(dim=None, codim=None)

Return points by the codimension of the containing face in the convex hull.

EXAMPLES:

sage: triangle = PointConfiguration([[-1,0], [0,0], [1,-1], [1,0], [1,1]]);
sage: triangle.face_interior()
((1,), (3,), (0, 2, 4))
sage: triangle.face_interior(dim=0)    # the vertices of the convex hull
(0, 2, 4)
sage: triangle.face_interior(codim=1)  # interior of facets
(3,)
farthest_point(points, among=None)

Return the point with the most distance from points.

INPUT:

  • points – a list of points.
  • among – a list of points or None (default). The set of points from which to pick the farthest one. By default, all points of the configuration are considered.

OUTPUT:

A Point with largest minimal distance from all given points.

EXAMPLES:

sage: pc = PointConfiguration([(0,0),(1,0),(1,1),(0,1)])
sage: pc.farthest_point([ pc.point(0) ])
P(1, 1)
lexicographic_triangulation()

Return the lexicographic triangulation.

The algorithm was taken from [PUNTOS].

EXAMPLES:

sage: p = PointConfiguration([(0,0),(+1,0),(-1,0),(0,+1),(0,-1)])
sage: p.lexicographic_triangulation()
(<1,3,4>, <2,3,4>)

TESTS:

sage: U=matrix([
...      [ 0, 0, 0, 0, 0, 2, 4,-1, 1, 1, 0, 0, 1, 0],
...      [ 0, 0, 0, 1, 0, 0,-1, 0, 0, 0, 0, 0, 0, 0],
...      [ 0, 2, 0, 0, 0, 0,-1, 0, 1, 0, 1, 0, 0, 1],
...      [ 0, 1, 1, 0, 0, 1, 0,-2, 1, 0, 0,-1, 1, 1],
...      [ 0, 0, 0, 0, 1, 0,-1, 0, 0, 0, 0, 0, 0, 0]
...   ])
sage: pc = PointConfiguration(U.columns())
sage: pc.lexicographic_triangulation()
(<1,3,4,7,10,13>, <1,3,4,8,10,13>, <1,3,6,7,10,13>, <1,3,6,8,10,13>,
 <1,4,6,7,10,13>, <1,4,6,8,10,13>, <2,3,4,6,7,12>, <2,3,4,7,12,13>,
 <2,3,6,7,12,13>, <2,4,6,7,12,13>, <3,4,5,6,9,12>, <3,4,5,8,9,12>,
 <3,4,6,7,11,12>, <3,4,6,9,11,12>, <3,4,7,10,11,13>, <3,4,7,11,12,13>,
 <3,4,8,9,10,12>, <3,4,8,10,12,13>, <3,4,9,10,11,12>, <3,4,10,11,12,13>,
 <3,5,6,8,9,12>, <3,6,7,10,11,13>, <3,6,7,11,12,13>, <3,6,8,9,10,12>,
 <3,6,8,10,12,13>, <3,6,9,10,11,12>, <3,6,10,11,12,13>, <4,5,6,8,9,12>,
 <4,6,7,10,11,13>, <4,6,7,11,12,13>, <4,6,8,9,10,12>, <4,6,8,10,12,13>,
 <4,6,9,10,11,12>, <4,6,10,11,12,13>)
sage: len(_)
34
placing_triangulation(point_order=None)

Construct the placing (pushing) triangulation.

INPUT:

  • point_order – list of points or integers. The order in which the points are to be placed.

OUTPUT:

A Triangulation.

EXAMPLES:

sage: pc = PointConfiguration([(0,0),(1,0),(2,1),(1,2),(0,1)])
sage: pc.placing_triangulation()
(<0,1,2>, <0,2,4>, <2,3,4>)

sage: U=matrix([
...      [ 0, 0, 0, 0, 0, 2, 4,-1, 1, 1, 0, 0, 1, 0],
...      [ 0, 0, 0, 1, 0, 0,-1, 0, 0, 0, 0, 0, 0, 0],
...      [ 0, 2, 0, 0, 0, 0,-1, 0, 1, 0, 1, 0, 0, 1],
...      [ 0, 1, 1, 0, 0, 1, 0,-2, 1, 0, 0,-1, 1, 1],
...      [ 0, 0, 0, 0, 1, 0,-1, 0, 0, 0, 0, 0, 0, 0]
...   ])
sage: p = PointConfiguration(U.columns())
sage: triangulation = p.placing_triangulation();  triangulation
(<0,2,3,4,6,7>, <0,2,3,4,6,12>, <0,2,3,4,7,13>, <0,2,3,4,12,13>,
 <0,2,3,6,7,13>, <0,2,3,6,12,13>, <0,2,4,6,7,13>, <0,2,4,6,12,13>,
 <0,3,4,6,7,12>, <0,3,4,7,12,13>, <0,3,6,7,12,13>, <0,4,6,7,12,13>,
 <1,3,4,5,6,12>, <1,3,4,6,11,12>, <1,3,4,7,11,13>, <1,3,4,11,12,13>,
 <1,3,6,7,11,13>, <1,3,6,11,12,13>, <1,4,6,7,11,13>, <1,4,6,11,12,13>,
 <3,4,6,7,11,12>, <3,4,7,11,12,13>, <3,6,7,11,12,13>, <4,6,7,11,12,13>)
sage: sum(p.volume(t) for t in triangulation)
42
positive_circuits(*negative)

Returns the positive part of circuits with fixed negative part.

A circuit is a pair \((C_+, C_-)\), each consisting of a subset (actually, an ordered tuple) of point indices.

INPUT:

  • *negative – integer. The indices of points.

OUTPUT:

A tuple of all circuits with \(C_-\) = negative.

EXAMPLE:

sage: p = PointConfiguration([(1,0,0),(0,1,0),(0,0,1),(-2,0,-1),(-2,-1,0),(-3,-1,-1),(1,1,1),(-1,0,0),(0,0,0)])
sage: p.positive_circuits(8)
((0, 7), (0, 1, 4), (0, 2, 3), (0, 5, 6), (0, 1, 2, 5), (0, 3, 4, 6))
sage: p.positive_circuits(0,5,6)
((8,),)
pushing_triangulation(point_order=None)

Construct the placing (pushing) triangulation.

INPUT:

  • point_order – list of points or integers. The order in which the points are to be placed.

OUTPUT:

A Triangulation.

EXAMPLES:

sage: pc = PointConfiguration([(0,0),(1,0),(2,1),(1,2),(0,1)])
sage: pc.placing_triangulation()
(<0,1,2>, <0,2,4>, <2,3,4>)

sage: U=matrix([
...      [ 0, 0, 0, 0, 0, 2, 4,-1, 1, 1, 0, 0, 1, 0],
...      [ 0, 0, 0, 1, 0, 0,-1, 0, 0, 0, 0, 0, 0, 0],
...      [ 0, 2, 0, 0, 0, 0,-1, 0, 1, 0, 1, 0, 0, 1],
...      [ 0, 1, 1, 0, 0, 1, 0,-2, 1, 0, 0,-1, 1, 1],
...      [ 0, 0, 0, 0, 1, 0,-1, 0, 0, 0, 0, 0, 0, 0]
...   ])
sage: p = PointConfiguration(U.columns())
sage: triangulation = p.placing_triangulation();  triangulation
(<0,2,3,4,6,7>, <0,2,3,4,6,12>, <0,2,3,4,7,13>, <0,2,3,4,12,13>,
 <0,2,3,6,7,13>, <0,2,3,6,12,13>, <0,2,4,6,7,13>, <0,2,4,6,12,13>,
 <0,3,4,6,7,12>, <0,3,4,7,12,13>, <0,3,6,7,12,13>, <0,4,6,7,12,13>,
 <1,3,4,5,6,12>, <1,3,4,6,11,12>, <1,3,4,7,11,13>, <1,3,4,11,12,13>,
 <1,3,6,7,11,13>, <1,3,6,11,12,13>, <1,4,6,7,11,13>, <1,4,6,11,12,13>,
 <3,4,6,7,11,12>, <3,4,7,11,12,13>, <3,6,7,11,12,13>, <4,6,7,11,12,13>)
sage: sum(p.volume(t) for t in triangulation)
42
restrict_to_connected_triangulations(connected=True)

Restrict to connected triangulations.

NOTE:

Finding non-connected triangulations requires the optional TOPCOM package.

INPUT:

  • connected – boolean. Whether to restrict to triangulations that are connected by bistellar flips to the regular triangulations.

OUTPUT:

A new PointConfiguration with the same points, but whose triangulations will all be in the connected component. See PointConfiguration for details.

EXAMPLES:

sage: p = PointConfiguration([[0,0],[0,1],[1,0],[1,1],[-1,-1]])
sage: p
A point configuration in QQ^2 consisting of 5 points. The
triangulations of this point configuration are assumed to
be connected, not necessarily fine, not necessarily regular.
sage: len(p.triangulations_list())
4
sage: p_all = p.restrict_to_connected_triangulations(connected=False)  # optional - TOPCOM
sage: len(p_all.triangulations_list())                                 # optional - TOPCOM
4
sage: p == p_all.restrict_to_connected_triangulations(connected=True)  # optional - TOPCOM
True
restrict_to_fine_triangulations(fine=True)

Restrict to fine triangulations.

INPUT:

  • fine – boolean. Whether to restrict to fine triangulations.

OUTPUT:

A new PointConfiguration with the same points, but whose triangulations will all be fine. See PointConfiguration for details.

EXAMPLES:

sage: p = PointConfiguration([[0,0],[0,1],[1,0],[1,1],[-1,-1]])
sage: p
A point configuration in QQ^2 consisting of 5 points. The
triangulations of this point configuration are assumed to
be connected, not necessarily fine, not necessarily regular.
sage: len(p.triangulations_list())
4
sage: p_fine = p.restrict_to_fine_triangulations()
sage: len(p.triangulations_list())
4
sage: p == p_fine.restrict_to_fine_triangulations(fine=False)
True
restrict_to_regular_triangulations(regular=True)

Restrict to regular triangulations.

NOTE:

Regularity testing requires the optional TOPCOM package.

INPUT:

  • regularTrue, False, or None. Whether to restrict to regular triangulations, irregular triangulations, or lift any restrictions on regularity.

OUTPUT:

A new PointConfiguration with the same points, but whose triangulations will all be regular as specified. See PointConfiguration for details.

EXAMPLES:

sage: p = PointConfiguration([[0,0],[0,1],[1,0],[1,1],[-1,-1]])
sage: p
A point configuration in QQ^2 consisting of 5 points. The
triangulations of this point configuration are assumed to
be connected, not necessarily fine, not necessarily regular.
sage: len(p.triangulations_list())
4
sage: p_regular = p.restrict_to_regular_triangulations() # optional - TOPCOM
sage: len(p_regular.triangulations_list())               # optional - TOPCOM
4
sage: p == p_regular.restrict_to_regular_triangulations(regular=None) # optional - TOPCOM
True
restrict_to_star_triangulations(star)

Restrict to star triangulations with the given point as the center.

INPUT:

  • originNone or an integer or the coordinates of a point. An integer denotes the index of the central point. If None is passed, any restriction on the starshape will be removed.

OUTPUT:

A new PointConfiguration with the same points, but whose triangulations will all be star. See PointConfiguration for details.

EXAMPLES:

sage: p = PointConfiguration([[0,0],[0,1],[1,0],[1,1],[-1,-1]])
sage: len(list( p.triangulations() ))
4
sage: p_star =  p.restrict_to_star_triangulations(0)
sage: p_star is p.restrict_to_star_triangulations((0,0))
True
sage: p_star.triangulations_list()
[(<0,1,3>, <0,1,4>, <0,2,3>, <0,2,4>)]
sage: p_newstar = p_star.restrict_to_star_triangulations(1)  # pick different origin
sage: p_newstar.triangulations_list()
[(<1,2,3>, <1,2,4>)]
sage: p == p_star.restrict_to_star_triangulations(star=None)
True
restricted_automorphism_group()

Return the restricted automorphism group.

First, let the linear automorphism group be the subgroup of the Euclidean group \(E(d) = GL(d,\RR) \ltimes \RR^d\) preserving the \(d\)-dimensional point configuration. The Euclidean group acts in the usual way \(\vec{x}\mapsto A\vec{x}+b\) on the ambient space.

The restricted automorphism group is the subgroup of the linear automorphism group generated by permutations of points. See [BSS] for more details and a description of the algorithm.

OUTPUT:

A PermutationGroup that is isomorphic to the restricted automorphism group is returned.

Note that in Sage, permutation groups always act on positive integers while lists etc. are indexed by nonnegative integers. The indexing of the permutation group is chosen to be shifted by +1. That is, the transposition (i,j) in the permutation group corresponds to exchange of self[i-1] and self[j-1].

EXAMPLES:

sage: pyramid = PointConfiguration([[1,0,0],[0,1,1],[0,1,-1],[0,-1,-1],[0,-1,1]])
sage: pyramid.restricted_automorphism_group()
Permutation Group with generators [(3,5), (2,3)(4,5), (2,4)]
sage: DihedralGroup(4).is_isomorphic(_)
True

The square with an off-center point in the middle. Note thath the middle point breaks the restricted automorphism group \(D_4\) of the convex hull:

sage: square = PointConfiguration([(3/4,3/4),(1,1),(1,-1),(-1,-1),(-1,1)])
sage: square.restricted_automorphism_group()
Permutation Group with generators [(3,5)]
sage: DihedralGroup(1).is_isomorphic(_)
True
secondary_polytope()

Calculate the secondary polytope of the point configuration.

For a definition of the secondary polytope, see [GKZ] page 220 Definition 1.6.

Note that if you restricted the admissible triangulations of the point configuration then the output will be the corresponding face of the whole secondary polytope.

OUTPUT:

The secondary polytope of the point configuration as an instance of Polyhedron_base.

EXAMPLES:

sage: p = PointConfiguration([[0,0],[1,0],[2,1],[1,2],[0,1]])
sage: poly = p.secondary_polytope()
sage: poly.vertices_matrix()
[1 1 3 3 5]
[3 5 1 4 1]
[4 2 5 2 4]
[2 4 2 5 4]
[5 3 4 1 1]
sage: poly.Vrepresentation()
(A vertex at (1, 3, 4, 2, 5),
 A vertex at (1, 5, 2, 4, 3),
 A vertex at (3, 1, 5, 2, 4),
 A vertex at (3, 4, 2, 5, 1),
 A vertex at (5, 1, 4, 4, 1))
sage: poly.Hrepresentation()
(An equation (0, 0, 1, 2, 1) x - 13 == 0,
 An equation (1, 0, 0, 2, 2) x - 15 == 0,
 An equation (0, 1, 0, -3, -2) x + 13 == 0,
 An inequality (0, 0, 0, -1, -1) x + 7 >= 0,
 An inequality (0, 0, 0, 1, 0) x - 2 >= 0,
 An inequality (0, 0, 0, -2, -1) x + 11 >= 0,
 An inequality (0, 0, 0, 0, 1) x - 1 >= 0,
 An inequality (0, 0, 0, 3, 2) x - 14 >= 0)
classmethod set_engine(engine='auto')

Set the engine used to compute triangulations.

INPUT:

  • engine – either ‘auto’ (default), ‘internal’, or ‘TOPCOM’. The latter two 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.

EXAMPLES:

sage: p = PointConfiguration([[0,0],[0,1],[1,0],[1,1],[-1,-1]])
sage: p.set_engine('internal')   # to make doctests independent of TOPCOM
sage: p.triangulate()
(<1,3,4>, <2,3,4>)
sage: p.set_engine('TOPCOM')   # optional - TOPCOM
sage: p.triangulate()          # optional - TOPCOM
(<0,1,2>, <0,1,4>, <0,2,4>, <1,2,3>)
sage: p.set_engine('internal') # optional - TOPCOM
star_center()

Return the center used for star triangulations.

OUTPUT:

A Point if a distinguished star central point has been fixed. ValueError exception is raised otherwise.

EXAMPLES:

sage: pc = PointConfiguration([(1,0),(-1,0),(0,1),(0,2)], star=(0,1)); pc
A point configuration in QQ^2 consisting of 4 points. The
triangulations of this point configuration are assumed to be
connected, not necessarily fine, not necessarily regular, and
star with center P(0, 1).
sage: pc.star_center()
P(0, 1)

sage: pc_nostar = pc.restrict_to_star_triangulations(None)
sage: pc_nostar
A point configuration in QQ^2 consisting of 4 points. The
triangulations of this point configuration are assumed to be
connected, not necessarily fine, not necessarily regular.
sage: pc_nostar.star_center()
Traceback (most recent call last):
...
ValueError: The point configuration has no star center defined.
triangulate(verbose=False)

Return one (in no particular order) triangulation.

INPUT:

  • verbose – boolean. Whether to print out the TOPCOM interaction, if any.

OUTPUT:

A Triangulation satisfying all restrictions imposed. Raises a ValueError if no such triangulation exists.

EXAMPLES:

sage: p = PointConfiguration([[0,0],[0,1],[1,0],[1,1],[-1,-1]])
sage: p.triangulate()
(<1,3,4>, <2,3,4>)
sage: list( p.triangulate() )
[(1, 3, 4), (2, 3, 4)]

Using TOPCOM yields a different, but equally good, triangulation:

sage: p.set_engine('TOPCOM')           # optional - TOPCOM
sage: p.triangulate()                  # optional - TOPCOM
(<0,1,2>, <0,1,4>, <0,2,4>, <1,2,3>)
sage: list( p.triangulate() )          # optional - TOPCOM
[(0, 1, 2), (0, 1, 4), (0, 2, 4), (1, 2, 3)]
sage: p.set_engine('internal')         # optional - TOPCOM
triangulations(verbose=False)

Returns all triangulations.

  • verbose – boolean (default: False). Whether to print out the TOPCOM interaction, if any.

OUTPUT:

A generator for the triangulations satisfying all the restrictions imposed. Each triangulation is returned as a Triangulation object.

EXAMPLES:

   sage: p = PointConfiguration([[0,0],[0,1],[1,0],[1,1],[-1,-1]])
   sage: iter = p.triangulations()
   sage: iter.next()
   (<1,3,4>, <2,3,4>)
   sage: iter.next()
   (<0,1,3>, <0,1,4>, <0,2,3>, <0,2,4>)
   sage: iter.next()
   (<1,2,3>, <1,2,4>)
   sage: iter.next()
   (<0,1,2>, <0,1,4>, <0,2,4>, <1,2,3>)
   sage: p.triangulations_list()
   [(<1,3,4>, <2,3,4>),
    (<0,1,3>, <0,1,4>, <0,2,3>, <0,2,4>),
    (<1,2,3>, <1,2,4>),
    (<0,1,2>, <0,1,4>, <0,2,4>, <1,2,3>)]
   sage: p_fine = p.restrict_to_fine_triangulations()
   sage: p_fine.triangulations_list()
   [(<0,1,3>, <0,1,4>, <0,2,3>, <0,2,4>),
    (<0,1,2>, <0,1,4>, <0,2,4>, <1,2,3>)]

Note that we explicitly asked the internal algorithm to
compute the triangulations. Using TOPCOM, we obtain the same
triangulations but in a different order::

   sage: p.set_engine('TOPCOM')                       # optional - TOPCOM
   sage: iter = p.triangulations()                    # optional - TOPCOM
   sage: iter.next()                                  # optional - TOPCOM
   (<0,1,2>, <0,1,4>, <0,2,4>, <1,2,3>)
   sage: iter.next()                                  # optional - TOPCOM
   (<0,1,3>, <0,1,4>, <0,2,3>, <0,2,4>)
   sage: iter.next()                                  # optional - TOPCOM
   (<1,2,3>, <1,2,4>)
   sage: iter.next()                                  # optional - TOPCOM
   (<1,3,4>, <2,3,4>)
   sage: p.triangulations_list()                      # optional - TOPCOM
   [(<0,1,2>, <0,1,4>, <0,2,4>, <1,2,3>),
    (<0,1,3>, <0,1,4>, <0,2,3>, <0,2,4>),
    (<1,2,3>, <1,2,4>),
    (<1,3,4>, <2,3,4>)]
   sage: p_fine = p.restrict_to_fine_triangulations() # optional - TOPCOM
   sage: p_fine.set_engine('TOPCOM')                  # optional - TOPCOM
   sage: p_fine.triangulations_list()                 # optional - TOPCOM
   [(<0,1,2>, <0,1,4>, <0,2,4>, <1,2,3>),
    (<0,1,3>, <0,1,4>, <0,2,3>, <0,2,4>)]
   sage: p.set_engine('internal')                     # optional - TOPCOM
triangulations_list(verbose=False)

Return all triangulations.

INPUT:

  • verbose – boolean. Whether to print out the TOPCOM interaction, if any.

OUTPUT:

A list of triangulations (see Triangulation) satisfying all restrictions imposed previously.

EXAMPLES:

sage: p = PointConfiguration([[0,0],[0,1],[1,0],[1,1]])
sage: p.triangulations_list()
[(<0,1,2>, <1,2,3>), (<0,1,3>, <0,2,3>)]
sage: map(list, p.triangulations_list() )
[[(0, 1, 2), (1, 2, 3)], [(0, 1, 3), (0, 2, 3)]]
sage: p.set_engine('TOPCOM')       # optional - TOPCOM
sage: p.triangulations_list()      # optional - TOPCOM
[(<0,1,2>, <1,2,3>), (<0,1,3>, <0,2,3>)]
sage: p.set_engine('internal')     # optional - TOPCOM
volume(simplex=None)

Find n! times the n-volume of a simplex of dimension n.

INPUT:

  • simplex (optional argument) – a simplex from a triangulation T specified as a list of point indices.

OUTPUT:

  • If a simplex was passed as an argument: n!*(volume of simplex).
  • Without argument: n!*(the total volume of the convex hull).

EXAMPLES:

The volume of the standard simplex should always be 1:

sage: p = PointConfiguration([[0,0],[1,0],[0,1],[1,1]])
sage: p.volume( [0,1,2] )
1
sage: simplex = p.triangulate()[0]  # first simplex of triangulation
sage: p.volume(simplex)
1

The square can be triangulated into two minimal simplices, so in the “integral” normalization its volume equals two:

sage: p.volume()
2

Note

We return n!*(metric volume of the simplex) to ensure that the volume is an integer. Essentially, this normalizes things so that the volume of the standard n-simplex is 1. See [GKZ] page 182.

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