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; see topcom: Compute triangulations of point configurations and oriented matroids.

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

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

Python

>>> from sage.all import *
>>> PointConfiguration.set_engine('internal')   # to make doctests independent of TOPCOM

A 2-dimensional point configuration:

Sage

sage: p = PointConfiguration([[0,0], [0,1], [1,0], [1,1], [-1,-1]]); p
A point configuration in affine 2-space over Integer Ring consisting
of 5 points. The triangulations of this point configuration are
assumed to be connected, not necessarily fine, not necessarily regular.

Python

>>> from sage.all import *
>>> p = PointConfiguration([[Integer(0),Integer(0)], [Integer(0),Integer(1)], [Integer(1),Integer(0)], [Integer(1),Integer(1)], [-Integer(1),-Integer(1)]]); p
A point configuration in affine 2-space over Integer Ring consisting
of 5 points. The triangulations of this point configuration are
assumed to be connected, not necessarily fine, not necessarily regular.

../../../_images/point_configuration-1.svg

A triangulation of it:

Sage

sage: t = p.triangulate(); t  # a single triangulation
(<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)                                                            # needs sage.plot
Graphics object consisting of 12 graphics primitives

Python

>>> from sage.all import *
>>> t = p.triangulate(); t  # a single triangulation
(<1,3,4>, <2,3,4>)
>>> len(t)
2
>>> t[Integer(0)]
(1, 3, 4)
>>> t[Integer(1)]
(2, 3, 4)
>>> list(t)
[(1, 3, 4), (2, 3, 4)]
>>> t.plot(axes=False)                                                            # needs sage.plot
Graphics object consisting of 12 graphics primitives

../../../_images/point_configuration-2.svg

List triangulations of it:

Sage

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(); p_fine
A point configuration in affine 2-space over Integer Ring 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>)]

Python

>>> from sage.all import *
>>> 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>)]
>>> p_fine = p.restrict_to_fine_triangulations(); p_fine
A point configuration in affine 2-space over Integer Ring consisting
of 5 points. The triangulations of this point configuration are
assumed to be connected, fine, not necessarily regular.
>>> 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

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)                                                       # needs sage.plot
Graphics3d Object

Python

>>> from sage.all import *
>>> p = [[Integer(0),-Integer(1),-Integer(1)], [Integer(0),Integer(0),Integer(1)], [Integer(0),Integer(1),Integer(0)], [Integer(1),-Integer(1),-Integer(1)], [Integer(1),Integer(0),Integer(1)], [Integer(1),Integer(1),Integer(0)]]
>>> points = PointConfiguration(p)
>>> triang = points.triangulate()
>>> triang.plot(axes=False)                                                       # needs sage.plot
Graphics3d Object

../../../_images/point_configuration-3.svg

The standard example of a non-regular triangulation (requires TOPCOM):

Sage

sage: # optional - topcom
sage: PointConfiguration.set_engine('topcom')
sage: p = PointConfiguration([[-1,-5/9], [0,10/9], [1,-5/9],
....:                         [-2,-10/9], [0,20/9], [2,-10/9]])
sage: p_regular = p.restrict_to_regular_triangulations(True)
sage: regular = p_regular.triangulations_list()
sage: p_nonregular = p.restrict_to_regular_triangulations(False)
sage: nonregular = p_nonregular.triangulations_list()
sage: len(regular)
16
sage: len(nonregular)
2
sage: nonregular[0].plot(aspect_ratio=1, axes=False)                                # needs sage.plot
Graphics object consisting of 25 graphics primitives
sage: PointConfiguration.set_engine('internal')   # to make doctests independent of TOPCOM

Python

>>> from sage.all import *
>>> # optional - topcom
>>> PointConfiguration.set_engine('topcom')
>>> p = PointConfiguration([[-Integer(1),-Integer(5)/Integer(9)], [Integer(0),Integer(10)/Integer(9)], [Integer(1),-Integer(5)/Integer(9)],
...                         [-Integer(2),-Integer(10)/Integer(9)], [Integer(0),Integer(20)/Integer(9)], [Integer(2),-Integer(10)/Integer(9)]])
>>> p_regular = p.restrict_to_regular_triangulations(True)
>>> regular = p_regular.triangulations_list()
>>> p_nonregular = p.restrict_to_regular_triangulations(False)
>>> nonregular = p_nonregular.triangulations_list()
>>> len(regular)
16
>>> len(nonregular)
2
>>> nonregular[Integer(0)].plot(aspect_ratio=Integer(1), axes=False)                                # needs sage.plot
Graphics object consisting of 25 graphics primitives
>>> PointConfiguration.set_engine('internal')   # to make doctests independent of 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

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

Python

>>> from sage.all import *
>>> points = [[Integer(0),Integer(0),Integer(0),Integer(1)], [Integer(0),Integer(3),Integer(0),Integer(1)], [Integer(3),Integer(0),Integer(0),Integer(1)], [Integer(0),Integer(0),Integer(1),Integer(1)],
...           [Integer(0),Integer(3),Integer(1),Integer(1)], [Integer(3),Integer(0),Integer(1),Integer(1)], [Integer(1),Integer(1),Integer(2),Integer(1)]]
>>> points = [p + [Integer(1),Integer(2),Integer(3)] for p in points]
>>> pc = PointConfiguration(points)
>>> pc.ambient_dim()
7
>>> pc.dim()
3
>>> 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>)
>>> _ in pc.triangulations()
True
>>> len(pc.triangulations_list())
26

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 categories. 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)[source]¶

Bases: UniqueRepresentation, 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

sage: p = PointConfiguration([[0,0], [0,1], [1,0], [1,1], [-1,-1]]); p
A point configuration in affine 2-space over Integer Ring
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>)

Python

>>> from sage.all import *
>>> p = PointConfiguration([[Integer(0),Integer(0)], [Integer(0),Integer(1)], [Integer(1),Integer(0)], [Integer(1),Integer(1)], [-Integer(1),-Integer(1)]]); p
A point configuration in affine 2-space over Integer Ring
consisting of 5 points. The triangulations of this point
configuration are assumed to be connected, not necessarily fine,
not necessarily regular.
>>> p.triangulate()  # a single triangulation
(<1,3,4>, <2,3,4>)

Element[source]¶: alias of Triangulation

Gale_transform(points=None)[source]¶

Return the Gale transform of self.

INPUT:

points – 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

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]

Python

>>> from sage.all import *
>>> pc = PointConfiguration([(Integer(0),Integer(0)), (Integer(1),Integer(0)), (Integer(2),Integer(1)), (Integer(1),Integer(1)), (Integer(0),Integer(1))])
>>> pc.Gale_transform()
[ 1 -1  0  1 -1]
[ 0  0  1 -2  1]

>>> pc.Gale_transform((Integer(0),Integer(1),Integer(3),Integer(4)))
[ 1 -1  1 -1]

>>> points = (pc.point(Integer(0)), pc.point(Integer(1)), pc.point(Integer(3)), pc.point(Integer(4)))
>>> pc.Gale_transform(points)
[ 1 -1  1 -1]

an_element()[source]¶: Synonymous for triangulate().

bistellar_flips()[source]¶

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

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)                                                 # needs sage.plot
Graphics object consisting of 11 graphics primitives
sage: Tneg.plot(axes=False)                                                 # needs sage.plot
Graphics object consisting of 11 graphics primitives

Python

>>> from sage.all import *
>>> pc = PointConfiguration([(Integer(0),Integer(0)),(Integer(1),Integer(0)),(Integer(0),Integer(1)),(Integer(1),Integer(1))])
>>> pc.bistellar_flips()
(((<0,1,3>, <0,2,3>), (<0,1,2>, <1,2,3>)),)
>>> Tpos, Tneg = pc.bistellar_flips()[Integer(0)]
>>> Tpos.plot(axes=False)                                                 # needs sage.plot
Graphics object consisting of 11 graphics primitives
>>> Tneg.plot(axes=False)                                                 # needs sage.plot
Graphics object consisting of 11 graphics primitives

The 3d analog:

Sage

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>)),)

Python

>>> from sage.all import *
>>> pc = PointConfiguration([(Integer(0),Integer(0),Integer(0)),(Integer(0),Integer(2),Integer(0)),(Integer(0),Integer(0),Integer(2)),(-Integer(1),Integer(0),Integer(0)),(Integer(1),Integer(1),Integer(1))])
>>> 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

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)                                                 # needs sage.plot
Graphics3d Object

Python

>>> from sage.all import *
>>> pc = PointConfiguration([(Integer(0),Integer(0),Integer(0)),(Integer(0),Integer(2),Integer(0)),(Integer(0),Integer(0),Integer(2)),(Integer(0),Integer(2),Integer(2)),(Integer(1),Integer(1),Integer(1))])
>>> pc.bistellar_flips()
(((<0,1,3>, <0,2,3>), (<0,1,2>, <1,2,3>)),)
>>> Tpos, Tneg = pc.bistellar_flips()[Integer(0)]
>>> Tpos.plot(axes=False)                                                 # needs sage.plot
Graphics3d Object

circuits()[source]¶

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

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

Python

>>> from sage.all import *
>>> p = PointConfiguration([(Integer(0),Integer(0)), (+Integer(1),Integer(0)), (-Integer(1),Integer(0)), (Integer(0),+Integer(1)), (Integer(0),-Integer(1))])
>>> sorted(p.circuits())
[((0,), (1, 2), (3, 4)), ((0,), (3, 4), (1, 2)), ((1, 2), (0,), (3, 4))]

circuits_support()[source]¶

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

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

Python

>>> from sage.all import *
>>> p = PointConfiguration([(Integer(0),Integer(0)), (+Integer(1),Integer(0)), (-Integer(1),Integer(0)), (Integer(0),+Integer(1)), (Integer(0),-Integer(1))])
>>> sorted(p.circuits_support())
[(0, 1, 2), (0, 3, 4), (1, 2, 3, 4)]

contained_simplex(large=True, initial_point=None, point_order=None)[source]¶

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.
point_order – list or tuple of (some or all) Point s or None (default)

OUTPUT:

A tuple of points that span a simplex of dimension dim(). If large==True, the simplex is constructed by successively picking the farthest point. This will ensure that the simplex is not unnecessarily small, but will in general not return a maximal simplex. If a point_order is specified, the simplex is greedily constructed by considering the points in this order. The large option and initial_point is ignored in this case. The point_order may contain only a subset of the points; in this case, the dimension of the simplex will be the dimension of this subset.

EXAMPLES:

Sage

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(2))
(P(2, 1), P(0, 0), 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))
sage: pc.contained_simplex(point_order=[pc[1], pc[3], pc[4], pc[2], pc[0]])
(P(0, 1), P(1, 1), P(-1, -1))

Python

>>> from sage.all import *
>>> pc = PointConfiguration([(Integer(0),Integer(0)), (Integer(1),Integer(0)), (Integer(2),Integer(1)), (Integer(1),Integer(1)), (Integer(0),Integer(1))])
>>> pc.contained_simplex()
(P(0, 1), P(2, 1), P(1, 0))
>>> pc.contained_simplex(large=False)
(P(0, 1), P(1, 1), P(1, 0))
>>> pc.contained_simplex(initial_point=pc.point(Integer(2)))
(P(2, 1), P(0, 0), P(1, 0))

>>> pc = PointConfiguration([[Integer(0),Integer(0)], [Integer(0),Integer(1)], [Integer(1),Integer(0)], [Integer(1),Integer(1)], [-Integer(1),-Integer(1)]])
>>> pc.contained_simplex()
(P(-1, -1), P(1, 1), P(0, 1))
>>> pc.contained_simplex(point_order=[pc[Integer(1)], pc[Integer(3)], pc[Integer(4)], pc[Integer(2)], pc[Integer(0)]])
(P(0, 1), P(1, 1), P(-1, -1))

Lower-dimensional example:

Sage

sage: pc.contained_simplex(point_order=[pc[0], pc[3], pc[4]])
(P(0, 0), P(1, 1))

Python

>>> from sage.all import *
>>> pc.contained_simplex(point_order=[pc[Integer(0)], pc[Integer(3)], pc[Integer(4)]])
(P(0, 0), P(1, 1))

convex_hull()[source]¶

Return the convex hull of the point configuration.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> p = PointConfiguration([[Integer(0),Integer(0)], [Integer(0),Integer(1)], [Integer(1),Integer(0)], [Integer(1),Integer(1)], [-Integer(1),-Integer(1)]])
>>> p.convex_hull()
A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 4 vertices

distance(x, y)[source]¶

Return 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

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]

Python

>>> from sage.all import *
>>> pc = PointConfiguration([(Integer(0),Integer(0)), (Integer(1),Integer(0)), (Integer(2),Integer(1)), (Integer(1),Integer(2)), (Integer(0),Integer(1))])
>>> [pc.distance(pc.point(Integer(0)), p) for p in pc.points()]
[0, 1, 5, 5, 1]

>>> pc = PointConfiguration([(Integer(0),Integer(0),Integer(1)), (Integer(1),Integer(0),Integer(1)), (Integer(2),Integer(1),Integer(1)), (Integer(1),Integer(2),Integer(1)), (Integer(0),Integer(1),Integer(1))],
...                         projective=True)
>>> [pc.distance(pc.point(Integer(0)), p) for p in pc.points()]
[0, 1/2, 5/6, 5/6, 1/2]

distance_FS(x, y)[source]¶

Return 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

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]

Python

>>> from sage.all import *
>>> pc = PointConfiguration([(Integer(0),Integer(0)), (Integer(1),Integer(0)), (Integer(2),Integer(1)), (Integer(1),Integer(2)), (Integer(0),Integer(1))])
>>> [pc.distance_FS(pc.point(Integer(0)), p) for p in pc.points()]
[0, 1/2, 5/6, 5/6, 1/2]

distance_affine(x, y)[source]¶

Return 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

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]

Python

>>> from sage.all import *
>>> pc = PointConfiguration([(Integer(0),Integer(0)),(Integer(1),Integer(0)),(Integer(2),Integer(1)),(Integer(1),Integer(2)),(Integer(0),Integer(1))])
>>> [pc.distance_affine(pc.point(Integer(0)), p) for p in pc.points()]
[0, 1, 5, 5, 1]

exclude_points(point_idx_list)[source]¶

Return a new point configuration with the given points removed.

INPUT:

point_idx_list – list of integers; the indices of points to exclude

OUTPUT:

A new PointConfiguration with the given points removed.

EXAMPLES:

Sage

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))

Python

>>> from sage.all import *
>>> p = PointConfiguration([[-Integer(1),Integer(0)], [Integer(0),Integer(0)], [Integer(1),-Integer(1)], [Integer(1),Integer(0)], [Integer(1),Integer(1)]])
>>> list(p)
[P(-1, 0), P(0, 0), P(1, -1), P(1, 0), P(1, 1)]
>>> q = p.exclude_points([Integer(3)])
>>> list(q)
[P(-1, 0), P(0, 0), P(1, -1), P(1, 1)]
>>> p.exclude_points(p.face_interior(codim=Integer(1))).points()
(P(-1, 0), P(0, 0), P(1, -1), P(1, 1))

face_codimension(point)[source]¶

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

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> triangle = PointConfiguration([[Integer(0),Integer(0)], [Integer(1),-Integer(1)], [Integer(1),Integer(0)], [Integer(1),Integer(1)]])
>>> triangle.point(Integer(2))
P(1, 0)
>>> triangle.face_codimension(Integer(2))
1
>>> triangle.face_codimension([Integer(1),Integer(0)])
1

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

Sage

sage: pyramid = PointConfiguration([[1,0,0], [0,1,1], [0,1,-1],
....:                               [0,-1,-1], [0,-1,1]])
sage: pyramid.face_codimension(0)
3

Python

>>> from sage.all import *
>>> pyramid = PointConfiguration([[Integer(1),Integer(0),Integer(0)], [Integer(0),Integer(1),Integer(1)], [Integer(0),Integer(1),-Integer(1)],
...                               [Integer(0),-Integer(1),-Integer(1)], [Integer(0),-Integer(1),Integer(1)]])
>>> pyramid.face_codimension(Integer(0))
3

face_interior(dim=None, codim=None)[source]¶

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

EXAMPLES:

Sage

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,)

Python

>>> from sage.all import *
>>> triangle = PointConfiguration([[-Integer(1),Integer(0)], [Integer(0),Integer(0)], [Integer(1),-Integer(1)], [Integer(1),Integer(0)], [Integer(1),Integer(1)]])
>>> triangle.face_interior()
((1,), (3,), (0, 2, 4))
>>> triangle.face_interior(dim=Integer(0))    # the vertices of the convex hull
(0, 2, 4)
>>> triangle.face_interior(codim=Integer(1))  # interior of facets
(3,)

farthest_point(points, among=None)[source]¶

Return the point with the most distance from points.

INPUT:

points – list of points
among – 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

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

Python

>>> from sage.all import *
>>> pc = PointConfiguration([(Integer(0),Integer(0)), (Integer(1),Integer(0)), (Integer(1),Integer(1)), (Integer(0),Integer(1))])
>>> pc.farthest_point([pc.point(Integer(0))])
P(1, 1)

lexicographic_triangulation()[source]¶

Return the lexicographic triangulation.

The algorithm was taken from [PUNTOS].

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> p = PointConfiguration([(Integer(0),Integer(0)), (+Integer(1),Integer(0)), (-Integer(1),Integer(0)), (Integer(0),+Integer(1)), (Integer(0),-Integer(1))])
>>> p.lexicographic_triangulation()
(<1,3,4>, <2,3,4>)

placing_triangulation(point_order=None)[source]¶

Construct the placing (pushing) triangulation.

INPUT:

point_order – list of points or integers. The order in which the points are to be placed. If not given, the points will be placed in some arbitrary order that attempts to produce a small number of simplices.

OUTPUT: a Triangulation

EXAMPLES:

Sage

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: pc.placing_triangulation(point_order=(3,2,1,4,0))
(<0,1,4>, <1,2,3>, <1,3,4>)
sage: pc.placing_triangulation(point_order=[pc[1], pc[3], pc[4], pc[0]])
(<0,1,4>, <1,3,4>)
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
sage: p0 = PointConfiguration([(0,0), (+1,0), (-1,0), (0,+1), (0,-1)])
sage: p0.pushing_triangulation(point_order=[1,2,0,3,4])
(<1,2,3>, <1,2,4>)
sage: p0.pushing_triangulation(point_order=[0,1,2,3,4])
(<0,1,3>, <0,1,4>, <0,2,3>, <0,2,4>)

Python

>>> from sage.all import *
>>> pc = PointConfiguration([(Integer(0),Integer(0)), (Integer(1),Integer(0)), (Integer(2),Integer(1)), (Integer(1),Integer(2)), (Integer(0),Integer(1))])
>>> pc.placing_triangulation()
(<0,1,2>, <0,2,4>, <2,3,4>)
>>> pc.placing_triangulation(point_order=(Integer(3),Integer(2),Integer(1),Integer(4),Integer(0)))
(<0,1,4>, <1,2,3>, <1,3,4>)
>>> pc.placing_triangulation(point_order=[pc[Integer(1)], pc[Integer(3)], pc[Integer(4)], pc[Integer(0)]])
(<0,1,4>, <1,3,4>)
>>> U = matrix([
...    [ Integer(0), Integer(0), Integer(0), Integer(0), Integer(0), Integer(2), Integer(4),-Integer(1), Integer(1), Integer(1), Integer(0), Integer(0), Integer(1), Integer(0)],
...    [ Integer(0), Integer(0), Integer(0), Integer(1), Integer(0), Integer(0),-Integer(1), Integer(0), Integer(0), Integer(0), Integer(0), Integer(0), Integer(0), Integer(0)],
...    [ Integer(0), Integer(2), Integer(0), Integer(0), Integer(0), Integer(0),-Integer(1), Integer(0), Integer(1), Integer(0), Integer(1), Integer(0), Integer(0), Integer(1)],
...    [ Integer(0), Integer(1), Integer(1), Integer(0), Integer(0), Integer(1), Integer(0),-Integer(2), Integer(1), Integer(0), Integer(0),-Integer(1), Integer(1), Integer(1)],
...    [ Integer(0), Integer(0), Integer(0), Integer(0), Integer(1), Integer(0),-Integer(1), Integer(0), Integer(0), Integer(0), Integer(0), Integer(0), Integer(0), Integer(0)]
... ])
>>> p = PointConfiguration(U.columns())
>>> 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>)
>>> sum(p.volume(t) for t in triangulation)
42
>>> p0 = PointConfiguration([(Integer(0),Integer(0)), (+Integer(1),Integer(0)), (-Integer(1),Integer(0)), (Integer(0),+Integer(1)), (Integer(0),-Integer(1))])
>>> p0.pushing_triangulation(point_order=[Integer(1),Integer(2),Integer(0),Integer(3),Integer(4)])
(<1,2,3>, <1,2,4>)
>>> p0.pushing_triangulation(point_order=[Integer(0),Integer(1),Integer(2),Integer(3),Integer(4)])
(<0,1,3>, <0,1,4>, <0,2,3>, <0,2,4>)

The same triangulation with renumbered points 0->4, 1->0, etc:

Sage

sage: p1 = PointConfiguration([(+1,0), (-1,0), (0,+1), (0,-1), (0,0)])
sage: p1.pushing_triangulation(point_order=[4,0,1,2,3])
(<0,2,4>, <0,3,4>, <1,2,4>, <1,3,4>)

Python

>>> from sage.all import *
>>> p1 = PointConfiguration([(+Integer(1),Integer(0)), (-Integer(1),Integer(0)), (Integer(0),+Integer(1)), (Integer(0),-Integer(1)), (Integer(0),Integer(0))])
>>> p1.pushing_triangulation(point_order=[Integer(4),Integer(0),Integer(1),Integer(2),Integer(3)])
(<0,2,4>, <0,3,4>, <1,2,4>, <1,3,4>)

plot(**kwds)[source]¶

Produce a graphical representation of the point configuration.

EXAMPLES:

Sage

sage: p = PointConfiguration([[0,0], [0,1], [1,0], [1,1], [-1,-1]])
sage: p.plot(axes=False)                                                    # needs sage.plot
Graphics object consisting of 5 graphics primitives

Python

>>> from sage.all import *
>>> p = PointConfiguration([[Integer(0),Integer(0)], [Integer(0),Integer(1)], [Integer(1),Integer(0)], [Integer(1),Integer(1)], [-Integer(1),-Integer(1)]])
>>> p.plot(axes=False)                                                    # needs sage.plot
Graphics object consisting of 5 graphics primitives

../../../_images/point_configuration-4.svg

positive_circuits(*negative)[source]¶

Return 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

EXAMPLES:

Sage

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: sorted(p.positive_circuits(8))
[(0, 1, 2, 5), (0, 1, 4), (0, 2, 3), (0, 3, 4, 6), (0, 5, 6), (0, 7)]
sage: p.positive_circuits(0,5,6)
((8,),)

Python

>>> from sage.all import *
>>> p = PointConfiguration([(Integer(1),Integer(0),Integer(0)), (Integer(0),Integer(1),Integer(0)), (Integer(0),Integer(0),Integer(1)), (-Integer(2),Integer(0),-Integer(1)), (-Integer(2),-Integer(1),Integer(0)),
...                         (-Integer(3),-Integer(1),-Integer(1)), (Integer(1),Integer(1),Integer(1)), (-Integer(1),Integer(0),Integer(0)), (Integer(0),Integer(0),Integer(0))])
>>> sorted(p.positive_circuits(Integer(8)))
[(0, 1, 2, 5), (0, 1, 4), (0, 2, 3), (0, 3, 4, 6), (0, 5, 6), (0, 7)]
>>> p.positive_circuits(Integer(0),Integer(5),Integer(6))
((8,),)

pushing_triangulation(point_order=None)[source]¶

Construct the placing (pushing) triangulation.

INPUT:

point_order – list of points or integers. The order in which the points are to be placed. If not given, the points will be placed in some arbitrary order that attempts to produce a small number of simplices.

OUTPUT: a Triangulation

EXAMPLES:

Sage

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: pc.placing_triangulation(point_order=(3,2,1,4,0))
(<0,1,4>, <1,2,3>, <1,3,4>)
sage: pc.placing_triangulation(point_order=[pc[1], pc[3], pc[4], pc[0]])
(<0,1,4>, <1,3,4>)
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
sage: p0 = PointConfiguration([(0,0), (+1,0), (-1,0), (0,+1), (0,-1)])
sage: p0.pushing_triangulation(point_order=[1,2,0,3,4])
(<1,2,3>, <1,2,4>)
sage: p0.pushing_triangulation(point_order=[0,1,2,3,4])
(<0,1,3>, <0,1,4>, <0,2,3>, <0,2,4>)

Python

>>> from sage.all import *
>>> pc = PointConfiguration([(Integer(0),Integer(0)), (Integer(1),Integer(0)), (Integer(2),Integer(1)), (Integer(1),Integer(2)), (Integer(0),Integer(1))])
>>> pc.placing_triangulation()
(<0,1,2>, <0,2,4>, <2,3,4>)
>>> pc.placing_triangulation(point_order=(Integer(3),Integer(2),Integer(1),Integer(4),Integer(0)))
(<0,1,4>, <1,2,3>, <1,3,4>)
>>> pc.placing_triangulation(point_order=[pc[Integer(1)], pc[Integer(3)], pc[Integer(4)], pc[Integer(0)]])
(<0,1,4>, <1,3,4>)
>>> U = matrix([
...    [ Integer(0), Integer(0), Integer(0), Integer(0), Integer(0), Integer(2), Integer(4),-Integer(1), Integer(1), Integer(1), Integer(0), Integer(0), Integer(1), Integer(0)],
...    [ Integer(0), Integer(0), Integer(0), Integer(1), Integer(0), Integer(0),-Integer(1), Integer(0), Integer(0), Integer(0), Integer(0), Integer(0), Integer(0), Integer(0)],
...    [ Integer(0), Integer(2), Integer(0), Integer(0), Integer(0), Integer(0),-Integer(1), Integer(0), Integer(1), Integer(0), Integer(1), Integer(0), Integer(0), Integer(1)],
...    [ Integer(0), Integer(1), Integer(1), Integer(0), Integer(0), Integer(1), Integer(0),-Integer(2), Integer(1), Integer(0), Integer(0),-Integer(1), Integer(1), Integer(1)],
...    [ Integer(0), Integer(0), Integer(0), Integer(0), Integer(1), Integer(0),-Integer(1), Integer(0), Integer(0), Integer(0), Integer(0), Integer(0), Integer(0), Integer(0)]
... ])
>>> p = PointConfiguration(U.columns())
>>> 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>)
>>> sum(p.volume(t) for t in triangulation)
42
>>> p0 = PointConfiguration([(Integer(0),Integer(0)), (+Integer(1),Integer(0)), (-Integer(1),Integer(0)), (Integer(0),+Integer(1)), (Integer(0),-Integer(1))])
>>> p0.pushing_triangulation(point_order=[Integer(1),Integer(2),Integer(0),Integer(3),Integer(4)])
(<1,2,3>, <1,2,4>)
>>> p0.pushing_triangulation(point_order=[Integer(0),Integer(1),Integer(2),Integer(3),Integer(4)])
(<0,1,3>, <0,1,4>, <0,2,3>, <0,2,4>)

The same triangulation with renumbered points 0->4, 1->0, etc:

Sage

sage: p1 = PointConfiguration([(+1,0), (-1,0), (0,+1), (0,-1), (0,0)])
sage: p1.pushing_triangulation(point_order=[4,0,1,2,3])
(<0,2,4>, <0,3,4>, <1,2,4>, <1,3,4>)

Python

>>> from sage.all import *
>>> p1 = PointConfiguration([(+Integer(1),Integer(0)), (-Integer(1),Integer(0)), (Integer(0),+Integer(1)), (Integer(0),-Integer(1)), (Integer(0),Integer(0))])
>>> p1.pushing_triangulation(point_order=[Integer(4),Integer(0),Integer(1),Integer(2),Integer(3)])
(<0,2,4>, <0,3,4>, <1,2,4>, <1,3,4>)

restrict_to_connected_triangulations(connected=True)[source]¶

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

sage: p = PointConfiguration([[0,0], [0,1], [1,0], [1,1], [-1,-1]]); p
A point configuration in affine 2-space over Integer Ring
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: PointConfiguration.set_engine('topcom')
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
sage: PointConfiguration.set_engine('internal')

Python

>>> from sage.all import *
>>> p = PointConfiguration([[Integer(0),Integer(0)], [Integer(0),Integer(1)], [Integer(1),Integer(0)], [Integer(1),Integer(1)], [-Integer(1),-Integer(1)]]); p
A point configuration in affine 2-space over Integer Ring
consisting of 5 points. The triangulations of this point
configuration are assumed to be connected, not necessarily
fine, not necessarily regular.
>>> len(p.triangulations_list())
4
>>> PointConfiguration.set_engine('topcom')
>>> p_all = p.restrict_to_connected_triangulations(connected=False)  # optional - topcom
>>> len(p_all.triangulations_list())                                 # optional - topcom
4
>>> p == p_all.restrict_to_connected_triangulations(connected=True)  # optional - topcom
True
>>> PointConfiguration.set_engine('internal')

restrict_to_fine_triangulations(fine=True)[source]¶

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

sage: p = PointConfiguration([[0,0], [0,1], [1,0], [1,1], [-1,-1]])
sage: p
A point configuration in affine 2-space over Integer Ring
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

Python

>>> from sage.all import *
>>> p = PointConfiguration([[Integer(0),Integer(0)], [Integer(0),Integer(1)], [Integer(1),Integer(0)], [Integer(1),Integer(1)], [-Integer(1),-Integer(1)]])
>>> p
A point configuration in affine 2-space over Integer Ring
consisting of 5 points. The triangulations of this point
configuration are assumed to be connected, not necessarily
fine, not necessarily regular.

>>> len(p.triangulations_list())
4
>>> p_fine = p.restrict_to_fine_triangulations()
>>> len(p.triangulations_list())
4
>>> p == p_fine.restrict_to_fine_triangulations(fine=False)
True

restrict_to_regular_triangulations(regular=True)[source]¶

Restrict to regular triangulations.

NOTE:

Regularity testing requires the optional TOPCOM package.

INPUT:

regular – True, 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

sage: p = PointConfiguration([[0,0], [0,1], [1,0], [1,1], [-1,-1]]); p
A point configuration in affine 2-space over Integer Ring
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: PointConfiguration.set_engine('topcom')
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
sage: PointConfiguration.set_engine('internal')

Python

>>> from sage.all import *
>>> p = PointConfiguration([[Integer(0),Integer(0)], [Integer(0),Integer(1)], [Integer(1),Integer(0)], [Integer(1),Integer(1)], [-Integer(1),-Integer(1)]]); p
A point configuration in affine 2-space over Integer Ring
consisting of 5 points. The triangulations of this point
configuration are assumed to be connected, not necessarily
fine, not necessarily regular.
>>> len(p.triangulations_list())
4
>>> PointConfiguration.set_engine('topcom')
>>> p_regular = p.restrict_to_regular_triangulations() # optional - topcom
>>> len(p_regular.triangulations_list())               # optional - topcom
4
>>> p == p_regular.restrict_to_regular_triangulations(regular=None) # optional - topcom
True
>>> PointConfiguration.set_engine('internal')

restrict_to_star_triangulations(star)[source]¶

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

INPUT:

origin – None 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

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

Python

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

restricted_automorphism_group()[source]¶

Return the restricted automorphism group.

First, let the linear automorphism group be the subgroup of the affine group \(AGL(d,\RR) = GL(d,\RR) \ltimes \RR^d\) preserving the \(d\)-dimensional point configuration. The affine 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 [BSS2009] 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

sage: pyramid = PointConfiguration([[1,0,0], [0,1,1], [0,1,-1],
....:                               [0,-1,-1], [0,-1,1]])
sage: G = pyramid.restricted_automorphism_group()                           # needs sage.graphs sage.groups
sage: G == PermutationGroup([[(3,5)], [(2,3),(4,5)], [(2,4)]])              # needs sage.graphs sage.groups
True
sage: DihedralGroup(4).is_isomorphic(G)                                     # needs sage.graphs sage.groups
True

Python

>>> from sage.all import *
>>> pyramid = PointConfiguration([[Integer(1),Integer(0),Integer(0)], [Integer(0),Integer(1),Integer(1)], [Integer(0),Integer(1),-Integer(1)],
...                               [Integer(0),-Integer(1),-Integer(1)], [Integer(0),-Integer(1),Integer(1)]])
>>> G = pyramid.restricted_automorphism_group()                           # needs sage.graphs sage.groups
>>> G == PermutationGroup([[(Integer(3),Integer(5))], [(Integer(2),Integer(3)),(Integer(4),Integer(5))], [(Integer(2),Integer(4))]])              # needs sage.graphs sage.groups
True
>>> DihedralGroup(Integer(4)).is_isomorphic(G)                                     # needs sage.graphs sage.groups
True

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

Sage

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

Python

>>> from sage.all import *
>>> square = PointConfiguration([(Integer(3)/Integer(4),Integer(3)/Integer(4)), (Integer(1),Integer(1)), (Integer(1),-Integer(1)), (-Integer(1),-Integer(1)), (-Integer(1),Integer(1))])
>>> square.restricted_automorphism_group()                                # needs sage.graphs sage.groups
Permutation Group with generators [(3,5)]
>>> DihedralGroup(Integer(1)).is_isomorphic(_)                                     # needs sage.graphs sage.groups
True

secondary_polytope()[source]¶

Calculate the secondary polytope of the point configuration.

For a definition of the secondary polytope, see [GKZ1994] 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

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)

Python

>>> from sage.all import *
>>> p = PointConfiguration([[Integer(0),Integer(0)], [Integer(1),Integer(0)], [Integer(2),Integer(1)], [Integer(1),Integer(2)], [Integer(0),Integer(1)]])
>>> poly = p.secondary_polytope()
>>> 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]
>>> 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))
>>> 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')[source]¶

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'), internal routines are used.

EXAMPLES:

Sage

sage: # optional - topcom
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')
sage: p.triangulate()
(<0,1,2>, <0,1,4>, <0,2,4>, <1,2,3>)
sage: p.set_engine('internal')

Python

>>> from sage.all import *
>>> # optional - topcom
>>> p = PointConfiguration([[Integer(0),Integer(0)], [Integer(0),Integer(1)], [Integer(1),Integer(0)], [Integer(1),Integer(1)], [-Integer(1),-Integer(1)]])
>>> p.set_engine('internal')   # to make doctests independent of TOPCOM
>>> p.triangulate()
(<1,3,4>, <2,3,4>)
>>> p.set_engine('topcom')
>>> p.triangulate()
(<0,1,2>, <0,1,4>, <0,2,4>, <1,2,3>)
>>> p.set_engine('internal')

star_center()[source]¶

Return the center used for star triangulations.