A triangulation
In Sage, the PointConfiguration and Triangulation satisfy a parent/element relationship. In particular, each triangulation refers back to its point configuration. If you want to triangulate a point configuration, you should construct a point configuration first and then use one of its methods to triangulate it according to your requirements. You should never have to construct a Triangulation object directly.
EXAMPLES:
First, we select the internal implementation for enumerating triangulations:
sage: PointConfiguration.set_engine('internal') # to make doctests independent of TOPCOM
Here is a simple example of how to triangulate a 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(); triang
(<0,1,2,5>, <0,1,3,5>, <1,3,4,5>)
sage: triang.plot(axes=False)
See sage.geometry.triangulation.point_configuration for more details.
Bases: sage.structure.element.Element
A triangulation of a PointConfiguration.
Warning
You should never create Triangulation objects manually. See triangulate() and triangulations() to triangulate point configurations.
Returns a graph showing which simplices are adjacent in the triangulation
OUTPUT:
A graph consisting of vertices referring to the simplices in the triangulation, and edges showing which simplices are adjacent to each other.
See also
AUTHORS:
EXAMPLES:
sage: p = PointConfiguration([[1,0,0], [0,1,0], [0,0,1], [-1,0,1],
....: [1,0,-1], [-1,0,0], [0,-1,0], [0,0,-1]])
sage: t = p.triangulate()
sage: t.adjacency_graph()
Graph on 8 vertices
Return the boundary of the triangulation.
OUTPUT:
The outward-facing boundary simplices (of dimension \(d-1\)) of the \(d\)-dimensional triangulation as a set. Each boundary is returned by a tuple of point indices.
EXAMPLES:
sage: triangulation = polytopes.n_cube(3).triangulate(engine='internal')
sage: triangulation
(<0,1,2,7>, <0,1,4,7>, <0,2,4,7>, <1,2,3,7>, <1,4,5,7>, <2,4,6,7>)
sage: triangulation.boundary()
frozenset([(1, 3, 7), (4, 5, 7), (1, 2, 3), (0, 1, 2), (2, 4, 6), (2, 6, 7),
(2, 3, 7), (1, 5, 7), (0, 1, 4), (1, 4, 5), (4, 6, 7), (0, 2, 4)])
sage: triangulation.interior_facets()
frozenset([(1, 4, 7), (1, 2, 7), (2, 4, 7), (0, 1, 7), (0, 4, 7), (0, 2, 7)])
Return the enumerated simplices.
OUTPUT:
A tuple of integers that uniquely specifies the triangulation.
EXAMPLES:
sage: pc = PointConfiguration(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]
... ]).columns())
sage: triangulation = pc.lexicographic_triangulation()
sage: triangulation.enumerate_simplices()
(1678, 1688, 1769, 1779, 1895, 1905, 2112, 2143, 2234, 2360, 2555, 2580,
2610, 2626, 2650, 2652, 2654, 2661, 2663, 2667, 2685, 2755, 2757, 2759,
2766, 2768, 2772, 2811, 2881, 2883, 2885, 2892, 2894, 2898)
You can recreate the triangulation from this list by passing it to the constructor:
sage: from sage.geometry.triangulation.point_configuration import Triangulation
sage: Triangulation([1678, 1688, 1769, 1779, 1895, 1905, 2112, 2143,
... 2234, 2360, 2555, 2580, 2610, 2626, 2650, 2652, 2654, 2661, 2663,
... 2667, 2685, 2755, 2757, 2759, 2766, 2768, 2772, 2811, 2881, 2883,
... 2885, 2892, 2894, 2898], pc)
(<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>)
Construct the fan of cones over the simplices of the triangulation.
INPUT:
OUTPUT:
A RationalPolyhedralFan. The coordinates of the points are shifted so that the apex of the fan is the origin of the coordinate system.
Note
If the set of cones over the simplices is not a fan, a suitable exception is raised.
EXAMPLES:
sage: pc = PointConfiguration([(0,0), (1,0), (0,1), (-1,-1)], star=0, fine=True)
sage: triangulation = pc.triangulate()
sage: fan = triangulation.fan(); fan
Rational polyhedral fan in 2-d lattice N
sage: fan.is_equivalent( toric_varieties.P2().fan() )
True
Toric diagrams (the \(\ZZ_5\) hyperconifold):
sage: vertices=[(0, 1, 0), (0, 3, 1), (0, 2, 3), (0, 0, 2)]
sage: interior=[(0, 1, 1), (0, 1, 2), (0, 2, 1), (0, 2, 2)]
sage: points = vertices+interior
sage: pc = PointConfiguration(points, fine=True)
sage: triangulation = pc.triangulate()
sage: fan = triangulation.fan( (-1,0,0) )
sage: fan
Rational polyhedral fan in 3-d lattice N
sage: fan.rays()
N(1, 1, 0),
N(1, 3, 1),
N(1, 2, 3),
N(1, 0, 2),
N(1, 1, 1),
N(1, 1, 2),
N(1, 2, 1),
N(1, 2, 2)
in 3-d lattice N
Calculate the GKZ phi vector of the triangulation.
The phi vector is a vector of length equals to the number of points in the point configuration. For a fixed triangulation \(T\), the entry corresponding to the \(i\)-th point \(p_i\) is
that is, the total volume of all simplices containing \(p_i\). See also [GKZ] page 220 equation 1.4.
OUTPUT:
The phi vector of self.
EXAMPLES:
sage: p = PointConfiguration([[0,0],[1,0],[2,1],[1,2],[0,1]])
sage: p.triangulate().gkz_phi()
(3, 1, 5, 2, 4)
sage: p.lexicographic_triangulation().gkz_phi()
(1, 3, 4, 2, 5)
Return the interior facets of the triangulation.
OUTPUT:
The inward-facing boundary simplices (of dimension \(d-1\)) of the \(d\)-dimensional triangulation as a set. Each boundary is returned by a tuple of point indices.
EXAMPLES:
sage: triangulation = polytopes.n_cube(3).triangulate(engine='internal')
sage: triangulation
(<0,1,2,7>, <0,1,4,7>, <0,2,4,7>, <1,2,3,7>, <1,4,5,7>, <2,4,6,7>)
sage: triangulation.boundary()
frozenset([(1, 3, 7), (4, 5, 7), (1, 2, 3), (0, 1, 2), (2, 4, 6), (2, 6, 7),
(2, 3, 7), (1, 5, 7), (0, 1, 4), (1, 4, 5), (4, 6, 7), (0, 2, 4)])
sage: triangulation.interior_facets()
frozenset([(1, 4, 7), (1, 2, 7), (2, 4, 7), (0, 1, 7), (0, 4, 7), (0, 2, 7)])
Return the (closure of the) normal cone of the triangulation.
Recall that a regular triangulation is one that equals the “crease lines” of a convex piecewise-linear function. This support function is not unique, for example, you can scale it by a positive constant. The set of all piecewise-linear functions with fixed creases forms an open cone. This cone can be interpreted as the cone of normal vectors at a point of the secondary polytope, which is why we call it normal cone. See [GKZ] Section 7.1 for details.
OUTPUT:
The closure of the normal cone. The \(i\)-th entry equals the value of the piecewise-linear function at the \(i\)-th point of the configuration.
For an irregular triangulation, the normal cone is empty. In this case, a single point (the origin) is returned.
EXAMPLES:
sage: triangulation = polytopes.n_cube(2).triangulate(engine='internal')
sage: triangulation
(<0,1,3>, <0,2,3>)
sage: N = triangulation.normal_cone(); N
4-d cone in 4-d lattice
sage: N.rays()
(-1, 0, 0, 0),
( 1, 0, 1, 0),
(-1, 0, -1, 0),
( 1, 0, 0, -1),
(-1, 0, 0, 1),
( 1, 1, 0, 0),
(-1, -1, 0, 0)
in Ambient free module of rank 4
over the principal ideal domain Integer Ring
sage: N.dual().rays()
(-1, 1, 1, -1)
in Ambient free module of rank 4
over the principal ideal domain Integer Ring
TESTS:
sage: polytopes.n_simplex(2).triangulate().normal_cone()
3-d cone in 3-d lattice
sage: _.dual().is_trivial()
True
Produce a graphical representation of the triangulation.
EXAMPLES:
sage: p = PointConfiguration([[0,0],[0,1],[1,0],[1,1],[-1,-1]])
sage: triangulation = p.triangulate()
sage: triangulation
(<1,3,4>, <2,3,4>)
sage: triangulation.plot(axes=False)
Returns the point configuration underlying the triangulation.
EXAMPLES:
sage: pconfig = PointConfiguration([[0,0],[0,1],[1,0]])
sage: pconfig
A point configuration in QQ^2 consisting of 3 points. The
triangulations of this point configuration are assumed to
be connected, not necessarily fine, not necessarily regular.
sage: triangulation = pconfig.triangulate()
sage: triangulation
(<0,1,2>)
sage: triangulation.point_configuration()
A point configuration in QQ^2 consisting of 3 points. The
triangulations of this point configuration are assumed to
be connected, not necessarily fine, not necessarily regular.
sage: pconfig == triangulation.point_configuration()
True
Return a simplicial complex from a triangulation of the point configuration.
OUTPUT:
EXAMPLES:
sage: p = polytopes.cuboctahedron()
sage: sc = p.triangulate(engine='internal').simplicial_complex()
sage: sc
Simplicial complex with 12 vertices and 16 facets
Any convex set is contractable, so its reduced homology groups vanish:
sage: sc.homology()
{0: 0, 1: 0, 2: 0, 3: 0}
Return a graphical representation of a 2-d triangulation.
INPUT:
OUTPUT:
A 2-d graphics object.
EXAMPLES:
sage: points = PointConfiguration([[0,0],[0,1],[1,0],[1,1],[-1,-1]])
sage: triang = points.triangulate()
sage: triang.plot(axes=False, aspect_ratio=1) # indirect doctest
Return a graphical representation of a 3-d triangulation.
INPUT:
OUTPUT:
A 3-d graphics object.
EXAMPLES:
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) # indirect doctest