Fast Lattice Polygons using PPL¶

See ppl_lattice_polytope for the implementation of arbitrary-dimensional lattice polytopes. This module is about the specialization to 2 dimensions. To be more precise, the LatticePolygon_PPL_class is used if the ambient space is of dimension 2 or less. These all allow you to cyclically order (see LatticePolygon_PPL_class.ordered_vertices()) the vertices, which is in general not possible in higher dimensions.

class sage.geometry.polyhedron.ppl_lattice_polygon.LatticePolygon_PPL_class[source]¶

Bases: LatticePolytope_PPL_class

A lattice polygon.

This includes 2-dimensional polytopes as well as degenerate (0 and 1-dimensional) lattice polygons. Any polytope in 2d is a polygon.

find_isomorphism(polytope)[source]¶

Return a lattice isomorphism with polytope.

INPUT:

polytope – a polytope, potentially higher-dimensional

OUTPUT:

A LatticeEuclideanGroupElement. It is not necessarily invertible if the affine dimension of self or polytope is not two. A LatticePolytopesNotIsomorphicError is raised if no such isomorphism exists.

EXAMPLES:

Sage

sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: L1 = LatticePolytope_PPL((1,0),(0,1),(0,0))
sage: L2 = LatticePolytope_PPL((1,0,3),(0,1,0),(0,0,1))
sage: iso = L1.find_isomorphism(L2)
sage: iso(L1) == L2
True

sage: L1 = LatticePolytope_PPL((0, 1), (3, 0), (0, 3), (1, 0))
sage: L2 = LatticePolytope_PPL((0,0,2,1),(0,1,2,0),(2,0,0,3),(2,3,0,0))
sage: iso = L1.find_isomorphism(L2)
sage: iso(L1) == L2
True

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
>>> L1 = LatticePolytope_PPL((Integer(1),Integer(0)),(Integer(0),Integer(1)),(Integer(0),Integer(0)))
>>> L2 = LatticePolytope_PPL((Integer(1),Integer(0),Integer(3)),(Integer(0),Integer(1),Integer(0)),(Integer(0),Integer(0),Integer(1)))
>>> iso = L1.find_isomorphism(L2)
>>> iso(L1) == L2
True

>>> L1 = LatticePolytope_PPL((Integer(0), Integer(1)), (Integer(3), Integer(0)), (Integer(0), Integer(3)), (Integer(1), Integer(0)))
>>> L2 = LatticePolytope_PPL((Integer(0),Integer(0),Integer(2),Integer(1)),(Integer(0),Integer(1),Integer(2),Integer(0)),(Integer(2),Integer(0),Integer(0),Integer(3)),(Integer(2),Integer(3),Integer(0),Integer(0)))
>>> iso = L1.find_isomorphism(L2)
>>> iso(L1) == L2
True

Sage Live

from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
L1 = LatticePolytope_PPL((1,0),(0,1),(0,0))
L2 = LatticePolytope_PPL((1,0,3),(0,1,0),(0,0,1))
iso = L1.find_isomorphism(L2)
iso(L1) == L2
L1 = LatticePolytope_PPL((0, 1), (3, 0), (0, 3), (1, 0))
L2 = LatticePolytope_PPL((0,0,2,1),(0,1,2,0),(2,0,0,3),(2,3,0,0))
iso = L1.find_isomorphism(L2)
iso(L1) == L2

The following polygons are isomorphic over \(\QQ\), but not as lattice polytopes:

Sage

sage: L1 = LatticePolytope_PPL((1,0),(0,1),(-1,-1))
sage: L2 = LatticePolytope_PPL((0, 0), (0, 1), (1, 0))
sage: L1.find_isomorphism(L2)
Traceback (most recent call last):
...
LatticePolytopesNotIsomorphicError: different number of integral points
sage: L2.find_isomorphism(L1)
Traceback (most recent call last):
...
LatticePolytopesNotIsomorphicError: different number of integral points

Python

>>> from sage.all import *
>>> L1 = LatticePolytope_PPL((Integer(1),Integer(0)),(Integer(0),Integer(1)),(-Integer(1),-Integer(1)))
>>> L2 = LatticePolytope_PPL((Integer(0), Integer(0)), (Integer(0), Integer(1)), (Integer(1), Integer(0)))
>>> L1.find_isomorphism(L2)
Traceback (most recent call last):
...
LatticePolytopesNotIsomorphicError: different number of integral points
>>> L2.find_isomorphism(L1)
Traceback (most recent call last):
...
LatticePolytopesNotIsomorphicError: different number of integral points

Sage Live

L1 = LatticePolytope_PPL((1,0),(0,1),(-1,-1))
L2 = LatticePolytope_PPL((0, 0), (0, 1), (1, 0))
L1.find_isomorphism(L2)
L2.find_isomorphism(L1)

is_isomorphic(polytope)[source]¶

Test if self and polytope are isomorphic.

INPUT:

polytope – a lattice polytope

OUTPUT: boolean

EXAMPLES:

Sage

sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: L1 = LatticePolytope_PPL((1,0),(0,1),(0,0))
sage: L2 = LatticePolytope_PPL((1,0,3),(0,1,0),(0,0,1))
sage: L1.is_isomorphic(L2)
True

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
>>> L1 = LatticePolytope_PPL((Integer(1),Integer(0)),(Integer(0),Integer(1)),(Integer(0),Integer(0)))
>>> L2 = LatticePolytope_PPL((Integer(1),Integer(0),Integer(3)),(Integer(0),Integer(1),Integer(0)),(Integer(0),Integer(0),Integer(1)))
>>> L1.is_isomorphic(L2)
True

Sage Live

from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
L1 = LatticePolytope_PPL((1,0),(0,1),(0,0))
L2 = LatticePolytope_PPL((1,0,3),(0,1,0),(0,0,1))
L1.is_isomorphic(L2)

ordered_vertices()[source]¶

Return the vertices of a lattice polygon in cyclic order.

OUTPUT:

A tuple of vertices ordered along the perimeter of the polygon. The first point is arbitrary.

EXAMPLES:

Sage

sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: square = LatticePolytope_PPL((0,0), (1,1), (0,1), (1,0))
sage: square.vertices()
((0, 0), (0, 1), (1, 0), (1, 1))
sage: square.ordered_vertices()
((0, 0), (1, 0), (1, 1), (0, 1))

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
>>> square = LatticePolytope_PPL((Integer(0),Integer(0)), (Integer(1),Integer(1)), (Integer(0),Integer(1)), (Integer(1),Integer(0)))
>>> square.vertices()
((0, 0), (0, 1), (1, 0), (1, 1))
>>> square.ordered_vertices()
((0, 0), (1, 0), (1, 1), (0, 1))

Sage Live

from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
square = LatticePolytope_PPL((0,0), (1,1), (0,1), (1,0))
square.vertices()
square.ordered_vertices()

plot()[source]¶

Plot the lattice polygon.

OUTPUT: a graphics object

EXAMPLES:

Sage

sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: P = LatticePolytope_PPL((1,0), (0,1), (0,0), (2,2))
sage: P.plot()                                                              # needs sage.plot
Graphics object consisting of 6 graphics primitives
sage: LatticePolytope_PPL([0], [1]).plot()                                  # needs sage.plot
Graphics object consisting of 3 graphics primitives
sage: LatticePolytope_PPL([0]).plot()                                       # needs sage.plot
Graphics object consisting of 2 graphics primitives

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
>>> P = LatticePolytope_PPL((Integer(1),Integer(0)), (Integer(0),Integer(1)), (Integer(0),Integer(0)), (Integer(2),Integer(2)))
>>> P.plot()                                                              # needs sage.plot
Graphics object consisting of 6 graphics primitives
>>> LatticePolytope_PPL([Integer(0)], [Integer(1)]).plot()                                  # needs sage.plot
Graphics object consisting of 3 graphics primitives
>>> LatticePolytope_PPL([Integer(0)]).plot()                                       # needs sage.plot
Graphics object consisting of 2 graphics primitives

Sage Live

from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
P = LatticePolytope_PPL((1,0), (0,1), (0,0), (2,2))
P.plot()                                                              # needs sage.plot
LatticePolytope_PPL([0], [1]).plot()                                  # needs sage.plot
LatticePolytope_PPL([0]).plot()                                       # needs sage.plot

sub_polytopes()[source]¶

Return a list of all lattice sub-polygons up to isomorphism.

OUTPUT:

All non-empty sub-lattice polytopes up to isomorphism. This includes self as improper sub-polytope, but excludes the empty polytope. Isomorphic sub-polytopes that can be embedded in different places are only returned once.

EXAMPLES:

Sage

sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: P1xP1 = LatticePolytope_PPL((1,0), (0,1), (-1,0), (0,-1))
sage: P1xP1.sub_polytopes()
(A 2-dimensional lattice polytope in ZZ^2 with 4 vertices,
 A 2-dimensional lattice polytope in ZZ^2 with 3 vertices,
 A 2-dimensional lattice polytope in ZZ^2 with 3 vertices,
 A 1-dimensional lattice polytope in ZZ^2 with 2 vertices,
 A 1-dimensional lattice polytope in ZZ^2 with 2 vertices,
 A 0-dimensional lattice polytope in ZZ^2 with 1 vertex)

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
>>> P1xP1 = LatticePolytope_PPL((Integer(1),Integer(0)), (Integer(0),Integer(1)), (-Integer(1),Integer(0)), (Integer(0),-Integer(1)))
>>> P1xP1.sub_polytopes()
(A 2-dimensional lattice polytope in ZZ^2 with 4 vertices,
 A 2-dimensional lattice polytope in ZZ^2 with 3 vertices,
 A 2-dimensional lattice polytope in ZZ^2 with 3 vertices,
 A 1-dimensional lattice polytope in ZZ^2 with 2 vertices,
 A 1-dimensional lattice polytope in ZZ^2 with 2 vertices,
 A 0-dimensional lattice polytope in ZZ^2 with 1 vertex)

Sage Live

from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
P1xP1 = LatticePolytope_PPL((1,0), (0,1), (-1,0), (0,-1))
P1xP1.sub_polytopes()

sage.geometry.polyhedron.ppl_lattice_polygon.polar_P1xP1_polytope()[source]¶

The polar of the \(P^1 \times P^1\) polytope.

EXAMPLES:

Sage

sage: from sage.geometry.polyhedron.ppl_lattice_polygon import polar_P1xP1_polytope
sage: polar_P1xP1_polytope()
A 2-dimensional lattice polytope in ZZ^2 with 4 vertices
sage: _.vertices()
((0, 0), (0, 2), (2, 0), (2, 2))

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.ppl_lattice_polygon import polar_P1xP1_polytope
>>> polar_P1xP1_polytope()
A 2-dimensional lattice polytope in ZZ^2 with 4 vertices
>>> _.vertices()
((0, 0), (0, 2), (2, 0), (2, 2))

Sage Live

from sage.geometry.polyhedron.ppl_lattice_polygon import polar_P1xP1_polytope
polar_P1xP1_polytope()
_.vertices()

sage.geometry.polyhedron.ppl_lattice_polygon.polar_P2_112_polytope()[source]¶

The polar of the \(P^2[1,1,2]\) polytope.

EXAMPLES:

Sage

sage: from sage.geometry.polyhedron.ppl_lattice_polygon import polar_P2_112_polytope
sage: polar_P2_112_polytope()
A 2-dimensional lattice polytope in ZZ^2 with 3 vertices
sage: _.vertices()
((0, 0), (0, 2), (4, 0))

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.ppl_lattice_polygon import polar_P2_112_polytope
>>> polar_P2_112_polytope()
A 2-dimensional lattice polytope in ZZ^2 with 3 vertices
>>> _.vertices()
((0, 0), (0, 2), (4, 0))

Sage Live

from sage.geometry.polyhedron.ppl_lattice_polygon import polar_P2_112_polytope
polar_P2_112_polytope()
_.vertices()

sage.geometry.polyhedron.ppl_lattice_polygon.polar_P2_polytope()[source]¶

The polar of the \(P^2\) polytope.

EXAMPLES:

Sage

sage: from sage.geometry.polyhedron.ppl_lattice_polygon import polar_P2_polytope
sage: polar_P2_polytope()
A 2-dimensional lattice polytope in ZZ^2 with 3 vertices
sage: _.vertices()
((0, 0), (0, 3), (3, 0))

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.ppl_lattice_polygon import polar_P2_polytope
>>> polar_P2_polytope()
A 2-dimensional lattice polytope in ZZ^2 with 3 vertices
>>> _.vertices()
((0, 0), (0, 3), (3, 0))

Sage Live

from sage.geometry.polyhedron.ppl_lattice_polygon import polar_P2_polytope
polar_P2_polytope()
_.vertices()

sage.geometry.polyhedron.ppl_lattice_polygon.sub_reflexive_polygons()[source]¶

Return all lattice sub-polygons of reflexive polygons.

OUTPUT:

A tuple of all lattice sub-polygons. Each sub-polygon is returned as a pair sub-polygon, containing reflexive polygon.

EXAMPLES:

Sage

sage: from sage.geometry.polyhedron.ppl_lattice_polygon import sub_reflexive_polygons
sage: l = sub_reflexive_polygons(); l[5]
(A 2-dimensional lattice polytope in ZZ^2 with 6 vertices,
 A 2-dimensional lattice polytope in ZZ^2 with 3 vertices)
sage: len(l)
33

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.ppl_lattice_polygon import sub_reflexive_polygons
>>> l = sub_reflexive_polygons(); l[Integer(5)]
(A 2-dimensional lattice polytope in ZZ^2 with 6 vertices,
 A 2-dimensional lattice polytope in ZZ^2 with 3 vertices)
>>> len(l)
33

Sage Live

from sage.geometry.polyhedron.ppl_lattice_polygon import sub_reflexive_polygons
l = sub_reflexive_polygons(); l[5]
len(l)

sage.geometry.polyhedron.ppl_lattice_polygon.subpolygons_of_polar_P1xP1()[source]¶

The lattice sub-polygons of the polar \(P^1 \times P^1\) polytope.

OUTPUT: a tuple of lattice polytopes

EXAMPLES:

Sage

sage: from sage.geometry.polyhedron.ppl_lattice_polygon import subpolygons_of_polar_P1xP1
sage: len(subpolygons_of_polar_P1xP1())
20

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.ppl_lattice_polygon import subpolygons_of_polar_P1xP1
>>> len(subpolygons_of_polar_P1xP1())
20

Sage Live

from sage.geometry.polyhedron.ppl_lattice_polygon import subpolygons_of_polar_P1xP1
len(subpolygons_of_polar_P1xP1())

sage.geometry.polyhedron.ppl_lattice_polygon.subpolygons_of_polar_P2()[source]¶

The lattice sub-polygons of the polar \(P^2\) polytope.

OUTPUT: a tuple of lattice polytopes

EXAMPLES:

Sage

sage: from sage.geometry.polyhedron.ppl_lattice_polygon import subpolygons_of_polar_P2
sage: len(subpolygons_of_polar_P2())
27

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.ppl_lattice_polygon import subpolygons_of_polar_P2
>>> len(subpolygons_of_polar_P2())
27

Sage Live

from sage.geometry.polyhedron.ppl_lattice_polygon import subpolygons_of_polar_P2
len(subpolygons_of_polar_P2())

sage.geometry.polyhedron.ppl_lattice_polygon.subpolygons_of_polar_P2_112()[source]¶

The lattice sub-polygons of the polar \(P^2[1,1,2]\) polytope.

OUTPUT: a tuple of lattice polytopes

EXAMPLES:

Sage

sage: from sage.geometry.polyhedron.ppl_lattice_polygon import subpolygons_of_polar_P2_112
sage: len(subpolygons_of_polar_P2_112())
28

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.ppl_lattice_polygon import subpolygons_of_polar_P2_112
>>> len(subpolygons_of_polar_P2_112())
28

Sage Live

from sage.geometry.polyhedron.ppl_lattice_polygon import subpolygons_of_polar_P2_112
len(subpolygons_of_polar_P2_112())