Point collections
This module was designed as a part of framework for toric varieties (variety, fano_variety).
AUTHORS:
EXAMPLES:
The idea behind point collections is to have a container for points of the same space that
behaves like a tuple without significant performance penalty:
sage: c = Cone([(0,0,1), (1,0,1), (0,1,1), (1,1,1)]).rays()
sage: c[1]
N(1, 0, 1)
sage: for point in c: point
N(0, 0, 1)
N(1, 0, 1)
N(0, 1, 1)
N(1, 1, 1)
prints in a convenient way and with clear indication of the ambient space:
sage: c
N(0, 0, 1),
N(1, 0, 1),
N(0, 1, 1),
N(1, 1, 1)
in 3-d lattice N
allows (cached) access to alternative representations:
sage: c.set()
frozenset([N(0, 1, 1), N(1, 1, 1), N(0, 0, 1), N(1, 0, 1)])
allows introduction of additional methods:
sage: c.basis()
N(0, 0, 1),
N(1, 0, 1),
N(0, 1, 1)
in 3-d lattice N
Examples of natural point collections include ray and line generators of cones, vertices and points of polytopes, normals to facets, their subcollections, etc.
Using this class for all of the above cases allows for unified interface and cache sharing. Suppose that \(\Delta\) is a reflexive polytope. Then the same point collection can be linked as
If all these objects are in use and, say, a matrix representation was computed for one of them, it becomes available to all others as well, eliminating the need to spend time and memory four times.
Bases: sage.structure.sage_object.SageObject
Create a point collection.
Warning
No correctness check or normalization is performed on the input data. This class is designed for internal operations and you probably should not use it directly.
Point collections are immutable, but cache most of the returned values.
INPUT:
OUTPUT:
Return a linearly independent subset of points of self.
OUTPUT:
EXAMPLES:
sage: c = Cone([(0,0,1), (1,0,1), (0,1,1), (1,1,1)]).rays()
sage: c.basis()
N(0, 0, 1),
N(1, 0, 1),
N(0, 1, 1)
in 3-d lattice N
Calling this method twice will always return exactly the same point collection:
sage: c.basis().basis() is c.basis()
True
Return the number of points in self.
OUTPUT:
EXAMPLES:
sage: c = Cone([(0,0,1), (1,0,1), (0,1,1), (1,1,1)]).rays()
sage: c.cardinality()
4
Return the Cartesian product of self with other.
INPUT:
OUTPUT:
EXAMPLES:
sage: c = Cone([(0,0,1), (1,1,1)]).rays()
sage: c.cartesian_product(c)
N+N(0, 0, 1, 0, 0, 1),
N+N(1, 1, 1, 0, 0, 1),
N+N(0, 0, 1, 1, 1, 1),
N+N(1, 1, 1, 1, 1, 1)
in 6-d lattice N+N
Return a matrix whose columns are points of self.
OUTPUT:
EXAMPLES:
sage: c = Cone([(0,0,1), (1,0,1), (0,1,1), (1,1,1)]).rays()
sage: c.column_matrix()
[0 1 0 1]
[0 0 1 1]
[1 1 1 1]
Return the dimension of the space spanned by points of self.
Note
You can use either dim() or dimension().
OUTPUT:
EXAMPLES:
sage: c = Cone([(0,0,1), (1,1,1)]).rays()
sage: c.dimension()
2
sage: c.dim()
2
Return the dimension of the space spanned by points of self.
Note
You can use either dim() or dimension().
OUTPUT:
EXAMPLES:
sage: c = Cone([(0,0,1), (1,1,1)]).rays()
sage: c.dimension()
2
sage: c.dim()
2
Return the dual of the ambient module of self.
OUTPUT:
EXAMPLES:
sage: c = Cone([(0,0,1), (1,0,1), (0,1,1), (1,1,1)]).rays()
sage: c.dual_module()
3-d lattice M
Return the index of the first occurrence of point in self.
INPUT:
OUTPUT:
EXAMPLES:
sage: c = Cone([(0,0,1), (1,0,1), (0,1,1), (1,1,1)]).rays()
sage: c.index((0,1,1))
Traceback (most recent call last):
...
ValueError: tuple.index(x): x not in tuple
Note that this was not a mistake: the tuple (0,1,1) is not a point of c! We need to pass actual element of the ambient module of c to get their indices:
sage: N = c.module()
sage: c.index(N(0,1,1))
2
sage: c[2]
N(0, 1, 1)
Return a matrix whose rows are points of self.
OUTPUT:
EXAMPLES:
sage: c = Cone([(0,0,1), (1,0,1), (0,1,1), (1,1,1)]).rays()
sage: c.matrix()
[0 0 1]
[1 0 1]
[0 1 1]
[1 1 1]
Return the ambient module of self.
OUTPUT:
EXAMPLES:
sage: c = Cone([(0,0,1), (1,0,1), (0,1,1), (1,1,1)]).rays()
sage: c.module()
3-d lattice N
Return or set the output format for ALL point collections.
INPUT:
OUTPUT:
This function affects both regular and LaTeX output.
EXAMPLES:
sage: c = Cone([(0,0,1), (1,0,1), (0,1,1), (1,1,1)]).rays()
sage: c
N(0, 0, 1),
N(1, 0, 1),
N(0, 1, 1),
N(1, 1, 1)
in 3-d lattice N
sage: c.output_format()
'default'
sage: c.output_format("tuple")
sage: c
(N(0, 0, 1), N(1, 0, 1), N(0, 1, 1), N(1, 1, 1))
in 3-d lattice N
sage: c.output_format("matrix")
sage: c
[0 0 1]
[1 0 1]
[0 1 1]
[1 1 1]
in 3-d lattice N
sage: c.output_format("column matrix")
sage: c
[0 1 0 1]
[0 0 1 1]
[1 1 1 1]
in 3-d lattice N
sage: c.output_format("separated column matrix")
sage: c
[0 1 0 1]
[0 0 1 1]
[1 1 1 1]
in 3-d lattice N
Note that the last two outpus are identical, separators are only inserted in the LaTeX mode:
sage: latex(c)
\left(\begin{array}{r|r|r|r}
0 & 1 & 0 & 1 \\
0 & 0 & 1 & 1 \\
1 & 1 & 1 & 1
\end{array}\right)_{N}
Since this is a static method, you can call it for the class directly:
sage: from sage.geometry.point_collection import PointCollection
sage: PointCollection.output_format("default")
sage: c
N(0, 0, 1),
N(1, 0, 1),
N(0, 1, 1),
N(1, 1, 1)
in 3-d lattice N
Return points of self as a frozenset.
OUTPUT:
EXAMPLES:
sage: c = Cone([(0,0,1), (1,0,1), (0,1,1), (1,1,1)]).rays()
sage: c.set()
frozenset([N(0, 1, 1), N(1, 1, 1), N(0, 0, 1), N(1, 0, 1)])
Check if x is a point collection.
INPUT:
OUTPUT:
EXAMPLES:
sage: from sage.geometry.point_collection import is_PointCollection
sage: is_PointCollection(1)
False
sage: c = Cone([(0,0,1), (1,0,1), (0,1,1), (1,1,1)])
sage: is_PointCollection(c.rays())
True