# Point collections¶

This module was designed as a part of framework for toric varieties (variety, fano_variety).

AUTHORS:

• Andrey Novoseltsev (2011-04-25): initial version, based on cone module.
• Andrey Novoseltsev (2012-03-06): additions and doctest changes while switching cones to use point collections.

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
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


sage: c.set()
frozenset({N(0, 0, 1), N(0, 1, 1), N(1, 0, 1), N(1, 1, 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

1. vertices of $$\Delta$$;
2. facet normals of its polar $$\Delta^\circ$$;
3. ray generators of the face fan of $$\Delta$$;
4. ray generators of the normal fan of $$\Delta$$.

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.

class sage.geometry.point_collection.PointCollection

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:

• points – an iterable structure of immutable elements of module, if points are already accessible to you as a tuple, it is preferable to use it for speed and memory consumption reasons;
• module – an ambient module for points. If None, it will be determined as parent() of the first point. Of course, this cannot be done if there are no points, so in this case you must give an appropriate module directly. Note that None is not the default value - you always must give this argument explicitly, even if it is None.

OUTPUT:

• a point collection.
basis()

Return a linearly independent subset of points of self.

OUTPUT:

• a point collection giving a random (but fixed) choice of an $$\RR$$-basis for the vector space spanned by the points of self.

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

cardinality()

Return the number of points in self.

OUTPUT:

• an integer.

EXAMPLES:

sage: c = Cone([(0,0,1), (1,0,1), (0,1,1), (1,1,1)]).rays()
sage: c.cardinality()
4

cartesian_product(other, module=None)

Return the Cartesian product of self with other.

INPUT:

• other – a point collection;
• module – (optional) the ambient module for the result. By default, the direct sum of the ambient modules of self and other is constructed.

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

column_matrix()

Return a matrix whose columns are points of self.

OUTPUT:

• a matrix.

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]

dim()

Return the dimension of the space spanned by points of self.

Note

You can use either dim() or dimension().

OUTPUT:

• an integer.

EXAMPLES:

sage: c = Cone([(0,0,1), (1,1,1)]).rays()
sage: c.dimension()
2
sage: c.dim()
2

dimension()

Return the dimension of the space spanned by points of self.

Note

You can use either dim() or dimension().

OUTPUT:

• an integer.

EXAMPLES:

sage: c = Cone([(0,0,1), (1,1,1)]).rays()
sage: c.dimension()
2
sage: c.dim()
2

dual_module()

Return the dual of the ambient module of self.

OUTPUT:

• a module. If possible (that is, if the ambient module() $$M$$ of self has a dual() method), the dual module is returned. Otherwise, $$R^n$$ is returned, where $$n$$ is the dimension of $$M$$ and $$R$$ is its base ring.

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

index(*args)

Return the index of the first occurrence of point in self.

INPUT:

• point – a point of self;
• start – (optional) an integer, if given, the search will start at this position;
• stop – (optional) an integer, if given, the search will stop at this position.

OUTPUT:

• an integer if point is in self[start:stop], otherwise a ValueError exception is raised.

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
N(0, 1, 1)

matrix()

Return a matrix whose rows are points of self.

OUTPUT:

• a matrix.

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]

module()

Return the ambient module of self.

OUTPUT:

• a module.

EXAMPLES:

sage: c = Cone([(0,0,1), (1,0,1), (0,1,1), (1,1,1)]).rays()
sage: c.module()
3-d lattice N

static output_format(format=None)

Return or set the output format for ALL point collections.

INPUT:

• format – (optional) if given, must be one of the strings
• “default” – output one point per line with vertical alignment of coordinates in text mode, same as “tuple” for LaTeX;
• “tuple” – output tuple(self) with lattice information;
• “matrix” – output matrix() with lattice information;
• “column matrix” – output column_matrix() with lattice information;
• “separated column matrix” – same as “column matrix” for text mode, for LaTeX separate columns by lines (not shown by jsMath).

OUTPUT:

• a string with the current format (only if format was omitted).

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 outputs 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

set()

Return points of self as a frozenset.

OUTPUT:

• a frozenset.

EXAMPLES:

sage: c = Cone([(0,0,1), (1,0,1), (0,1,1), (1,1,1)]).rays()
sage: c.set()
frozenset({N(0, 0, 1), N(0, 1, 1), N(1, 0, 1), N(1, 1, 1)})

write_for_palp(f)

Write self into an open file f in PALP format.

INPUT:

• f – a file opened for writing.

EXAMPLES:

sage: o = lattice_polytope.cross_polytope(3)
sage: from six import StringIO
sage: f = StringIO()
sage: o.vertices().write_for_palp(f)
sage: print(f.getvalue())
6 3
1 0 0
0 1 0
0 0 1
-1 0 0
0 -1 0
0 0 -1

sage.geometry.point_collection.is_PointCollection(x)

Check if x is a point collection.

INPUT:

• x – anything.

OUTPUT:

• True if x is a point collection and False otherwise.

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

sage.geometry.point_collection.read_palp_point_collection(f, lattice=None, permutation=False)

Read and return a point collection from an opened file.

Data must be in PALP format:

• the first input line starts with two integers $$m$$ and $$n$$, the number of points and the number of components of each;
• the rest of the first line may contain a permutation;
• the next $$m$$ lines contain $$n$$ numbers each.

Note

If $$m$$ < $$n$$, it is assumed (for compatibility with PALP) that the matrix is transposed, i.e. that each column is a point.

INPUT:

• f – an opened file with PALP output.
• lattice – the lattice for points. If not given, the toric lattice $$M$$ of dimension $$n$$ will be used.
• permutation – (default: False) if True, try to retrieve the permutation. This parameter makes sense only when PALP computed the normal form of a lattice polytope.

OUTPUT:

EXAMPLES:

sage: data = "3 2 regular\n1 2\n3 4\n5 6\n2 3 transposed\n1 2 3\n4 5 6"
sage: print(data)
3 2 regular
1 2
3 4
5 6
2 3 transposed
1 2 3
4 5 6
sage: from six import StringIO
sage: f = StringIO(data)
sage: from sage.geometry.point_collection \