Point collections - Combinatorial and Discrete Geometry

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# 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

      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)

  Python

      >>> from sage.all import *
      >>> c = Cone([(Integer(0),Integer(0),Integer(1)), (Integer(1),Integer(0),Integer(1)), (Integer(0),Integer(1),Integer(1)), (Integer(1),Integer(1),Integer(1))]).rays()
      >>> c[Integer(1)]
      N(1, 0, 1)
      >>> 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

      sage: c
      N(0, 0, 1),
      N(1, 0, 1),
      N(0, 1, 1),
      N(1, 1, 1)
      in 3-d lattice N

  Python

      >>> from sage.all import *
      >>> 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

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

  Python

      >>> from sage.all import *
      >>> 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

      sage: c.basis()
      N(0, 0, 1),
      N(1, 0, 1),
      N(0, 1, 1)
      in 3-d lattice N

  Python

      >>> from sage.all import *
      >>> 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

  Bases: `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:

  - `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` (the default), 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.

  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

        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

    Python

        >>> from sage.all import *
        >>> c = Cone([(Integer(0),Integer(0),Integer(1)), (Integer(1),Integer(0),Integer(1)), (Integer(0),Integer(1),Integer(1)), (Integer(1),Integer(1),Integer(1))]).rays()
        >>> 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

        sage: c.basis().basis() is c.basis()
        True

    [Python]

        >>> from sage.all import *
        >>> c.basis().basis() is c.basis()
        True

  cardinality()

    Return the number of points in `self`.

    OUTPUT: integer

    EXAMPLES:

    Sage

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

    Python

        >>> from sage.all import *
        >>> c = Cone([(Integer(0),Integer(0),Integer(1)), (Integer(1),Integer(0),Integer(1)), (Integer(0),Integer(1),Integer(1)), (Integer(1),Integer(1),Integer(1))]).rays()
        >>> 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: a `point collection`

    EXAMPLES:

    Sage

        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

    Python

        >>> from sage.all import *
        >>> c = Cone([(Integer(0),Integer(0),Integer(1)), (Integer(1),Integer(1),Integer(1))]).rays()
        >>> 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

        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]

    Python

        >>> from sage.all import *
        >>> c = Cone([(Integer(0),Integer(0),Integer(1)), (Integer(1),Integer(0),Integer(1)), (Integer(0),Integer(1),Integer(1)), (Integer(1),Integer(1),Integer(1))]).rays()
        >>> c.column_matrix()
        [0 1 0 1]
        [0 0 1 1]
        [1 1 1 1]

  dim()

    alias of `dimension()`.

  dimension()

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

    Note

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

    OUTPUT: integer

    EXAMPLES:

    Sage

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

    Python

        >>> from sage.all import *
        >>> c = Cone([(Integer(0),Integer(0),Integer(1)), (Integer(1),Integer(1),Integer(1))]).rays()
        >>> c.dimension()
        2
        >>> 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

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

    Python

        >>> from sage.all import *
        >>> c = Cone([(Integer(0),Integer(0),Integer(1)), (Integer(1),Integer(0),Integer(1)), (Integer(0),Integer(1),Integer(1)), (Integer(1),Integer(1),Integer(1))]).rays()
        >>> 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

        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

    Python

        >>> from sage.all import *
        >>> c = Cone([(Integer(0),Integer(0),Integer(1)), (Integer(1),Integer(0),Integer(1)), (Integer(0),Integer(1),Integer(1)), (Integer(1),Integer(1),Integer(1))]).rays()
        >>> c.index((Integer(0),Integer(1),Integer(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

        sage: N = c.module()
        sage: c.index(N(0,1,1))
        2
        sage: c[2]
        N(0, 1, 1)

    [Python]

        >>> from sage.all import *
        >>> N = c.module()
        >>> c.index(N(Integer(0),Integer(1),Integer(1)))
        2
        >>> c[Integer(2)]
        N(0, 1, 1)

  matrix()

    Return a matrix whose rows are points of `self`.

    OUTPUT: a `matrix`

    EXAMPLES:

    Sage

        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]

    Python

        >>> from sage.all import *
        >>> c = Cone([(Integer(0),Integer(0),Integer(1)), (Integer(1),Integer(0),Integer(1)), (Integer(0),Integer(1),Integer(1)), (Integer(1),Integer(1),Integer(1))]).rays()
        >>> 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

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

    Python

        >>> from sage.all import *
        >>> c = Cone([(Integer(0),Integer(0),Integer(1)), (Integer(1),Integer(0),Integer(1)), (Integer(0),Integer(1),Integer(1)), (Integer(1),Integer(1),Integer(1))]).rays()
        >>> 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

        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

    Python

        >>> from sage.all import *
        >>> c = Cone([(Integer(0),Integer(0),Integer(1)), (Integer(1),Integer(0),Integer(1)), (Integer(0),Integer(1),Integer(1)), (Integer(1),Integer(1),Integer(1))]).rays()
        >>> c
        N(0, 0, 1),
        N(1, 0, 1),
        N(0, 1, 1),
        N(1, 1, 1)
        in 3-d lattice N
        >>> c.output_format()
        'default'
        >>> c.output_format("tuple")
        >>> c
        (N(0, 0, 1), N(1, 0, 1), N(0, 1, 1), N(1, 1, 1))
        in 3-d lattice N
        >>> c.output_format("matrix")
        >>> c
        [0 0 1]
        [1 0 1]
        [0 1 1]
        [1 1 1]
        in 3-d lattice N
        >>> c.output_format("column matrix")
        >>> c
        [0 1 0 1]
        [0 0 1 1]
        [1 1 1 1]
        in 3-d lattice N
        >>> c.output_format("separated column matrix")
        >>> 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

        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}

    [Python]

        >>> from sage.all import *
        >>> 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

        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

    Python

        >>> from sage.all import *
        >>> from sage.geometry.point_collection import PointCollection
        >>> PointCollection.output_format("default")
        >>> 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

        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)})

    Python

        >>> from sage.all import *
        >>> c = Cone([(Integer(0),Integer(0),Integer(1)), (Integer(1),Integer(0),Integer(1)), (Integer(0),Integer(1),Integer(1)), (Integer(1),Integer(1),Integer(1))]).rays()
        >>> 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

        sage: o = lattice_polytope.cross_polytope(3)
        sage: from io 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

    Python

        >>> from sage.all import *
        >>> o = lattice_polytope.cross_polytope(Integer(3))
        >>> from io import StringIO
        >>> f = StringIO()
        >>> o.vertices().write_for_palp(f)
        >>> 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

      sage: from sage.geometry.point_collection import PointCollection
      sage: isinstance(1, PointCollection)
      False
      sage: c = Cone([(0,0,1), (1,0,1), (0,1,1), (1,1,1)])
      sage: isinstance(c.rays(), PointCollection)
      True

  Python

      >>> from sage.all import *
      >>> from sage.geometry.point_collection import PointCollection
      >>> isinstance(Integer(1), PointCollection)
      False
      >>> c = Cone([(Integer(0),Integer(0),Integer(1)), (Integer(1),Integer(0),Integer(1)), (Integer(0),Integer(1),Integer(1)), (Integer(1),Integer(1),Integer(1))])
      >>> isinstance(c.rays(), PointCollection)
      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` – boolean (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:

  - a `point collection`, optionally followed by a permutation. `None` if EOF is reached.

  EXAMPLES:

  Sage

      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 io import StringIO
      sage: f = StringIO(data)
      sage: from sage.geometry.point_collection \
      ....:     import read_palp_point_collection
      sage: read_palp_point_collection(f)
      M(1, 2),
      M(3, 4),
      M(5, 6)
      in 2-d lattice M
      sage: read_palp_point_collection(f)
      M(1, 4),
      M(2, 5),
      M(3, 6)
      in 2-d lattice M
      sage: read_palp_point_collection(f) is None
      True

  Python

      >>> from sage.all import *
      >>> data = "3 2 regular\n1 2\n3 4\n5 6\n2 3 transposed\n1 2 3\n4 5 6"
      >>> print(data)
      3 2 regular
      1 2
      3 4
      5 6
      2 3 transposed
      1 2 3
      4 5 6
      >>> from io import StringIO
      >>> f = StringIO(data)
      >>> from sage.geometry.point_collection     import read_palp_point_collection
      >>> read_palp_point_collection(f)
      M(1, 2),
      M(3, 4),
      M(5, 6)
      in 2-d lattice M
      >>> read_palp_point_collection(f)
      M(1, 4),
      M(2, 5),
      M(3, 6)
      in 2-d lattice M
      >>> read_palp_point_collection(f) is None
      True