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# Regions in fundamental domains of period lattices

Regions in fundamental domains of period lattices

This module is used to represent sub-regions of a fundamental parallelogram of the period lattice of an elliptic curve, used in computing minimum height bounds.

In particular, these are the approximating sets `S^{(v)}` in section 3.2 of Thotsaphon Thongjunthug’s Ph.D. Thesis and paper [TT].

AUTHORS:

- Robert Bradshaw (2010): initial version
- John Cremona (2014): added some docstrings and doctests

REFERENCES:

[T] | T. Thongjunthug, Computing a lower bound for the canonical height on elliptic curves over number fields, Math. Comp. 79 (2010), pages 2431-2449.

class `sage.schemes.elliptic_curves.period_lattice_region.``PeriodicRegion`

  Bases: `object`

  EXAMPLE:

      sage: import numpy as np
      sage: from sage.schemes.elliptic_curves.period_lattice_region import PeriodicRegion
      sage: S = PeriodicRegion(CDF(2), CDF(2*I), np.zeros((4, 4)))
      sage: S.plot()
      Graphics object consisting of 1 graphics primitive
      sage: data = np.zeros((4, 4))
      sage: data[1,1] = True
      sage: S = PeriodicRegion(CDF(2), CDF(2*I+1), data)
      sage: S.plot()
      Graphics object consisting of 5 graphics primitives

  `border`(raw=True)

    Returns the boundary of this region as set of tile boundaries.

    If raw is true, returns a list with respect to the internal bitmap, otherwise returns complex intervals covering the border.

    EXAMPLES:

        sage: import numpy as np
        sage: from sage.schemes.elliptic_curves.period_lattice_region import PeriodicRegion
        sage: data = np.zeros((4, 4))
        sage: data[1, 1] = True
        sage: PeriodicRegion(CDF(1), CDF(I), data).border()
        [(1, 1, 0), (2, 1, 0), (1, 1, 1), (1, 2, 1)]
        sage: PeriodicRegion(CDF(2), CDF(I-1/2), data).border()
        [(1, 1, 0), (2, 1, 0), (1, 1, 1), (1, 2, 1)]

        sage: PeriodicRegion(CDF(1), CDF(I), data).border(raw=False)
        [0.25000000000000000? + 1.?*I,
         0.50000000000000000? + 1.?*I,
         1.? + 0.25000000000000000?*I,
         1.? + 0.50000000000000000?*I]
        sage: PeriodicRegion(CDF(2), CDF(I-1/2), data).border(raw=False)
        [0.3? + 1.?*I,
         0.8? + 1.?*I,
         1.? + 0.25000000000000000?*I,
         1.? + 0.50000000000000000?*I]

        sage: data[1:3, 2] = True
        sage: PeriodicRegion(CDF(1), CDF(I), data).border()
        [(1, 1, 0), (2, 1, 0), (1, 1, 1), (1, 2, 0), (1, 3, 1), (3, 2, 0), (2, 2, 1), (2, 3, 1)]

  `contract`(corners=True)

    Opposite (but not inverse) of expand; removes neighbors of complement.

    EXAMPLES:

        sage: import numpy as np
        sage: from sage.schemes.elliptic_curves.period_lattice_region import PeriodicRegion
        sage: data = np.zeros((10, 10))
        sage: data[1:4,1:4] = True
        sage: S = PeriodicRegion(CDF(1), CDF(I + 1/2), data)
        sage: S.plot()
        Graphics object consisting of 13 graphics primitives
        sage: S.contract().plot()
        Graphics object consisting of 5 graphics primitives
        sage: S.contract().data.sum()
        1
        sage: S.contract().contract().is_empty()
        True

  `data`
  `ds`()

    Returns the sides of each parallelogram tile.

    EXAMPLES:

        sage: import numpy as np
        sage: from sage.schemes.elliptic_curves.period_lattice_region import PeriodicRegion
        sage: data = np.zeros((4, 4))
        sage: S = PeriodicRegion(CDF(2), CDF(2*I), data, full=False)
        sage: S.ds()
        (0.5, 0.25*I)
        sage: _ = S._ensure_full()
        sage: S.ds()
        (0.5, 0.25*I)

        sage: data = np.zeros((8, 8))
        sage: S = PeriodicRegion(CDF(1), CDF(I + 1/2), data)
        sage: S.ds()
        (0.125, 0.0625 + 0.125*I)

  `expand`(corners=True)

    Returns a region containing this region by adding all neighbors of internal tiles.

    EXAMPLES:

        sage: import numpy as np
        sage: from sage.schemes.elliptic_curves.period_lattice_region import PeriodicRegion
        sage: data = np.zeros((4, 4))
        sage: data[1,1] = True
        sage: S = PeriodicRegion(CDF(1), CDF(I + 1/2), data)
        sage: S.plot()
        Graphics object consisting of 5 graphics primitives
        sage: S.expand().plot()
        Graphics object consisting of 13 graphics primitives
        sage: S.expand().data
        array([[1, 1, 1, 0],
               [1, 1, 1, 0],
               [1, 1, 1, 0],
               [0, 0, 0, 0]], dtype=int8)
        sage: S.expand(corners=False).plot()
        Graphics object consisting of 13 graphics primitives
        sage: S.expand(corners=False).data
        array([[0, 1, 0, 0],
               [1, 1, 1, 0],
               [0, 1, 0, 0],
               [0, 0, 0, 0]], dtype=int8)

  `full`
  `innermost_point`()

    Returns a point well inside the region, specifically the center of (one of) the last tile(s) to be removed on contraction.

    EXAMPLES:

        sage: import numpy as np
        sage: from sage.schemes.elliptic_curves.period_lattice_region import PeriodicRegion
        sage: data = np.zeros((10, 10))
        sage: data[1:4, 1:4] = True
        sage: data[1, 0:8] = True
        sage: S = PeriodicRegion(CDF(1), CDF(I+1/2), data)
        sage: S.innermost_point()
        0.375 + 0.25*I
        sage: S.plot() + point(S.innermost_point())
        Graphics object consisting of 24 graphics primitives

  `is_empty`()

    Returns whether this region is empty.

    EXAMPLES:

        sage: import numpy as np
        sage: from sage.schemes.elliptic_curves.period_lattice_region import PeriodicRegion
        sage: data = np.zeros((4, 4))
        sage: PeriodicRegion(CDF(2), CDF(2*I), data).is_empty()
        True
        sage: data[1,1] = True
        sage: PeriodicRegion(CDF(2), CDF(2*I), data).is_empty()
        False

  `plot`(**kwds)

    Plots this region in the fundamental lattice. If full is False plots only the lower half. Note that the true nature of this region is periodic.

    EXAMPLES:

        sage: import numpy as np
        sage: from sage.schemes.elliptic_curves.period_lattice_region import PeriodicRegion
        sage: data = np.zeros((10, 10))
        sage: data[2, 2:8] = True
        sage: data[2:5, 2] = True
        sage: data[3, 3] = True
        sage: S = PeriodicRegion(CDF(1), CDF(I + 1/2), data)
        sage: plot(S) + plot(S.expand(), rgbcolor=(1, 0, 1), thickness=2)
        Graphics object consisting of 46 graphics primitives

  `refine`(condition=None, times=1)

    Recursive function to refine the current tiling.

    INPUT:

    - `condition` (function, default None) - if not None, only keep tiles in the refinement which satisfy the condition.
    - `times` (int, default 1) - the number of times to refine; each refinement step halves the mesh size.

    OUTPUT:

    The refined PeriodicRegion.

    EXAMPLES:

        sage: import numpy as np
        sage: from sage.schemes.elliptic_curves.period_lattice_region import PeriodicRegion
        sage: data = np.zeros((4, 4))
        sage: S = PeriodicRegion(CDF(2), CDF(2*I), data, full=False)
        sage: S.ds()
        (0.5, 0.25*I)
        sage: S = S.refine()
        sage: S.ds()
        (0.25, 0.125*I)
        sage: S = S.refine(2)
        sage: S.ds()
        (0.125, 0.0625*I)

  `verify`(condition)

    Given a condition that should hold for every line segment on the boundary, verify that it actually does so.

    INPUT:

    - `condition` (function) - a boolean-valued function on \(\CC\).

    OUTPUT:

    True or False according to whether the condition holds for all lines on the boundary.

    EXAMPLES:

        sage: import numpy as np
        sage: from sage.schemes.elliptic_curves.period_lattice_region import PeriodicRegion
        sage: data = np.zeros((4, 4))
        sage: data[1, 1] = True
        sage: S = PeriodicRegion(CDF(1), CDF(I), data)
        sage: S.border()
        [(1, 1, 0), (2, 1, 0), (1, 1, 1), (1, 2, 1)]
        sage: condition = lambda z: z.real().abs()<0.5
        sage: S.verify(condition)
        False
        sage: condition = lambda z: z.real().abs()<1
        sage: S.verify(condition)
        True

  `w1`
  `w2`

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