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r""" Metric Spaces """ #***************************************************************************** # Copyright (C) 2015 Travis Scrimshaw <tscrim at ucdavis.edu> # # Distributed under the terms of the GNU General Public License (GPL) # http://www.gnu.org/licenses/ #******************************************************************************
from sage.misc.cachefunc import cached_method from sage.categories.category import Category from sage.categories.category_with_axiom import CategoryWithAxiom from sage.categories.covariant_functorial_construction import RegressiveCovariantConstructionCategory from sage.categories.with_realizations import WithRealizationsCategory
class MetricSpacesCategory(RegressiveCovariantConstructionCategory):
_functor_category = "Metric"
@classmethod def default_super_categories(cls, category): """ Return the default super categories of ``category.Metric()``.
Mathematical meaning: if `A` is a metric space in the category `C`, then `A` is also a topological space.
INPUT:
- ``cls`` -- the class ``MetricSpaces`` - ``category`` -- a category `Cat`
OUTPUT:
A (join) category
In practice, this returns ``category.Metric()``, joined together with the result of the method :meth:`RegressiveCovariantConstructionCategory.default_super_categories() <sage.categories.covariant_functorial_construction.RegressiveCovariantConstructionCategory.default_super_categories>` (that is the join of ``category`` and ``cat.Metric()`` for each ``cat`` in the super categories of ``category``).
EXAMPLES:
Consider ``category=Groups()``. Then, a group `G` with a metric is simultaneously a topological group by itself, and a metric space::
sage: Groups().Metric().super_categories() [Category of topological groups, Category of metric spaces]
This resulted from the following call::
sage: sage.categories.metric_spaces.MetricSpacesCategory.default_super_categories(Groups()) Join of Category of topological groups and Category of metric spaces """ super(MetricSpacesCategory, cls).default_super_categories(category)])
# We currently don't have a use for this, but we probably will def _repr_object_names(self): """ EXAMPLES::
sage: Groups().Metric() # indirect doctest Join of Category of topological groups and Category of metric spaces """ return "metric {}".format(self.base_category()._repr_object_names())
class MetricSpaces(MetricSpacesCategory): r""" The category of metric spaces.
A *metric* on a set `S` is a function `d : S \times S \to \RR` such that:
- `d(a, b) \geq 0`, - `d(a, b) = 0` if and only if `a = b`.
A metric space is a set `S` with a distinguished metric.
.. RUBRIC:: Implementation
Objects in this category must implement either a ``dist`` on the parent or the elements or ``metric`` on the parent; otherwise this will cause an infinite recursion.
.. TODO::
- Implement a general geodesics class. - Implement a category for metric additive groups and move the generic distance `d(a, b) = |a - b|` there. - Incorporate the length of a geodesic as part of the default distance cycle.
EXAMPLES::
sage: from sage.categories.metric_spaces import MetricSpaces sage: C = MetricSpaces() sage: C Category of metric spaces sage: TestSuite(C).run() """ def _repr_object_names(self): """ EXAMPLES::
sage: Sets().Metric() # indirect doctest Category of metric spaces """
class ParentMethods: def _test_metric(self, **options): r""" Test that this metric space has a properly implemented metric.
INPUT:
- ``options`` -- any keyword arguments accepted by :meth:`_tester`
EXAMPLES::
sage: UHP = HyperbolicPlane().UHP() sage: UHP._test_metric() sage: elts = [UHP.random_element() for i in range(5)] sage: UHP._test_metric(some_elements=elts) """ else:
def metric(self): """ Return the metric of ``self``.
EXAMPLES::
sage: UHP = HyperbolicPlane().UHP() sage: m = UHP.metric() sage: p1 = UHP.get_point(5 + 7*I) sage: p2 = UHP.get_point(1.0 + I) sage: m(p1, p2) 2.23230104635820 """
def dist(self, a, b): """ Return the distance between ``a`` and ``b`` in ``self``.
EXAMPLES::
sage: UHP = HyperbolicPlane().UHP() sage: p1 = UHP.get_point(5 + 7*I) sage: p2 = UHP.get_point(1.0 + I) sage: UHP.dist(p1, p2) 2.23230104635820
sage: PD = HyperbolicPlane().PD() sage: PD.dist(PD.get_point(0), PD.get_point(I/2)) arccosh(5/3)
TESTS::
sage: RR.dist(-1, pi) 4.14159265358979 sage: RDF.dist(1, -1/2) 1.5 sage: CC.dist(3, 2) 1.00000000000000 sage: CC.dist(-1, I) 1.41421356237310 sage: CDF.dist(-1, I) 1.4142135623730951 """
class ElementMethods: def abs(self): """ Return the absolute value of ``self``.
EXAMPLES::
sage: CC(I).abs() 1.00000000000000 """ P = self.parent() return P.metric()(self, P.zero())
def dist(self, b): """ Return the distance between ``self`` and ``other``.
EXAMPLES::
sage: UHP = HyperbolicPlane().UHP() sage: p1 = UHP.get_point(5 + 7*I) sage: p2 = UHP.get_point(1 + I) sage: p1.dist(p2) arccosh(33/7) """
class WithRealizations(WithRealizationsCategory): class ParentMethods: def dist(self, a, b): """ Return the distance between ``a`` and ``b`` by converting them to a realization of ``self`` and doing the computation.
EXAMPLES::
sage: H = HyperbolicPlane() sage: PD = H.PD() sage: p1 = PD.get_point(0) sage: p2 = PD.get_point(I/2) sage: H.dist(p1, p2) arccosh(5/3) """
class SubcategoryMethods: @cached_method def Complete(self): """ Return the full subcategory of the complete objects of ``self``.
EXAMPLES::
sage: Sets().Metric().Complete() Category of complete metric spaces
TESTS::
sage: TestSuite(Sets().Metric().Complete()).run() sage: Sets().Metric().Complete.__module__ 'sage.categories.metric_spaces' """
class Complete(CategoryWithAxiom): """ The category of complete metric spaces. """
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