Coverage for local/lib/python2.7/site-packages/sage/geometry/hyperbolic_space/hyperbolic_interface.py : 97%

# -*- coding: utf-8 -*- 

r""" 

Interface to Hyperbolic Models 

This module provides a convenient interface for interacting with models 

of hyperbolic space as well as their points, geodesics, and isometries. 

The primary point of this module is to allow the code that implements 

hyperbolic space to be sufficiently decoupled while still providing a 

convenient user experience. 

The interfaces are by default given abbreviated names. For example, 

UHP (upper half plane model), PD (Poincaré disk model), KM (Klein disk 

model), and HM (hyperboloid model). 

.. NOTE:: 

All of the current models of 2 dimensional hyperbolic space 

use the upper half plane model for their computations. This can 

lead to some problems, such as long coordinate strings for symbolic 

points. For example, the vector ``(1, 0, sqrt(2))`` defines a point 

in the hyperboloid model. Performing mapping this point to the upper 

half plane and performing computations there may return with vector 

whose components are unsimplified strings have several ``sqrt(2)``'s. 

Presently, this drawback is outweighted by the rapidity with which new 

models can be implemented. 

AUTHORS: 

- Greg Laun (2013): Initial version. 

- Rania Amer, Jean-Philippe Burelle, Bill Goldman, Zach Groton, 

Jeremy Lent, Leila Vaden, Derrick Wigglesworth (2011): many of the 

methods spread across the files. 

EXAMPLES:: 

sage: HyperbolicPlane().UHP().get_point(2 + I) 

Point in UHP I + 2 

sage: HyperbolicPlane().PD().get_point(1/2 + I/2) 

Point in PD 1/2*I + 1/2 

""" 

#*********************************************************************** 

# 

# Copyright (C) 2013 Greg Laun <glaun@math.umd.edu> 

# 

# 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# as published by the Free Software Foundation; either version 2 of 

# the License, or (at your option) any later version. 

# http://www.gnu.org/licenses/ 

#*********************************************************************** 

from __future__ import division 

from sage.structure.unique_representation import UniqueRepresentation 

from sage.structure.parent import Parent 

from sage.misc.abstract_method import abstract_method 

from sage.categories.sets_cat import Sets 

from sage.categories.realizations import Realizations, Category_realization_of_parent 

from sage.geometry.hyperbolic_space.hyperbolic_model import ( 

HyperbolicModelUHP, HyperbolicModelPD, 

HyperbolicModelHM, HyperbolicModelKM) 

def HyperbolicSpace(n): 

""" 

Return ``n`` dimensional hyperbolic space. 

EXAMPLES:: 

sage: from sage.geometry.hyperbolic_space.hyperbolic_interface import HyperbolicSpace 

sage: HyperbolicSpace(2) 

Hyperbolic plane 

""" 

if n == 2: 

return HyperbolicPlane() 

raise NotImplementedError("currently only implemented in dimension 2") 

class HyperbolicPlane(Parent, UniqueRepresentation): 

""" 

The hyperbolic plane `\mathbb{H}^2`. 

Here are the models currently implemented: 

- ``UHP`` -- upper half plane 

- ``PD`` -- Poincaré disk 

- ``KM`` -- Klein disk 

- ``HM`` -- hyperboloid model 

""" 

def __init__(self): 

""" 

Initialize ``self``. 

EXAMPLES:: 

sage: H = HyperbolicPlane() 

sage: TestSuite(H).run() 

""" 

Parent.__init__(self, category=Sets().Metric().WithRealizations()) 

self.a_realization() # We create a realization so at least one is known 

def _repr_(self): 

""" 

Return a string representation of ``self``. 

EXAMPLES:: 

sage: HyperbolicPlane() 

Hyperbolic plane 

""" 

return "Hyperbolic plane" 

def a_realization(self): 

""" 

Return a realization of ``self``. 

EXAMPLES:: 

sage: H = HyperbolicPlane() 

sage: H.a_realization() 

Hyperbolic plane in the Upper Half Plane Model 

""" 

return self.UHP() 

UHP = HyperbolicModelUHP 

UpperHalfPlane = UHP 

PD = HyperbolicModelPD 

PoincareDisk = PD 

KM = HyperbolicModelKM 

KleinDisk = KM 

HM = HyperbolicModelHM 

Hyperboloid = HM 

class HyperbolicModels(Category_realization_of_parent): 

r""" 

The category of hyperbolic models of hyperbolic space. 

""" 

def __init__(self, base): 

r""" 

Initialize the hyperbolic models of hyperbolic space. 

INPUT: 

- ``base`` -- a hyperbolic space 

TESTS:: 

sage: from sage.geometry.hyperbolic_space.hyperbolic_interface import HyperbolicModels 

sage: H = HyperbolicPlane() 

sage: models = HyperbolicModels(H) 

sage: H.UHP() in models 

True 

""" 

Category_realization_of_parent.__init__(self, base) 

def _repr_(self): 

r""" 

Return the representation of ``self``. 

EXAMPLES:: 

sage: from sage.geometry.hyperbolic_space.hyperbolic_interface import HyperbolicModels 

sage: H = HyperbolicPlane() 

sage: HyperbolicModels(H) 

Category of hyperbolic models of Hyperbolic plane 

""" 

return "Category of hyperbolic models of {}".format(self.base()) 

def super_categories(self): 

r""" 

The super categories of ``self``. 

EXAMPLES:: 

sage: from sage.geometry.hyperbolic_space.hyperbolic_interface import HyperbolicModels 

sage: H = HyperbolicPlane() 

sage: models = HyperbolicModels(H) 

sage: models.super_categories() 

[Category of metric spaces, 

Category of realizations of Hyperbolic plane] 

""" 

return [Sets().Metric(), Realizations(self.base())] 

class ParentMethods: 

def _an_element_(self): 

""" 

Return an element of ``self``. 

EXAMPLES:: 

sage: H = HyperbolicPlane() 

sage: H.UHP().an_element() 

Point in UHP I 

sage: H.PD().an_element() 

Point in PD 0 

sage: H.KM().an_element() 

Point in KM (0, 0) 

sage: H.HM().an_element() 

Point in HM (0, 0, 1) 

""" 

from sage.rings.integer_ring import ZZ 

return self(self.realization_of().PD().get_point(ZZ.zero()))