# Hyperbolic Models¶

In this module, a hyperbolic model is a collection of data that allow the user to implement new models of hyperbolic space with minimal effort. The data include facts about the underlying set (such as whether the model is bounded), facts about the metric (such as whether the model is conformal), facts about the isometry group (such as whether it is a linear or projective group), and more. Generally speaking, any data or method that pertains to the model itself – rather than the points, geodesics, or isometries of the model – is implemented in this module.

Abstractly, a model of hyperbolic space is a connected, simply connected manifold equipped with a complete Riemannian metric of constant curvature $$-1$$. This module records information sufficient to enable computations in hyperbolic space without explicitly specifying the underlying set or its Riemannian metric. Although, see the SageManifolds project if you would like to take this approach.

This module implements the abstract base class for a model of hyperbolic space of arbitrary dimension. It also contains the implementations of specific models of hyperbolic geometry.

AUTHORS:

• Greg Laun (2013): Initial version.

EXAMPLES:

We illustrate how the classes in this module encode data by comparing the upper half plane (UHP), Poincaré disk (PD) and hyperboloid (HM) models. First we create:

sage: U = HyperbolicPlane().UHP()
sage: P = HyperbolicPlane().PD()
sage: H = HyperbolicPlane().HM()


We note that the UHP and PD models are bounded while the HM model is not:

sage: U.is_bounded() and P.is_bounded()
True
sage: H.is_bounded()
False


The isometry groups of UHP and PD are projective, while that of HM is linear:

sage: U.is_isometry_group_projective()
True
sage: H.is_isometry_group_projective()
False


The models are responsible for determining if the coordinates of points and the matrix of linear maps are appropriate for constructing points and isometries in hyperbolic space:

sage: U.point_in_model(2 + I)
True
sage: U.point_in_model(2 - I)
False
sage: U.point_in_model(2)
False
sage: U.boundary_point_in_model(2)
True

class sage.geometry.hyperbolic_space.hyperbolic_model.HyperbolicModel(space, name, short_name, bounded, conformal, dimension, isometry_group, isometry_group_is_projective)

Abstract base class for hyperbolic models.

Element
bdry_point_test(p)

Test whether a point is in the model. If the point is in the model, do nothing; otherwise raise a ValueError.

EXAMPLES:

sage: HyperbolicPlane().UHP().bdry_point_test(2)
sage: HyperbolicPlane().UHP().bdry_point_test(1 + I)
Traceback (most recent call last):
...
ValueError: I + 1 is not a valid boundary point in the UHP model

boundary_point_in_model(p)

Return True if the point is on the ideal boundary of hyperbolic space and False otherwise.

INPUT:

• any object that can converted into a complex number

OUTPUT:

• boolean

EXAMPLES:

sage: HyperbolicPlane().UHP().boundary_point_in_model(I)
False

dist(a, b)

Calculate the hyperbolic distance between a and b.

INPUT:

• a, b – a point or geodesic

OUTPUT:

• the hyperbolic distance

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: p1 = PD.get_point(0)
sage: p2 = PD.get_point(I/2)
sage: PD.dist(p1, p2)
arccosh(5/3)

sage: UHP(p1).dist(UHP(p2))
arccosh(5/3)

sage: KM = HyperbolicPlane().KM()
sage: p1 = KM.get_point((0, 0))
sage: p2 = KM.get_point((1/2, 1/2))
sage: numerical_approx(KM.dist(p1, p2))
0.881373587019543

sage: HM = HyperbolicPlane().HM()
sage: p1 = HM.get_point((0,0,1))
sage: p2 = HM.get_point((1,0,sqrt(2)))
sage: numerical_approx(HM.dist(p1, p2))
0.881373587019543


Distance between a point and itself is 0:

sage: p = UHP.get_point(47 + I)
sage: UHP.dist(p, p)
0


Points on the boundary are infinitely far from interior points:

sage: UHP.get_point(3).dist(UHP.get_point(I))
+Infinity

get_geodesic(start, end=None, **graphics_options)

Return a geodesic in the appropriate model.

EXAMPLES:

sage: HyperbolicPlane().UHP().get_geodesic(I, 2*I)
Geodesic in UHP from I to 2*I

sage: HyperbolicPlane().PD().get_geodesic(0, I/2)
Geodesic in PD from 0 to 1/2*I

sage: HyperbolicPlane().KM().get_geodesic((1/2, 1/2), (0,0))
Geodesic in KM from (1/2, 1/2) to (0, 0)

sage: HyperbolicPlane().HM().get_geodesic((0,0,1), (1,0, sqrt(2)))
Geodesic in HM from (0, 0, 1) to (1, 0, sqrt(2))

get_isometry(A)

Return an isometry in self from the matrix A in the isometry group of self.

EXAMPLES:

sage: HyperbolicPlane().UHP().get_isometry(identity_matrix(2))
Isometry in UHP
[1 0]
[0 1]

sage: HyperbolicPlane().PD().get_isometry(identity_matrix(2))
Isometry in PD
[1 0]
[0 1]

sage: HyperbolicPlane().KM().get_isometry(identity_matrix(3))
Isometry in KM
[1 0 0]
[0 1 0]
[0 0 1]

sage: HyperbolicPlane().HM().get_isometry(identity_matrix(3))
Isometry in HM
[1 0 0]
[0 1 0]
[0 0 1]

get_point(coordinates, is_boundary=None, **graphics_options)

Return a point in self.

Automatically determine the type of point to return given either:

1. the coordinates of a point in the interior or ideal boundary of hyperbolic space, or
2. a HyperbolicPoint object.

INPUT:

• a point in hyperbolic space or on the ideal boundary

OUTPUT:

EXAMPLES:

We can create an interior point via the coordinates:

sage: HyperbolicPlane().UHP().get_point(2*I)
Point in UHP 2*I


Or we can create a boundary point via the coordinates:

sage: HyperbolicPlane().UHP().get_point(23)
Boundary point in UHP 23


However we cannot create points outside of our model:

sage: HyperbolicPlane().UHP().get_point(12 - I)
Traceback (most recent call last):
...
ValueError: -I + 12 is not a valid point in the UHP model

sage: HyperbolicPlane().UHP().get_point(2 + 3*I)
Point in UHP 3*I + 2

sage: HyperbolicPlane().PD().get_point(0)
Point in PD 0

sage: HyperbolicPlane().KM().get_point((0,0))
Point in KM (0, 0)

sage: HyperbolicPlane().HM().get_point((0,0,1))
Point in HM (0, 0, 1)

sage: p = HyperbolicPlane().UHP().get_point(I, color="red")
sage: p.graphics_options()
{'color': 'red'}

sage: HyperbolicPlane().UHP().get_point(12)
Boundary point in UHP 12

sage: HyperbolicPlane().UHP().get_point(infinity)
Boundary point in UHP +Infinity

sage: HyperbolicPlane().PD().get_point(I)
Boundary point in PD I

sage: HyperbolicPlane().KM().get_point((0,-1))
Boundary point in KM (0, -1)

is_bounded()

Return True if self is a bounded model.

EXAMPLES:

sage: HyperbolicPlane().UHP().is_bounded()
True
sage: HyperbolicPlane().PD().is_bounded()
True
sage: HyperbolicPlane().KM().is_bounded()
True
sage: HyperbolicPlane().HM().is_bounded()
False

is_conformal()

Return True if self is a conformal model.

EXAMPLES:

sage: UHP = HyperbolicPlane().UHP()
sage: UHP.is_conformal()
True

is_isometry_group_projective()

Return True if the isometry group of self is projective.

EXAMPLES:

sage: UHP = HyperbolicPlane().UHP()
sage: UHP.is_isometry_group_projective()
True

isometry_from_fixed_points(repel, attract)

Given two fixed points repel and attract as hyperbolic points return a hyperbolic isometry with repel as repelling fixed point and attract as attracting fixed point.

EXAMPLES:

sage: UHP = HyperbolicPlane().UHP()
sage: PD = HyperbolicPlane().PD()
sage: PD.isometry_from_fixed_points(-i, i)
Isometry in PD
[   3/4  1/4*I]
[-1/4*I    3/4]

sage: p, q = PD.get_point(1/2 + I/2), PD.get_point(6/13 + 9/13*I)
sage: PD.isometry_from_fixed_points(p, q)
Traceback (most recent call last):
...
ValueError: fixed points of hyperbolic elements must be ideal

sage: p, q = PD.get_point(4/5 + 3/5*I), PD.get_point(-I)
sage: PD.isometry_from_fixed_points(p, q)
Isometry in PD
[ 1/6*I - 2/3 -1/3*I - 1/6]
[ 1/3*I - 1/6 -1/6*I - 2/3]

isometry_in_model(A)

Return True if the input matrix represents an isometry of the given model and False otherwise.

INPUT:

• a matrix that represents an isometry in the appropriate model

OUTPUT:

• boolean

EXAMPLES:

sage: HyperbolicPlane().UHP().isometry_in_model(identity_matrix(2))
True

sage: HyperbolicPlane().UHP().isometry_in_model(identity_matrix(3))
False

isometry_test(A)

Test whether an isometry A is in the model.

If the isometry is in the model, do nothing. Otherwise, raise a ValueError.

EXAMPLES:

sage: HyperbolicPlane().UHP().isometry_test(identity_matrix(2))
sage: HyperbolicPlane().UHP().isometry_test(matrix(2, [I,1,2,1]))
Traceback (most recent call last):
...
ValueError:
[I 1]
[2 1] is not a valid isometry in the UHP model

name()

Return the name of this model.

EXAMPLES:

sage: UHP = HyperbolicPlane().UHP()
sage: UHP.name()
'Upper Half Plane Model'

point_in_model(p)

Return True if the point p is in the interior of the given model and False otherwise.

INPUT:

• any object that can converted into a complex number

OUTPUT:

• boolean

EXAMPLES:

sage: HyperbolicPlane().UHP().point_in_model(I)
True
sage: HyperbolicPlane().UHP().point_in_model(-I)
False

point_test(p)

Test whether a point is in the model. If the point is in the model, do nothing. Otherwise, raise a ValueError.

EXAMPLES:

sage: from sage.geometry.hyperbolic_space.hyperbolic_model import HyperbolicModelUHP
sage: HyperbolicPlane().UHP().point_test(2 + I)
sage: HyperbolicPlane().UHP().point_test(2 - I)
Traceback (most recent call last):
...
ValueError: -I + 2 is not a valid point in the UHP model

random_element(**kwargs)

Return a random point in self.

The points are uniformly distributed over the rectangle $$[-10, 10] \times [0, 10 i]$$ in the upper half plane model.

EXAMPLES:

sage: p = HyperbolicPlane().UHP().random_element()
sage: bool((p.coordinates().imag()) > 0)
True

sage: p = HyperbolicPlane().PD().random_element()
sage: HyperbolicPlane().PD().point_in_model(p.coordinates())
True

sage: p = HyperbolicPlane().KM().random_element()
sage: HyperbolicPlane().KM().point_in_model(p.coordinates())
True

sage: p = HyperbolicPlane().HM().random_element().coordinates()
sage: bool((p[0]**2 + p[1]**2 - p[2]**2 - 1) < 10**-8)
True

random_geodesic(**kwargs)

Return a random hyperbolic geodesic.

Return the geodesic between two random points.

EXAMPLES:

sage: h = HyperbolicPlane().PD().random_geodesic()
sage: bool((h.endpoints()[0].coordinates()).imag() >= 0)
True

random_isometry(preserve_orientation=True, **kwargs)

Return a random isometry in the model of self.

INPUT:

• preserve_orientation – if True return an orientation-preserving isometry

OUTPUT:

• a hyperbolic isometry

EXAMPLES:

sage: A = HyperbolicPlane().PD().random_isometry()
sage: A.preserves_orientation()
True
sage: B = HyperbolicPlane().PD().random_isometry(preserve_orientation=False)
sage: B.preserves_orientation()
False

random_point(**kwargs)

Return a random point of self.

The points are uniformly distributed over the rectangle $$[-10, 10] \times [0, 10 i]$$ in the upper half plane model.

EXAMPLES:

sage: p = HyperbolicPlane().UHP().random_point()
sage: bool((p.coordinates().imag()) > 0)
True

sage: PD = HyperbolicPlane().PD()
sage: p = PD.random_point()
sage: PD.point_in_model(p.coordinates())
True

short_name()

Return the short name of this model.

EXAMPLES:

sage: UHP = HyperbolicPlane().UHP()
sage: UHP.short_name()
'UHP'

class sage.geometry.hyperbolic_space.hyperbolic_model.HyperbolicModelHM(space)

Hyperboloid Model.

boundary_point_in_model(p)

Return False since the Hyperboloid model has no boundary points.

EXAMPLES:

sage: HM = HyperbolicPlane().HM()
sage: HM.boundary_point_in_model((0,0,1))
False
sage: HM.boundary_point_in_model((1,0,sqrt(2)))
False
sage: HM.boundary_point_in_model((1,2,1))
False

get_background_graphic(**bdry_options)

Return a graphic object that makes the model easier to visualize. For the hyperboloid model, the background object is the hyperboloid itself.

EXAMPLES:

sage: H = HyperbolicPlane().HM().get_background_graphic()

isometry_in_model(A)

Test that the matrix A is in the group $$SO(2,1)^+$$.

EXAMPLES:

sage: A = diagonal_matrix([1,1,-1])
sage: HyperbolicPlane().HM().isometry_in_model(A)
True

point_in_model(p)

Check whether a complex number lies in the hyperboloid.

EXAMPLES:

sage: HM = HyperbolicPlane().HM()
sage: HM.point_in_model((0,0,1))
True
sage: HM.point_in_model((1,0,sqrt(2)))
True
sage: HM.point_in_model((1,2,1))
False

class sage.geometry.hyperbolic_space.hyperbolic_model.HyperbolicModelKM(space)

Klein Model.

boundary_point_in_model(p)

Check whether a point lies in the unit circle, which corresponds to the ideal boundary of the hyperbolic plane in the Klein model.

EXAMPLES:

sage: KM = HyperbolicPlane().KM()
sage: KM.boundary_point_in_model((1, 0))
True
sage: KM.boundary_point_in_model((1/2, 1/2))
False
sage: KM.boundary_point_in_model((1, .2))
False

get_background_graphic(**bdry_options)

Return a graphic object that makes the model easier to visualize.

For the Klein model, the background object is the ideal boundary.

EXAMPLES:

sage: circ = HyperbolicPlane().KM().get_background_graphic()

isometry_in_model(A)

Check if the given matrix A is in the group $$SO(2,1)$$.

EXAMPLES:

sage: A = matrix(3, [[1, 0, 0], [0, 17/8, 15/8], [0, 15/8, 17/8]])
sage: HyperbolicPlane().KM().isometry_in_model(A)
True

point_in_model(p)

Check whether a point lies in the open unit disk.

EXAMPLES:

sage: KM = HyperbolicPlane().KM()
sage: KM.point_in_model((1, 0))
False
sage: KM.point_in_model((1/2, 1/2))
True
sage: KM.point_in_model((1, .2))
False

class sage.geometry.hyperbolic_space.hyperbolic_model.HyperbolicModelPD(space)

Poincaré Disk Model.

boundary_point_in_model(p)

Check whether a complex number lies in the open unit disk.

EXAMPLES:

sage: PD = HyperbolicPlane().PD()
sage: PD.boundary_point_in_model(1.00)
True
sage: PD.boundary_point_in_model(1/2 + I/2)
False
sage: PD.boundary_point_in_model(1 + .2*I)
False

get_background_graphic(**bdry_options)

Return a graphic object that makes the model easier to visualize.

For the Poincaré disk, the background object is the ideal boundary.

EXAMPLES:

sage: circ = HyperbolicPlane().PD().get_background_graphic()

isometry_in_model(A)

Check if the given matrix A is in the group $$U(1,1)$$.

EXAMPLES:

sage: z = [CC.random_element() for k in range(2)]; z.sort(key=abs)
sage: A = matrix(2,[z[1], z[0],z[0].conjugate(),z[1].conjugate()])
sage: HyperbolicPlane().PD().isometry_in_model(A)
True

point_in_model(p)

Check whether a complex number lies in the open unit disk.

EXAMPLES:

sage: PD = HyperbolicPlane().PD()
sage: PD.point_in_model(1.00)
False
sage: PD.point_in_model(1/2 + I/2)
True
sage: PD.point_in_model(1 + .2*I)
False

class sage.geometry.hyperbolic_space.hyperbolic_model.HyperbolicModelUHP(space)

Upper Half Plane model.

Element
boundary_point_in_model(p)

Check whether a complex number is a real number or \infty. In the UHP.model_name_name, this is the ideal boundary of hyperbolic space.

EXAMPLES:

sage: UHP = HyperbolicPlane().UHP()
sage: UHP.boundary_point_in_model(1 + I)
False
sage: UHP.boundary_point_in_model(infinity)
True
sage: UHP.boundary_point_in_model(CC(infinity))
True
sage: UHP.boundary_point_in_model(RR(infinity))
True
sage: UHP.boundary_point_in_model(1)
True
sage: UHP.boundary_point_in_model(12)
True
sage: UHP.boundary_point_in_model(1 - I)
False
sage: UHP.boundary_point_in_model(-2*I)
False
sage: UHP.boundary_point_in_model(0)
True
sage: UHP.boundary_point_in_model(I)
False

get_background_graphic(**bdry_options)

Return a graphic object that makes the model easier to visualize. For the upper half space, the background object is the ideal boundary.

EXAMPLES:

sage: hp = HyperbolicPlane().UHP().get_background_graphic()

isometry_from_fixed_points(repel, attract)

Given two fixed points repel and attract as complex numbers return a hyperbolic isometry with repel as repelling fixed point and attract as attracting fixed point.

EXAMPLES:

sage: UHP = HyperbolicPlane().UHP()
sage: UHP.isometry_from_fixed_points(2 + I, 3 + I)
Traceback (most recent call last):
...
ValueError: fixed points of hyperbolic elements must be ideal

sage: UHP.isometry_from_fixed_points(2, 0)
Isometry in UHP
[  -1    0]
[-1/3 -1/3]

isometry_in_model(A)

Check that A acts as an isometry on the upper half plane. That is, A must be an invertible $$2 \times 2$$ matrix with real entries.

EXAMPLES:

sage: UHP = HyperbolicPlane().UHP()
sage: A = matrix(2,[1,2,3,4])
sage: UHP.isometry_in_model(A)
True
sage: B = matrix(2,[I,2,4,1])
sage: UHP.isometry_in_model(B)
False


An example of a matrix $$A$$ such that $$\det(A) \neq 1$$, but the $$A$$ acts isometrically:

sage: C = matrix(2,[10,0,0,10])
sage: UHP.isometry_in_model(C)
True

point_in_model(p)

Check whether a complex number lies in the open upper half plane.

EXAMPLES:

sage: UHP = HyperbolicPlane().UHP()
sage: UHP.point_in_model(1 + I)
True
sage: UHP.point_in_model(infinity)
False
sage: UHP.point_in_model(CC(infinity))
False
sage: UHP.point_in_model(RR(infinity))
False
sage: UHP.point_in_model(1)
False
sage: UHP.point_in_model(12)
False
sage: UHP.point_in_model(1 - I)
False
sage: UHP.point_in_model(-2*I)
False
sage: UHP.point_in_model(I)
True
sage: UHP.point_in_model(0) # Not interior point
False

random_isometry(preserve_orientation=True, **kwargs)

Return a random isometry in the Upper Half Plane model.

INPUT:

• preserve_orientation – if True return an orientation-preserving isometry

OUTPUT:

• a hyperbolic isometry

EXAMPLES:

sage: A = HyperbolicPlane().UHP().random_isometry()
sage: B = HyperbolicPlane().UHP().random_isometry(preserve_orientation=False)
sage: B.preserves_orientation()
False

random_point(**kwargs)

Return a random point in the upper half plane. The points are uniformly distributed over the rectangle $$[-10, 10] \times [0, 10i]$$.

EXAMPLES:

sage: p = HyperbolicPlane().UHP().random_point().coordinates()
sage: bool((p.imag()) > 0)
True