Hyperbolic Points¶

This module implements points in hyperbolic space of arbitrary dimension. It also contains the implementations for specific models of hyperbolic geometry.

This module also implements ideal points in hyperbolic space of arbitrary dimension. It also contains the implementations for specific models of hyperbolic geometry.

Note that not all models of hyperbolic space are bounded, meaning that the ideal boundary is not the topological boundary of the set underlying tho model. For example, the unit disk model is bounded with boundary given by the unit sphere. The hyperboloid model is not bounded.

AUTHORS:

Greg Laun (2013): initial version

EXAMPLES:

We can construct points in the upper half plane model, abbreviated UHP for convenience:

Sage

sage: UHP = HyperbolicPlane().UHP()
sage: UHP.get_point(2 + I)
Point in UHP I + 2
sage: g = UHP.get_point(3 + I)
sage: g.dist(UHP.get_point(I))
arccosh(11/2)

Python

>>> from sage.all import *
>>> UHP = HyperbolicPlane().UHP()
>>> UHP.get_point(Integer(2) + I)
Point in UHP I + 2
>>> g = UHP.get_point(Integer(3) + I)
>>> g.dist(UHP.get_point(I))
arccosh(11/2)

Sage Live

UHP = HyperbolicPlane().UHP()
UHP.get_point(2 + I)
g = UHP.get_point(3 + I)
g.dist(UHP.get_point(I))

We can also construct boundary points in the upper half plane model:

Sage

sage: UHP.get_point(3)
Boundary point in UHP 3

Python

>>> from sage.all import *
>>> UHP.get_point(Integer(3))
Boundary point in UHP 3

Sage Live

UHP.get_point(3)

Some more examples:

Sage

sage: HyperbolicPlane().UHP().get_point(0)
Boundary point in UHP 0

sage: HyperbolicPlane().PD().get_point(I/2)
Point in PD 1/2*I

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

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

Python

>>> from sage.all import *
>>> HyperbolicPlane().UHP().get_point(Integer(0))
Boundary point in UHP 0

>>> HyperbolicPlane().PD().get_point(I/Integer(2))
Point in PD 1/2*I

>>> HyperbolicPlane().KM().get_point((Integer(0),Integer(1)))
Boundary point in KM (0, 1)

>>> HyperbolicPlane().HM().get_point((Integer(0),Integer(0),Integer(1)))
Point in HM (0, 0, 1)

Sage Live

HyperbolicPlane().UHP().get_point(0)
HyperbolicPlane().PD().get_point(I/2)
HyperbolicPlane().KM().get_point((0,1))
HyperbolicPlane().HM().get_point((0,0,1))

class sage.geometry.hyperbolic_space.hyperbolic_point.HyperbolicPoint(model, coordinates, is_boundary, check=True, **graphics_options)[source]¶

Bases: Element

Abstract base class for hyperbolic points. This class should never be instantiated.

INPUT:

model – the model of the hyperbolic space
coordinates – the coordinates of a hyperbolic point in the appropriate model
is_boundary – whether the point is a boundary point
check – boolean (default: True); if True, then check to make sure the coordinates give a valid point in the model

EXAMPLES:

Comparison between different models is performed via coercion:

Sage

sage: UHP = HyperbolicPlane().UHP()
sage: p = UHP.get_point(.2 + .3*I); p
Point in UHP 0.200000000000000 + 0.300000000000000*I

sage: PD = HyperbolicPlane().PD()
sage: q = PD.get_point(0.2 + 0.3*I); q
Point in PD 0.200000000000000 + 0.300000000000000*I

sage: p == q
False
sage: PD(p)
Point in PD 0.231213872832370 - 0.502890173410405*I

sage: bool(p.coordinates() == q.coordinates())
True

Python

>>> from sage.all import *
>>> UHP = HyperbolicPlane().UHP()
>>> p = UHP.get_point(RealNumber('.2') + RealNumber('.3')*I); p
Point in UHP 0.200000000000000 + 0.300000000000000*I

>>> PD = HyperbolicPlane().PD()
>>> q = PD.get_point(RealNumber('0.2') + RealNumber('0.3')*I); q
Point in PD 0.200000000000000 + 0.300000000000000*I

>>> p == q
False
>>> PD(p)
Point in PD 0.231213872832370 - 0.502890173410405*I

>>> bool(p.coordinates() == q.coordinates())
True

Sage Live

UHP = HyperbolicPlane().UHP()
p = UHP.get_point(.2 + .3*I); p
PD = HyperbolicPlane().PD()
q = PD.get_point(0.2 + 0.3*I); q
p == q
PD(p)
bool(p.coordinates() == q.coordinates())

Similarly for boundary points:

Sage

sage: p = UHP.get_point(-1); p
Boundary point in UHP -1

sage: q = PD.get_point(-1); q
Boundary point in PD -1

sage: p == q
True
sage: PD(p)
Boundary point in PD -1

Python

>>> from sage.all import *
>>> p = UHP.get_point(-Integer(1)); p
Boundary point in UHP -1

>>> q = PD.get_point(-Integer(1)); q
Boundary point in PD -1

>>> p == q
True
>>> PD(p)
Boundary point in PD -1

Sage Live

p = UHP.get_point(-1); p
q = PD.get_point(-1); q
p == q
PD(p)

It is an error to specify a point that does not lie in the appropriate model:

Sage

sage: HyperbolicPlane().UHP().get_point(0.2 - 0.3*I)
Traceback (most recent call last):
...
ValueError: 0.200000000000000 - 0.300000000000000*I is not a valid point in the UHP model

sage: HyperbolicPlane().PD().get_point(1.2)
Traceback (most recent call last):
...
ValueError: 1.20000000000000 is not a valid point in the PD model

sage: HyperbolicPlane().KM().get_point((1,1))
Traceback (most recent call last):
...
ValueError: (1, 1) is not a valid point in the KM model

sage: HyperbolicPlane().HM().get_point((1, 1, 1))
Traceback (most recent call last):
...
ValueError: (1, 1, 1) is not a valid point in the HM model

Python

>>> from sage.all import *
>>> HyperbolicPlane().UHP().get_point(RealNumber('0.2') - RealNumber('0.3')*I)
Traceback (most recent call last):
...
ValueError: 0.200000000000000 - 0.300000000000000*I is not a valid point in the UHP model

>>> HyperbolicPlane().PD().get_point(RealNumber('1.2'))
Traceback (most recent call last):
...
ValueError: 1.20000000000000 is not a valid point in the PD model

>>> HyperbolicPlane().KM().get_point((Integer(1),Integer(1)))
Traceback (most recent call last):
...
ValueError: (1, 1) is not a valid point in the KM model

>>> HyperbolicPlane().HM().get_point((Integer(1), Integer(1), Integer(1)))
Traceback (most recent call last):
...
ValueError: (1, 1, 1) is not a valid point in the HM model

Sage Live

HyperbolicPlane().UHP().get_point(0.2 - 0.3*I)
HyperbolicPlane().PD().get_point(1.2)
HyperbolicPlane().KM().get_point((1,1))
HyperbolicPlane().HM().get_point((1, 1, 1))

It is an error to specify an interior point of hyperbolic space as a boundary point:

Sage

sage: HyperbolicPlane().UHP().get_point(0.2 + 0.3*I, is_boundary=True)
Traceback (most recent call last):
...
ValueError: 0.200000000000000 + 0.300000000000000*I is not a valid boundary point in the UHP model

Python

>>> from sage.all import *
>>> HyperbolicPlane().UHP().get_point(RealNumber('0.2') + RealNumber('0.3')*I, is_boundary=True)
Traceback (most recent call last):
...
ValueError: 0.200000000000000 + 0.300000000000000*I is not a valid boundary point in the UHP model

Sage Live

HyperbolicPlane().UHP().get_point(0.2 + 0.3*I, is_boundary=True)

coordinates()[source]¶

Return the coordinates of the point.

EXAMPLES:

Sage

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

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

sage: HyperbolicPlane().KM().get_point((1/3, 1/4)).coordinates()
(1/3, 1/4)

sage: HyperbolicPlane().HM().get_point((0,0,1)).coordinates()
(0, 0, 1)

Python

>>> from sage.all import *
>>> HyperbolicPlane().UHP().get_point(Integer(2) + I).coordinates()
I + 2

>>> HyperbolicPlane().PD().get_point(Integer(1)/Integer(2) + Integer(1)/Integer(2)*I).coordinates()
1/2*I + 1/2

>>> HyperbolicPlane().KM().get_point((Integer(1)/Integer(3), Integer(1)/Integer(4))).coordinates()
(1/3, 1/4)

>>> HyperbolicPlane().HM().get_point((Integer(0),Integer(0),Integer(1))).coordinates()
(0, 0, 1)

Sage Live

HyperbolicPlane().UHP().get_point(2 + I).coordinates()
HyperbolicPlane().PD().get_point(1/2 + 1/2*I).coordinates()
HyperbolicPlane().KM().get_point((1/3, 1/4)).coordinates()
HyperbolicPlane().HM().get_point((0,0,1)).coordinates()

graphics_options()[source]¶

Return the graphics options of the current point.

EXAMPLES:

Sage

sage: p = HyperbolicPlane().UHP().get_point(2 + I, color='red')
sage: p.graphics_options()
{'color': 'red'}

Python

>>> from sage.all import *
>>> p = HyperbolicPlane().UHP().get_point(Integer(2) + I, color='red')
>>> p.graphics_options()
{'color': 'red'}

Sage Live

p = HyperbolicPlane().UHP().get_point(2 + I, color='red')
p.graphics_options()

is_boundary()[source]¶

Return True if self is a boundary point.

EXAMPLES:

Sage

sage: PD = HyperbolicPlane().PD()
sage: p = PD.get_point(0.5+.2*I)
sage: p.is_boundary()
False
sage: p = PD.get_point(I)
sage: p.is_boundary()
True

Python

>>> from sage.all import *
>>> PD = HyperbolicPlane().PD()
>>> p = PD.get_point(RealNumber('0.5')+RealNumber('.2')*I)
>>> p.is_boundary()
False
>>> p = PD.get_point(I)
>>> p.is_boundary()
True

Sage Live

PD = HyperbolicPlane().PD()
p = PD.get_point(0.5+.2*I)
p.is_boundary()
p = PD.get_point(I)
p.is_boundary()

model()[source]¶

Return the model to which the HyperbolicPoint belongs.

EXAMPLES:

Sage

sage: HyperbolicPlane().UHP().get_point(I).model()
Hyperbolic plane in the Upper Half Plane Model

sage: HyperbolicPlane().PD().get_point(0).model()
Hyperbolic plane in the Poincare Disk Model

sage: HyperbolicPlane().KM().get_point((0,0)).model()
Hyperbolic plane in the Klein Disk Model

sage: HyperbolicPlane().HM().get_point((0,0,1)).model()
Hyperbolic plane in the Hyperboloid Model

Python

>>> from sage.all import *
>>> HyperbolicPlane().UHP().get_point(I).model()
Hyperbolic plane in the Upper Half Plane Model

>>> HyperbolicPlane().PD().get_point(Integer(0)).model()
Hyperbolic plane in the Poincare Disk Model

>>> HyperbolicPlane().KM().get_point((Integer(0),Integer(0))).model()
Hyperbolic plane in the Klein Disk Model

>>> HyperbolicPlane().HM().get_point((Integer(0),Integer(0),Integer(1))).model()
Hyperbolic plane in the Hyperboloid Model

Sage Live

HyperbolicPlane().UHP().get_point(I).model()
HyperbolicPlane().PD().get_point(0).model()
HyperbolicPlane().KM().get_point((0,0)).model()
HyperbolicPlane().HM().get_point((0,0,1)).model()

show(boundary=True, **options)[source]¶

Plot self.

EXAMPLES:

Sage

sage: HyperbolicPlane().PD().get_point(0).show()                            # needs sage.plot
Graphics object consisting of 2 graphics primitives
sage: HyperbolicPlane().KM().get_point((0,0)).show()                        # needs sage.plot
Graphics object consisting of 2 graphics primitives
sage: HyperbolicPlane().HM().get_point((0,0,1)).show()                      # needs sage.plot
Graphics3d Object

Python

>>> from sage.all import *
>>> HyperbolicPlane().PD().get_point(Integer(0)).show()                            # needs sage.plot
Graphics object consisting of 2 graphics primitives
>>> HyperbolicPlane().KM().get_point((Integer(0),Integer(0))).show()                        # needs sage.plot
Graphics object consisting of 2 graphics primitives
>>> HyperbolicPlane().HM().get_point((Integer(0),Integer(0),Integer(1))).show()                      # needs sage.plot
Graphics3d Object

Sage Live

HyperbolicPlane().PD().get_point(0).show()                            # needs sage.plot
HyperbolicPlane().KM().get_point((0,0)).show()                        # needs sage.plot
HyperbolicPlane().HM().get_point((0,0,1)).show()                      # needs sage.plot

symmetry_involution()[source]¶

Return the involutory isometry fixing the given point.

EXAMPLES:

Sage

sage: z = HyperbolicPlane().UHP().get_point(3 + 2*I)
sage: z.symmetry_involution()
Isometry in UHP
[  3/2 -13/2]
[  1/2  -3/2]

sage: HyperbolicPlane().UHP().get_point(I).symmetry_involution()
Isometry in UHP
[ 0 -1]
[ 1  0]

sage: HyperbolicPlane().PD().get_point(0).symmetry_involution()
Isometry in PD
[-I  0]
[ 0  I]

sage: HyperbolicPlane().KM().get_point((0, 0)).symmetry_involution()
Isometry in KM
[-1  0  0]
[ 0 -1  0]
[ 0  0  1]

sage: HyperbolicPlane().HM().get_point((0,0,1)).symmetry_involution()
Isometry in HM
[-1  0  0]
[ 0 -1  0]
[ 0  0  1]

sage: p = HyperbolicPlane().UHP().random_element()
sage: A = p.symmetry_involution()
sage: p.dist(A*p)  # abs tol 1e-10
0

sage: A.preserves_orientation()
True

sage: A*A == HyperbolicPlane().UHP().get_isometry(identity_matrix(2))       # needs scipy
True

Python

>>> from sage.all import *
>>> z = HyperbolicPlane().UHP().get_point(Integer(3) + Integer(2)*I)
>>> z.symmetry_involution()
Isometry in UHP
[  3/2 -13/2]
[  1/2  -3/2]

>>> HyperbolicPlane().UHP().get_point(I).symmetry_involution()
Isometry in UHP
[ 0 -1]
[ 1  0]

>>> HyperbolicPlane().PD().get_point(Integer(0)).symmetry_involution()
Isometry in PD
[-I  0]
[ 0  I]

>>> HyperbolicPlane().KM().get_point((Integer(0), Integer(0))).symmetry_involution()
Isometry in KM
[-1  0  0]
[ 0 -1  0]
[ 0  0  1]

>>> HyperbolicPlane().HM().get_point((Integer(0),Integer(0),Integer(1))).symmetry_involution()
Isometry in HM
[-1  0  0]
[ 0 -1  0]
[ 0  0  1]

>>> p = HyperbolicPlane().UHP().random_element()
>>> A = p.symmetry_involution()
>>> p.dist(A*p)  # abs tol 1e-10
0

>>> A.preserves_orientation()
True

>>> A*A == HyperbolicPlane().UHP().get_isometry(identity_matrix(Integer(2)))       # needs scipy
True

Sage Live

z = HyperbolicPlane().UHP().get_point(3 + 2*I)
z.symmetry_involution()
HyperbolicPlane().UHP().get_point(I).symmetry_involution()
HyperbolicPlane().PD().get_point(0).symmetry_involution()
HyperbolicPlane().KM().get_point((0, 0)).symmetry_involution()
HyperbolicPlane().HM().get_point((0,0,1)).symmetry_involution()
p = HyperbolicPlane().UHP().random_element()
A = p.symmetry_involution()
p.dist(A*p)  # abs tol 1e-10
A.preserves_orientation()
A*A == HyperbolicPlane().UHP().get_isometry(identity_matrix(2))       # needs scipy

to_model(model)[source]¶

Convert self to the model.

INPUT:

other – (a string representing) the image model

EXAMPLES:

Sage

sage: UHP = HyperbolicPlane().UHP()
sage: PD = HyperbolicPlane().PD()
sage: PD.get_point(1/2+I/2).to_model(UHP)
Point in UHP I + 2
sage: PD.get_point(1/2+I/2).to_model('UHP')
Point in UHP I + 2

Python

>>> from sage.all import *
>>> UHP = HyperbolicPlane().UHP()
>>> PD = HyperbolicPlane().PD()
>>> PD.get_point(Integer(1)/Integer(2)+I/Integer(2)).to_model(UHP)
Point in UHP I + 2
>>> PD.get_point(Integer(1)/Integer(2)+I/Integer(2)).to_model('UHP')
Point in UHP I + 2

Sage Live

UHP = HyperbolicPlane().UHP()
PD = HyperbolicPlane().PD()
PD.get_point(1/2+I/2).to_model(UHP)
PD.get_point(1/2+I/2).to_model('UHP')

update_graphics(update=False, **options)[source]¶

Update the graphics options of a HyperbolicPoint. If update is True, update rather than overwrite.

EXAMPLES:

Sage

sage: p = HyperbolicPlane().UHP().get_point(I); p.graphics_options()
{}

sage: p.update_graphics(color = "red"); p.graphics_options()
{'color': 'red'}

sage: p.update_graphics(color = "blue"); p.graphics_options()
{'color': 'blue'}

sage: p.update_graphics(True, size = 20); p.graphics_options()
{'color': 'blue', 'size': 20}

Python

>>> from sage.all import *
>>> p = HyperbolicPlane().UHP().get_point(I); p.graphics_options()
{}

>>> p.update_graphics(color = "red"); p.graphics_options()
{'color': 'red'}

>>> p.update_graphics(color = "blue"); p.graphics_options()
{'color': 'blue'}

>>> p.update_graphics(True, size = Integer(20)); p.graphics_options()
{'color': 'blue', 'size': 20}

Sage Live

p = HyperbolicPlane().UHP().get_point(I); p.graphics_options()
p.update_graphics(color = "red"); p.graphics_options()
p.update_graphics(color = "blue"); p.graphics_options()
p.update_graphics(True, size = 20); p.graphics_options()

class sage.geometry.hyperbolic_space.hyperbolic_point.HyperbolicPointUHP(model, coordinates, is_boundary, check=True, **graphics_options)[source]¶

Bases: HyperbolicPoint

A point in the UHP model.

INPUT:

the coordinates of a point in the unit disk in the complex plane \(\CC\)

EXAMPLES:

Sage

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

sage: HyperbolicPlane().UHP().get_point(1)
Boundary point in UHP 1

Python

>>> from sage.all import *
>>> HyperbolicPlane().UHP().get_point(Integer(2)*I)
Point in UHP 2*I

>>> HyperbolicPlane().UHP().get_point(Integer(1))
Boundary point in UHP 1

Sage Live

HyperbolicPlane().UHP().get_point(2*I)
HyperbolicPlane().UHP().get_point(1)

show(boundary=True, **options)[source]¶

Plot self.

EXAMPLES:

Sage

sage: HyperbolicPlane().UHP().get_point(I).show()
Graphics object consisting of 2 graphics primitives
sage: HyperbolicPlane().UHP().get_point(0).show()                           # needs sage.plot
Graphics object consisting of 2 graphics primitives
sage: HyperbolicPlane().UHP().get_point(infinity).show()
Traceback (most recent call last):
...
NotImplementedError: can...t draw the point infinity

Python

>>> from sage.all import *
>>> HyperbolicPlane().UHP().get_point(I).show()
Graphics object consisting of 2 graphics primitives
>>> HyperbolicPlane().UHP().get_point(Integer(0)).show()                           # needs sage.plot
Graphics object consisting of 2 graphics primitives
>>> HyperbolicPlane().UHP().get_point(infinity).show()
Traceback (most recent call last):
...
NotImplementedError: can...t draw the point infinity

Sage Live

HyperbolicPlane().UHP().get_point(I).show()
HyperbolicPlane().UHP().get_point(0).show()                           # needs sage.plot
HyperbolicPlane().UHP().get_point(infinity).show()

symmetry_involution()[source]¶

Return the involutory isometry fixing the given point.

EXAMPLES:

Sage

sage: HyperbolicPlane().UHP().get_point(3 + 2*I).symmetry_involution()
Isometry in UHP
[  3/2 -13/2]
[  1/2  -3/2]

Python

>>> from sage.all import *
>>> HyperbolicPlane().UHP().get_point(Integer(3) + Integer(2)*I).symmetry_involution()
Isometry in UHP
[  3/2 -13/2]
[  1/2  -3/2]

Sage Live

HyperbolicPlane().UHP().get_point(3 + 2*I).symmetry_involution()