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Hyperbolic Geometry
Release 10.0
The Sage Development Team
May 25, 2023


ii

CHAPTER
ONE
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 implemen-
tations 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: 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)
We can also construct boundary points in the upper half plane model:
sage: UHP.get_point(3)
Boundary point in UHP 3
Some more examples:
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)
1
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