Convex geometry of the hyperbolic plane

[sfreedman67]

This is a draft of notes that we took at a meeting in Bordeaux in early 2022. Please extend these notes.

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Geometry on the hyperbolic plane is not really implemented in SageMath. It's needed for our implementation of #155. While it might eventually go upstream into SageMath, we want to first implement it in sage-flatsurf.

Here are some objects that we think we should implement:

class flatsurf.geometry.HyperbolicPlane:
    # over a fixed base ring. Probably does nothing but store that base ring.

class flatsurf.geometry.HyperbolicBoundaryPoint:
    # projective coordinates of a boundary point.

class flatsurf.geometry.HyperbolicGeodesic:

class flatsurf.geometry.HyperbolicEdge:
    # a geodesic segment

class flatsurf.geometry.HyperbolicHalfPlane:
    # a geodesic + an orientation. Implements sagemath containment for points.

class flatsurf.geometry.HyperbolicPoint:
    # a pair of geodesics. lazily computes cached exact coordinates, e.g., for hashing.

class flatsurf.geometry.HyperbolicPolytope:
    # intersection of half planes, as a list of edges.

It is not completely clear which hyperbolic model is the best to work with (probably the hyperboloid is more "linear").

[videlec]

I think that if we use the hyperboloid model (ie R^{2,1}) then there is no issue with taking square-roots for computing coordinates. The only drawback is that computations are now in 3 dimensions (but homogeneous cones, not general polytopes).

[videlec]

I also think that a boundary point is a special case of point (it is just that the quadratic form evaluates to zero). There is also "beyond ideal" points that are in the positive cone of the space.

[saraedum]

Fixed in #158.