Hyperbolic geometry functions

[aelzenaar]

Add functions to:

• construct regular polygons
• compute geometry of pairs of geodesics [B, section starting p.162]
• Find pencils fixed by mobius transforms and (inversely) find the mobius transform fixing a pencil

[aelzenaar]

More generally I want to revamp the parts of the library which do hyperbolic computations. We require:

• a class to represent conformal maps/anticonformal maps, which wraps matrices and also keeps information about precomposition of \overline{z};
• a class to represent families of circles which has different views (as centre/radius pairs, etc.)

then the maps should be able to act on different kinds of objects:

• hyperbolic space, in the upper half-space model
• circle families

We also would like to include n-dimensional hyperbolic space, but perhaps as a start only 2 and 3 dimensions.

Proposal: a new module, geometry.py, which includes the following classes:

• MobiusMap, which encapsulates matrices in PSL(2,C) and includes functions such as "act_on" (which is allowed to take any of the three following classes).
• CircleSet, which internally keeps track of circles/lines in the conformal boundary
• Pencil, which keeps track of an entire 1D pencil of circles
• HyperbolicPoint, which keeps track of a point in hyperbolic space.

We then have various geometric functions, e.g. cross-ratio, hyperbolic distance, construction of triangles, conformal product etc.

Next step is to produce a list of exactly which functions we will include with references to standard textbooks e.g. Beardon, Ratcliffe, Maskit.