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| * sage.geometry.hyperbolic_geometry (hyperbolic spaces) * sage.groups.surface_gps (abstract surface groups) Where do we put * representation of surface group into PSL(2,R) * fundamental domains of such groups / Poincare polygons / Dirichlet fundamental domains |
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| * computing fundamental group and homology * maps between flat surfaces * action of SL(2,R) and isomorphisms |
* computing fundamental group and relative homology and homology (as well as functors between them) * maps between flat surfaces (and functors to fundamental group and homologies) * action of SL(2,R) and isomorphisms (and functors) |
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| === Surface Group === They are needed from two point of vue: the group of the surface itself and its stabilizer under SL(2,R) or PSL(2,R) action. There must be some software for dealing with surface group. We need to look at * [[http://www.warwick.ac.uk/~mareg/download/kbmag2/|kbmag]]: Knuth-Bendix in Monoids and Automatic Groups implemented by Derek Holt |
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| This part is roughly implemented in [[http://trac.sagemath.org/sage_trac/ticket/9439|trac #9439]] | |
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| * tesselations (covering of HH by finite area convex polygonal domains) * Fuchsian groups, their fundamental domains and their associated tesselations |
* tessellations (covering of HH by finite area convex polygonal domains) * Fuchsian groups, their fundamental domains and their associated tessellations The [[http://egl.math.umd.edu/|Experimental Geometry Lab]] (university of Maryland) published a lot of Mathematica package/worksheets to deal with Kleinian adn Fuchsian groups, hyperbolic tessellations, etc... |
Flat surfaces in Sage
Introduction
For general mathematic references see the Flat surfaces wiki. A flat surface can be seen either
- as a union of polygons glued along pairs of parallel sides,
- as a flat metric with no holonomy on a compact surface,
- as a Riemann surface together with a non zero Abelian (or quadratic) differential,
- ...
This page is aimed to be a roadmap for the implementations of various algorithm related to flat surfaces and more generally geometry/combinatoric/dynamic of surfaces (Mapping class group, train track, pseudo-Anosov dynamic, ...). For the moment we share the sage-combinat repository with mercurial for the development.
General architecture
For now the main structure is as follows
- sage.combinat.flat_surfaces (which contains various generic objects)
- sage.combinat.flat_surfaces.iet (for interval exchange transformations stuff)
- sage.combinat.flat_surfaces.origamis (for origamis/square tiled surfaces stuff)
- sage.geometry.hyperbolic_geometry (hyperbolic spaces)
- sage.groups.surface_gps (abstract surface groups)
Where do we put
- representation of surface group into PSL(2,R)
- fundamental domains of such groups / Poincare polygons / Dirichlet fundamental domains
Roadmap
Port of other programs
- Joshua Bowman program on iso-Delaunay tessellations (written in Java)
- Finish Anton Zorich port of Interval Exchange Transformations and Linear Involutions (written in Mathematica)
- Anton Zorich program for computing approximation of various Lyapunov exponents (written in C and Mathematica)
- Alex Eskin program for analyzing saddle connections direction in a surface (written in C++)
Different representations/implementations for flat surfaces
- (convex) polygonal surface
- rectangulated surface
- suspension of iet (and li) (almost in Sage)
- Thurston-Veech construction
- triangulated surface
- Delaunay surface (?)
- rectangulated surface
- Algebraic curve with Abelian or quadratic differential
- Coverings (make it relative)... need to implement maps between translation surfaces
- square tiled surfaces/origamis (covering of the torus) (almost in Sage)
- hyperelliptic curves (specifying a double cover of the sphere)
- Unfoldings of rational billiards
Needed generic methods
- switch between representations (the one to which everybody can be converted is triangulated flat surface)
- computing fundamental group and relative homology and homology (as well as functors between them)
- maps between flat surfaces (and functors to fundamental group and homologies)
- action of SL(2,R) and isomorphisms (and functors)
- Siegel Veech constants
- Lyapunov exponents
Surface Group
They are needed from two point of vue: the group of the surface itself and its stabilizer under SL(2,R) or PSL(2,R) action. There must be some software for dealing with surface group. We need to look at
kbmag: Knuth-Bendix in Monoids and Automatic Groups implemented by Derek Holt
Hyperbolic geometry
This part is roughly implemented in trac #9439
- the three 2D models: hyperbolic plane, hyperbolic disc and the hyperboloïd
- points, geodesics and polygonal domains
- tessellations (covering of HH by finite area convex polygonal domains)
- Fuchsian groups, their fundamental domains and their associated tessellations
The Experimental Geometry Lab (university of Maryland) published a lot of Mathematica package/worksheets to deal with Kleinian adn Fuchsian groups, hyperbolic tessellations, etc...