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| = Flat surfaces in Sage = | #REDIRECT dynamics/FlatSurfaces = Flat surfaces = |
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| == Introduction == For general mathematic references see the [[https://lma.homelinux.org/wiki/FlatSurfaces/FlatSurfaces|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 [[http://wiki.sagemath.org/combinat|sage-combinat repository]] with mercurial for the development. This project take part in the wider [[SageDynamicsProject]] == 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 (?) * 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 * [[http://www.warwick.ac.uk/~mareg/download/kbmag2/|kbmag]]: Knuth-Bendix in Monoids and Automatic Groups implemented by Derek Holt === Hyperbolic geometry === This part is roughly implemented in [[http://trac.sagemath.org/sage_trac/ticket/9439|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 [[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... |
This page has moved to [[dynamics/FlatSurfaces]] |
