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| For general mathematic references see the [[https://lma.homelinux.org/wiki/FlatSurfaces/FlatSurfaces|Flat surfaces wiki]]. A flat surfaces can be seen either | For general mathematic references see the [[https://lma.homelinux.org/wiki/FlatSurfaces/FlatSurfaces|Flat surfaces wiki]]. A flat surface can be seen either |
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| * as a flat metric with no holonomy on a surface, * as a Riemann surface and a non zero Abelian (or quadratic) differential. |
* as a flat metric with no holonomy on a compact surface, * as a Riemann surface together with a non zero Abelian (or quadratic) differential. |
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| This page is aimed to be a roadmap for the implementations of various algorithm related to flat surfaces and more generally geometry of surfaces. For the moment we share the [[http://wiki.sagemath.org/combinat|sage-combinat repository]] with mercurial for the development. | 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. |
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| * hyperelliptic curves (specifying the double cover over the sphere) | * hyperelliptic curves (specifying a double cover of the sphere) |
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| * Siegel Veech constants * Lyapunov exponents |
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| * the three 2D models: hyperbolic plane '''HH''', hyperbolic disc '''DD''' and the hyperboloïd * polygonal domains |
* the three 2D models: hyperbolic plane, hyperbolic disc and the hyperboloïd * points, geodesics and polygonal domains |
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| * Fuchsian groups and fundamental domains | * Fuchsian groups, their fundamental domains and their associated tesselations |
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)
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 homology
- maps between flat surfaces
- action of SL(2,R) and isomorphisms
- Siegel Veech constants
- Lyapunov exponents
Hyperbolic geometry
- the three 2D models: hyperbolic plane, hyperbolic disc and the hyperboloïd
- points, geodesics and polygonal domains
- tesselations (covering of HH by finite area convex polygonal domains)
- Fuchsian groups, their fundamental domains and their associated tesselations