Diff for "SageFlatProject"

⇤ ← Revision 3 as of 2010-07-19 14:28:32 →

Size: 2571

Editor: vdelecroix

Comment:

← Revision 10 as of 2017-04-28 19:03:43 → ⇥

Size: 104

Editor: mrennekamp

Comment:

Deletions are marked like this.

Additions are marked like this.

= Flat surfaces in Sage =

#REDIRECT dynamics/FlatSurfaces
= Flat surfaces =

== Introduction ==

For general mathematic references see the [[https://lma.homelinux.org/wiki/FlatSurfaces/FlatSurfaces|Flat surfaces wiki]]. A flat surfaces can be seen either
* as a union of polygons glued along pairs of parallel sides,
* as a flat metric with no holonomy on a surface,
* as a Riemann surface and 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 of surfaces. For the moment we share the [[http://wiki.sagemath.org/combinat|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 (?)
* 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 the double cover over 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

=== Hyperbolic geometry ===
* the three 2D models: hyperbolic plane '''HH''', hyperbolic disc '''DD''' and the hyperboloïd
* polygonal domains
* tesselations (covering of HH by finite area convex polygonal domains)
* Fuchsian groups and fundamental domains

This page has moved to [[dynamics/FlatSurfaces]]

-  ⇤ ← Revision 3 as of 2010-07-19 14:28:32 → 
  Size: 2571
  Editor: vdelecroix
  Comment:
+   ← Revision 10 as of 2017-04-28 19:03:43 → ⇥
  Size: 104
  Editor: mrennekamp
  Comment:
-Deletions are marked like this.
+Additions are marked like this.
 Line 1:
-= Flat surfaces in Sage =
+#REDIRECT dynamics/FlatSurfaces
= Flat surfaces =
-Line 3:
+Line 4:
-== Introduction ==

For general mathematic references see the [[https://lma.homelinux.org/wiki/FlatSurfaces/FlatSurfaces|Flat surfaces wiki]]. A flat surfaces can be seen either
 * as a union of polygons glued along pairs of parallel sides,
 * as a flat metric with no holonomy on a surface,
 * as a Riemann surface and 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 of surfaces. For the moment we share the [[http://wiki.sagemath.org/combinat|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 (?)
 * 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 the double cover over 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

=== Hyperbolic geometry ===
 * the three 2D models: hyperbolic plane '''HH''', hyperbolic disc '''DD''' and the hyperboloïd
 * polygonal domains
 * tesselations (covering of HH by finite area convex polygonal domains)
 * Fuchsian groups and fundamental domains
+This page has moved to [[dynamics/FlatSurfaces]]