Triangulation with boundary (meromorphic differentials)

[videlec]

This PR implement triangulation with boundaries in order to handle meromorphic differentials (that naturally appears as degeneration of holomorphic differentials). It also does a lot of factorization and cleaning in the base code.

• add a generic Constellation class as a super class of Triangulation
• Triangulation now has an extra _bdry attribute to encode boundary
• VeeringTriangulation can now use the _bdry attribute to encode meromorphic differentials
• new StrebelGraph class (subclass of Constellation) with conversion back and forth from VeeringTriangulation via the methods VeeringTriangulation.strebel_graph() and StrebelGraph.veering_triangulations()
• new StrebelGraphLinearFamily and conversion from VeeringTriangulationLinearFamily
• new class DelaunayStrebelAutomaton: automaton with states of two types VeeringTriangulation and StrebelGraph (meromorphic case only)
• many examples

Follow-up

• explicit Strebel graphs: see Explicit StrebelGraph for each meromorphic stratum #52
• make FlatStructure more compatible with VeeringTriangulationLinearFamily: see FlatStructure versus VeeringTriangulationLinearFamily #51
• make FlatStructure a quadratic differential in sage-flatsurf and move all layout logic to sage-flatsurf: see Use sage-flatsurf and ipyvue-flatsurf for plotting #38
• Strebel flip graph: Flip of Strebel graphs #53
• Remove code duplications: Duplications between StrebelGraph, VeeringTriangulation and LinearFamily #54

[videlec]

@KaiFu2210 The following example does not work for me

from veerer import *
vt = VeeringTriangulation("(0,~2,1)(3,~1,~0)", boundary="(2:2,~3:2)", colouring="RBRR")
vt.strebel_graph()

Even though vt.is_strebel() returns True (no forward flippable edge). It seems to me that something is wrong in your code. Do you confirm?

[KaiFu2210]

@KaiFu2210 The following example does not work for me

from veerer import *
vt = VeeringTriangulation("(0,~2,1)(3,~1,~0)", boundary="(2:2,~3:2)", colouring="RBRR")
vt.strebel_graph()

Even though vt.is_strebel() returns True (no forward flippable edge). It seems to me that something is wrong in your code. Do you confirm?

Yes, I confirm. The issue is that the code did not find as many half-edges as expected. Let me see what went wrong.

[KaiFu2210]

@KaiFu2210 The following example does not work for me

from veerer import *
vt = VeeringTriangulation("(0,~2,1)(3,~1,~0)", boundary="(2:2,~3:2)", colouring="RBRR")
vt.strebel_graph()

Even though vt.is_strebel() returns True (no forward flippable edge). It seems to me that something is wrong in your code. Do you confirm?

The mistake is from the function "strebel_edges". I found the code:

if list_boundary[e] > 0: #ep[e] is in an internal edge
a = fp[ep[e]]
b = fp[ep[e]]

Here, I'd like to parameterize three edge of the internal face that the edge e adjacent to.
However, I should use

b = fp[a]

instead of b = fp[ep[e]].

I tested that if I modified this, the output is correct.

[videlec]

@KaiFu2210 I do not understand why the following example from residue_constraint_cone is valid

            sage: G0 = StrebelGraph("(0)(~0)")
            sage: G0.residue_constraint_cone([[1,1]], QQ, HORIZONTAL).rays()
            [[0, 1]]

Namely, the constraint forces the only edge to have length zero.

Actually, I do not understand the answer. The condition [1, 1] should mean "sum of residues" which is supposedly tautological.

[KaiFu2210]

The residue of these two faces have opposite signs. The coefficients [1,1] means the sum of residues is zero. The coefficients [1, -1] is the case where the edge is of length zero.

The cone is constructed in the Strebel polyhedron. For this Strebel graph, the cone is the right vertical half-plane and the ambient space is the entire plane. Will it be better to construct the cone only along x-coordinates?

[videlec]

There is no need to build the cone at all at this stage. I changed the functions to residue_matrix (which produces the linear map from residue to linear combinations of edges) and add_residue_constraints (which produces a linear family). It now works for both Strebel graphs and veering triangulations.

[KaiFu2210]

Thanks so much! I'm looking at the codes.