homology of saddle connection #308
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@saraedum For the interface I implemented it as Does that sound reasonable? |
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Documentation preview for this PR is ready! 🎉 |
flatsurf/geometry/surface_objects.py
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| if len(segments) == 1 and segments[0].is_edge(): | ||
| label = segments[0].polygon_label() | ||
| e = segments[0].edge() | ||
| return H((label, e)) |
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Is this necessary or just an optimization?
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It is necessary : the segment has a start in the polygon on the left side of the edge and the end on its right side. The generic code afterwards does not work.
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Here is an example
sage: S = translation_surfaces.mcmullen_L(1,1,1,1)
sage: traj = S.tangent_vector(0, (0, 0), (1, 0)).straight_line_trajectory()
sage: traj.flow(1)
sage: s = traj.segments()[0]
sage: s
Segment in polygon 0 starting at (0, 0) and ending at (1, 1)
Let me notice that the string representation of the segment is quite confusing : it would better have been ending at (1,0). The reason is that (0, 0) refers to polygon 0 while (1, 1) refers to polygon 1. This becomes apparent when looking at the start and end
sage: s.start()
SimilaritySurfaceTangentVector in polygon 0 based at (0, 0) with vector (1, 0)
sage: s.end()
SimilaritySurfaceTangentVector in polygon 1 based at (1, 1) with vector (-1, 0)
| h = H.zero() | ||
| for s in segments: | ||
| label = s.polygon_label() | ||
| n = len(surface.polygon(label).vertices()) | ||
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| start = s.start() | ||
| if start.position().is_in_edge_interior(): | ||
| # pick the next vertex clockwise | ||
| i = (start.position().get_edge() + 1) % n | ||
| else: | ||
| assert start.position().is_vertex() | ||
| i = start.vertex() | ||
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| end = s.end() | ||
| if end.position().is_in_edge_interior(): | ||
| # pick the previous vertex clockwise | ||
| j = end.position().get_edge() | ||
| else: | ||
| assert end.position().is_vertex() | ||
| j = end.vertex() | ||
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| if i < j: | ||
| h += sum(H((label, e)) for e in range(i, j)) | ||
| else: | ||
| h += sum(H((label, e)) for e in range(i, n)) | ||
| h += sum(H((label, e)) for e in range(j)) | ||
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| return h |
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Eventually, we could do the conversion via libflatsurf because there the saddle connections have a chain() naturally. (Especially with connections coming out of the flow decomposition code this won't work out anymore but let's do this in a 2.0 )
flatsurf/geometry/surface_objects.py
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| n = 10 | ||
| while not traj.is_saddle_connection(): | ||
| traj.flow(2**n) | ||
| n += 1 |
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| n = 10 | |
| while not traj.is_saddle_connection(): | |
| traj.flow(2**n) | |
| n += 1 | |
| traj.flow(oo) |
I think that also works actually.
| @@ -0,0 +1,3 @@ | |||
| **Added:** | |||
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| * homology of saddle connections | |||
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| * homology of saddle connections | |
| * Added homology of saddle connections |
(otherwise I have to modify it later when I do the release ;) )
Implement conversion of saddle connection as element of simplicial homology of the surface.
Checklist
doc/news/.