mappings

Mappings between translation surfaces.

class flatsurf.geometry.mappings.CanonicalizePolygonsMapping(s)

This is a mapping to a surface with the polygon vertices canonically determined. A canonical labeling is when the canonocal_first_vertex is the zero vertex.

pull_vector_back(tangent_vector)

Applies the pullback mapping to the provided vector.

push_vector_forward(tangent_vector)

Applies the mapping to the provided vector.

class flatsurf.geometry.mappings.GL2RMapping(s, m, category=None)

This class pushes a surface forward under a matrix.

Note that for matrices of negative determinant we need to relabel edges (because edges must have a counterclockwise cyclic order). For each n-gon in the surface, we relabel edges according to the involution \(e \mapsto n-1-e\).

EXAMPLE:

sage: from flatsurf import translation_surfaces
sage: s=translation_surfaces.veech_2n_gon(4)
sage: from flatsurf.geometry.mappings import GL2RMapping
sage: mat=Matrix([[2,1],[1,1]])
sage: m=GL2RMapping(s,mat)
sage: TestSuite(m.codomain()).run()

pull_vector_back(tangent_vector)

Applies the inverse of the mapping to the provided vector.

push_vector_forward(tangent_vector)

Applies the mapping to the provided vector.

class flatsurf.geometry.mappings.IdentityMapping(domain, codomain)

Construct an identity map between two “equal” surfaces.

pull_vector_back(tangent_vector)

Applies the pullback mapping to the provided vector.

push_vector_forward(tangent_vector)

Applies the mapping to the provided vector.

class flatsurf.geometry.mappings.ReindexMapping(s, relabler, new_base_label=None)

Apply a dictionary to relabel the polygons.

pull_vector_back(tangent_vector)

Applies the pullback mapping to the provided vector.

push_vector_forward(tangent_vector)

Applies the mapping to the provided vector.

class flatsurf.geometry.mappings.SimilarityJoinPolygonsMapping(s, p1, e1)

Return a SurfaceMapping joining two polygons together along the edge provided to the constructor.

EXAMPLES:

sage: from flatsurf import MutableOrientedSimilaritySurface, Polygon
sage: s = MutableOrientedSimilaritySurface(QQ)
sage: s.add_polygon(Polygon(edges=[(1,0),(0,1),(-1,-1)]))
0
sage: s.add_polygon(Polygon(edges=[(-1,0),(0,-1),(1,1)]))
1
sage: s.glue((0, 0), (1, 0))
sage: s.glue((0, 1), (1, 1))
sage: s.glue((0, 2), (1, 2))
sage: s.set_immutable()

sage: from flatsurf.geometry.mappings import SimilarityJoinPolygonsMapping
sage: m=SimilarityJoinPolygonsMapping(s, 0, 2)
sage: s2=m.codomain()
sage: s2.labels()
(0,)
sage: s2.polygons()
(Polygon(vertices=[(0, 0), (1, 0), (1, 1), (0, 1)]),)
sage: s2.gluings()
(((0, 0), (0, 2)), ((0, 1), (0, 3)), ((0, 2), (0, 0)), ((0, 3), (0, 1)))

glued_vertices()

Return the vertices of the newly glued polygon which bound the diagonal formed by the glue.

pull_vector_back(tangent_vector)

Applies the inverse of the mapping to the provided vector.

push_vector_forward(tangent_vector)

Applies the mapping to the provided vector.

removed_label()

Return the label that was removed in the joining process.

class flatsurf.geometry.mappings.SplitPolygonsMapping(s, p, v1, v2, new_label=None)

Class for cutting a polygon along a diagonal.

EXAMPLES:

sage: from flatsurf import translation_surfaces
sage: s=translation_surfaces.veech_2n_gon(4)
sage: from flatsurf.geometry.mappings import SplitPolygonsMapping
sage: m = SplitPolygonsMapping(s,0,0,2)
sage: s2=m.codomain()
sage: TestSuite(s2).run()
sage: s2.labels()
(0, 1)
sage: s2.polygons()
(Polygon(vertices=[(0, 0), (1/2*a + 1, 1/2*a), (1/2*a + 1, 1/2*a + 1), (1, a + 1), (0, a + 1), (-1/2*a, 1/2*a + 1), (-1/2*a, 1/2*a)]), Polygon(vertices=[(0, 0), (-1/2*a - 1, -1/2*a), (-1/2*a, -1/2*a)]))
sage: s2.gluings()
(((0, 0), (1, 0)), ((0, 1), (0, 5)), ((0, 2), (0, 6)), ((0, 3), (1, 1)), ((0, 4), (1, 2)), ((0, 5), (0, 1)), ((0, 6), (0, 2)), ((1, 0), (0, 0)), ((1, 1), (0, 3)), ((1, 2), (0, 4)))

pull_vector_back(tangent_vector)

Applies the pullback mapping to the provided vector.

push_vector_forward(tangent_vector)

Applies the mapping to the provided vector.

class flatsurf.geometry.mappings.SurfaceMapping(domain, codomain)

Abstract class for any mapping between surfaces.

codomain()

Return the range of the mapping.

domain()

Return the domain of the mapping.

pull_vector_back(tangent_vector)

Applies the inverse of the mapping to the provided vector.

push_vector_forward(tangent_vector)

Applies the mapping to the provided vector.

class flatsurf.geometry.mappings.SurfaceMappingComposition(mapping1, mapping2)

Composition of two mappings between surfaces.

factors()

Return the two factors of this surface mapping as a pair (f,g), where the original map is f o g.

pull_vector_back(tangent_vector)

Applies the inverse of the mapping to the provided vector.

push_vector_forward(tangent_vector)

Applies the mapping to the provided vector.

flatsurf.geometry.mappings.canonical_first_vertex(polygon)

Return the index of the vertex with smallest y-coordinate. If two vertices have the same y-coordinate, then the one with least x-coordinate is returned.

flatsurf.geometry.mappings.canonicalize_translation_surface_mapping(s)

Return the translation surface in a canonical form.

EXAMPLES:

sage: from flatsurf import translation_surfaces
sage: s = translation_surfaces.octagon_and_squares().canonicalize()

sage: TestSuite(s).run()

sage: a = s.base_ring().gen()  # a is the square root of 2.

sage: from flatsurf.geometry.mappings import GL2RMapping
sage: from flatsurf.geometry.mappings import canonicalize_translation_surface_mapping
sage: mat=Matrix([[1,2+a],[0,1]])
sage: from flatsurf.geometry.mappings import GL2RMapping
sage: m1=GL2RMapping(s, mat)
sage: m2=canonicalize_translation_surface_mapping(m1.codomain())
sage: m=m2*m1
sage: m.domain().cmp(m.codomain())
0
sage: TestSuite(m.codomain()).run()
sage: s=m.domain()
sage: v=s.tangent_vector(0,(0,0),(1,1))
sage: w=m.push_vector_forward(v)
sage: print(w)
SimilaritySurfaceTangentVector in polygon 0 based at (0, 0) with vector (a + 3, 1)

flatsurf.geometry.mappings.delaunay_decomposition_mapping(s)

Returns a mapping to a Delaunay decomposition or possibly None if the surface already is Delaunay.

flatsurf.geometry.mappings.delaunay_triangulation_mapping(s)

Returns a mapping to a Delaunay triangulation or None if the surface already is Delaunay triangulated.

flatsurf.geometry.mappings.flip_edge_mapping(s, p1, e1)

Return a mapping whose domain is s which flips the provided edge.

flatsurf.geometry.mappings.my_sgn(val)

flatsurf.geometry.mappings.one_delaunay_flip_mapping(s)

Returns one delaunay flip, or none if no flips are needed.

flatsurf.geometry.mappings.polygon_compare(poly1, poly2)

Compare two polygons first by area, then by number of sides, then by lexicographical ordering on edge vectors.

flatsurf.geometry.mappings.subdivide_a_polygon(s)

Return a SurfaceMapping which cuts one polygon along a diagonal or None if the surface is triangulated.

flatsurf.geometry.mappings.triangulation_mapping(s)

Return a SurfaceMapping triangulating s.

EXAMPLES:

sage: from flatsurf import translation_surfaces
sage: s=translation_surfaces.veech_2n_gon(4)
sage: from flatsurf.geometry.mappings import triangulation_mapping
sage: m=triangulation_mapping(s)
sage: s2=m.codomain()
sage: TestSuite(s2).run()
sage: s2.polygons()
(Polygon(vertices=[(0, 0), (-1/2*a, 1/2*a + 1), (-1/2*a, 1/2*a)]),
 Polygon(vertices=[(0, 0), (1/2*a, -1/2*a - 1), (1/2*a, 1/2*a)]),
 Polygon(vertices=[(0, 0), (-1/2*a - 1, -1/2*a - 1), (0, -1)]),
 Polygon(vertices=[(0, 0), (-1, -a - 1), (1/2*a, -1/2*a)]),
 Polygon(vertices=[(0, 0), (0, -a - 1), (1, 0)]),
 Polygon(vertices=[(0, 0), (-1/2*a - 1, -1/2*a), (-1/2*a, -1/2*a)]))