mappings#
Mappings between translation surfaces.
- class flatsurf.geometry.mappings.CanonicalizePolygonsMapping(s)[source]#
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.
- class flatsurf.geometry.mappings.IdentityMapping(domain, codomain)[source]#
Construct an identity map between two “equal” surfaces.
- class flatsurf.geometry.mappings.ReindexMapping(s, relabler, new_base_label=None)[source]#
Apply a dictionary to relabel the polygons.
- class flatsurf.geometry.mappings.SimilarityJoinPolygonsMapping(s, p1, e1)[source]#
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()[source]#
Return the vertices of the newly glued polygon which bound the diagonal formed by the glue.
- class flatsurf.geometry.mappings.SplitPolygonsMapping(s, p, v1, v2, new_label=None)[source]#
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)))
- class flatsurf.geometry.mappings.SurfaceMapping(domain, codomain)[source]#
Abstract class for any mapping between surfaces.
- class flatsurf.geometry.mappings.SurfaceMappingComposition(mapping1, mapping2)[source]#
Composition of two mappings between surfaces.
- factors()[source]#
Return the two factors of this surface mapping as a pair (f,g), where the original map is f o g.
- flatsurf.geometry.mappings.canonical_first_vertex(polygon)[source]#
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)[source]#
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.half_dilation_surface import GL2RMapping sage: from flatsurf.geometry.mappings import canonicalize_translation_surface_mapping sage: mat=Matrix([[1,2+a],[0,1]]) sage: from flatsurf.geometry.half_dilation_surface 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)[source]#
Returns a mapping to a Delaunay decomposition or possibly None if the surface already is Delaunay.
- flatsurf.geometry.mappings.delaunay_triangulation_mapping(s)[source]#
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)[source]#
Return a mapping whose domain is s which flips the provided edge.
- flatsurf.geometry.mappings.one_delaunay_flip_mapping(s)[source]#
Returns one delaunay flip, or none if no flips are needed.
- flatsurf.geometry.mappings.polygon_compare(poly1, poly2)[source]#
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)[source]#
Return a SurfaceMapping which cuts one polygon along a diagonal or None if the surface is triangulated.
- flatsurf.geometry.mappings.triangulation_mapping(s)[source]#
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)]))