dilation_surfaces

The category of dilation surfaces.

This module provides shared functionality for all surfaces in sage-flatsurf that are built from Euclidean polygons that are glued by translation followed by homothety, i.e., application of a diagonal matrix.

See flatsurf.geometry.categories for a general description of the category framework in sage-flatsurf.

Normally, you won’t create this (or any other) category directly. The correct category is automatically determined for immutable surfaces.

EXAMPLES:

sage: from flatsurf import Polygon, similarity_surfaces
sage: P = Polygon(vertices=[(0,0), (2,0), (1,4), (0,5)])
sage: S = similarity_surfaces.self_glued_polygon(P)
sage: C = S.category()

sage: from flatsurf.geometry.categories import DilationSurfaces
sage: C.is_subcategory(DilationSurfaces())
True

class flatsurf.geometry.categories.dilation_surfaces.DilationSurfaces

The category of surfaces built from polygons with edges identified by translations and homothety.

EXAMPLES:

sage: from flatsurf.geometry.categories import DilationSurfaces
sage: DilationSurfaces()
Category of dilation surfaces

class FiniteType(base_category)

The category of dilation surfaces built from a finite number of polygons.

EXAMPLES:

sage: from flatsurf import Polygon, similarity_surfaces
sage: P = Polygon(vertices=[(0,0), (2,0), (1,4), (0,5)])
sage: S = similarity_surfaces.self_glued_polygon(P)

sage: from flatsurf.geometry.categories import DilationSurfaces
sage: S in DilationSurfaces().FiniteType()
True

class ParentMethods

Provides methods available to all dilation surfaces built from finitely many polygons.

If you want to add functionality for such surfaces you most likely want to put it here.

l_infinity_delaunay_triangulation(triangulated=None, in_place=None, limit=None, direction=None)

Return an L-infinity Delaunay triangulation of a surface, or make some triangle flips to get closer to the Delaunay decomposition.

INPUT:

• triangulated – deprecated and ignored.

• in_place – deprecated and must be None (the default); otherwise an error is produced

• limit – optional (positive integer) If provided, then at most limit

  many diagonal flips will be done.

• direction – optional (vector). Used to determine labels when a pair of triangles is flipped. Each triangle has a unique separatrix which points in the provided direction or its negation. As such a vector determines a sign for each triangle. A pair of adjacent triangles have opposite signs. Labels are chosen so that this sign is preserved (as a function of labels).

EXAMPLES:

sage: from flatsurf import translation_surfaces
sage: s0 = translation_surfaces.veech_double_n_gon(5)
sage: field = s0.base_ring()
sage: a = field.gen()
sage: m = matrix(field, 2, [2,a,1,1])
sage: for _ in range(5): assert s0.triangulate().codomain().random_flip(5).l_infinity_delaunay_triangulation().is_veering_triangulated()

sage: s = m*s0
sage: s = s.l_infinity_delaunay_triangulation()
sage: assert s.is_veering_triangulated()
sage: TestSuite(s).run()

sage: s = (m**2)*s0
sage: s = s.l_infinity_delaunay_triangulation()
sage: assert s.is_veering_triangulated()
sage: TestSuite(s).run()

sage: s = (m**3)*s0
sage: s = s.l_infinity_delaunay_triangulation()
sage: assert s.is_veering_triangulated()
sage: TestSuite(s).run()

The octagon which has horizontal and vertical edges:

sage: t0 = translation_surfaces.regular_octagon()
sage: for _ in range(5): assert t0.triangulate().codomain().random_flip(5).l_infinity_delaunay_triangulation().is_veering_triangulated()
sage: r = matrix(t0.base_ring(), [
....:    [ sqrt(2)/2, -sqrt(2)/2 ],
....:    [ sqrt(2)/2, sqrt(2)/2 ]])
sage: p = matrix(t0.base_ring(), [
....:    [ 1, 2+2*sqrt(2) ],
....:    [ 0, 1 ]])

sage: t = (r * p * r * t0).l_infinity_delaunay_triangulation()
sage: systole_count = 0
sage: for l, e in t.edges():
....:     v = t.polygon(l).edge(e)
....:     systole_count += v[0]**2 + v[1]**2 == 1
sage: assert t.is_veering_triangulated() and systole_count == 8, (systole_count, {lab: t.polygon(lab) for lab in t.labels()}, t.gluings())

sage: t = (r**4 * p * r**5 * p**2 * r * t0).l_infinity_delaunay_triangulation()
sage: systole_count = 0
sage: for l, e in t.edges():
....:     v = t.polygon(l).edge(e)
....:     systole_count += v[0]**2 + v[1]**2 == 1
sage: assert t.is_veering_triangulated() and systole_count == 8, (systole_count, {lab: t.polygon(lab) for lab in t.labels()}, t.gluings())

veering_triangulation(l_infinity=False, limit=None, direction=None)

Return a veering triangulated surface, or make some triangle flips to get closer to the Delaunay decomposition.

INPUT:

• l_infinity – optional (boolean, default False). Whether to return a L^oo-Delaunay triangulation or any veering triangulation.

• limit – optional (positive integer) If provided, then at most limit

  many diagonal flips will be done.

• direction – optional (vector). Used to determine labels when a pair of triangles is flipped. Each triangle has a unique separatrix which points in the provided direction or its negation. As such a vector determines a sign for each triangle. A pair of adjacent triangles have opposite signs. Labels are chosen so that this sign is preserved (as a function of labels).

EXAMPLES:

sage: from flatsurf import translation_surfaces
sage: s0 = translation_surfaces.veech_double_n_gon(5)
sage: field = s0.base_ring()
sage: a = field.gen()
sage: m = matrix(field, 2, [2,a,1,1])
sage: for _ in range(5): assert s0.triangulate().codomain().random_flip(5).veering_triangulation().is_veering_triangulated()

sage: s = m*s0
sage: s = s.veering_triangulation()
sage: assert s.is_veering_triangulated()
sage: TestSuite(s).run()

sage: s = (m**2)*s0
sage: s = s.veering_triangulation()
sage: assert s.is_veering_triangulated()
sage: TestSuite(s).run()

sage: s = (m**3)*s0
sage: s = s.veering_triangulation()
sage: assert s.is_veering_triangulated()
sage: TestSuite(s).run()

The octagon which has horizontal and vertical edges:

sage: t0 = translation_surfaces.regular_octagon().triangulate().codomain()
sage: t0.is_veering_triangulated()
False
sage: t0.veering_triangulation().is_veering_triangulated()
True
sage: for _ in range(5): assert t0.random_flip(5).veering_triangulation().is_veering_triangulated()

sage: r = matrix(t0.base_ring(), [
....:    [ sqrt(2)/2, -sqrt(2)/2 ],
....:    [ sqrt(2)/2, sqrt(2)/2 ]])
sage: p = matrix(t0.base_ring(), [
....:    [ 1, 2+2*sqrt(2) ],
....:    [ 0, 1 ]])
sage: t = (r * p * r * t0).veering_triangulation()
sage: assert t.is_veering_triangulated()
sage: t = (r**4 * p * r**5 * p**2 * r * t0).veering_triangulation()
sage: assert t.is_veering_triangulated()

class ParentMethods

Provides methods available to all dilation surfaces.

If you want to add functionality for such surfaces you most likely want to put it here.

affine_automorphism_group()

Return the group of affine automorphisms of this surface, i.e., the group of homeomorphisms that can be locally expressed as affine transformations.

EXAMPLES:

sage: from flatsurf import dilation_surfaces
sage: S = dilation_surfaces.genus_two_square(1/2, 1/3, 1/4, 1/5)
sage: A = S.affine_automorphism_group(); A
AffineAutomorphismGroup(Genus 2 Positive Dilation Surface built from 2 right triangles and a hexagon)

is_dilation_surface(positive=False)

Return whether this surface is a dilation surface.

See similarity_surfaces.SimilaritySurfaces.ParentMethods.is_dilation_surface() for details.

EXAMPLES:

sage: from flatsurf import translation_surfaces
sage: S = translation_surfaces.infinite_staircase()

sage: S.is_dilation_surface(positive=True)
True
sage: S.is_dilation_surface(positive=False)
True

is_veering_triangulated(certificate=False)

Return whether this dilation surface is given by a veering triangulation.

A triangulation is veering if none of its triangles are made of three edges of the same slope (positive or negative).

EXAMPLES:

sage: from flatsurf import MutableOrientedSimilaritySurface, Polygon

sage: s = MutableOrientedSimilaritySurface(QQ)
sage: s.add_polygon(Polygon(vertices=[(0,0), (1,1), (-1,2)]))
0
sage: s.add_polygon(Polygon(vertices=[(0,0), (-1,2), (-2,1)]))
1
sage: s.glue((0, 0), (1, 1))
sage: s.glue((0, 1), (1, 2))
sage: s.glue((0, 2), (1, 0))
sage: s.set_immutable()
sage: s.is_veering_triangulated()
True

sage: m = matrix(ZZ, 2, 2, [5, 3, 3, 2])
sage: (m * s).is_veering_triangulated()
False

veech_group()

Return the Veech group of this surface, i.e., the group of matrices that fix the vertices of this surface.

EXAMPLES:

sage: from flatsurf import dilation_surfaces
sage: S = dilation_surfaces.genus_two_square(1/2, 1/3, 1/4, 1/5)
sage: V = S.veech_group(); V
VeechGroup(Genus 2 Positive Dilation Surface built from 2 right triangles and a hexagon)

class Positive(base_category)

The axiom satisfied by dilation surfaces that use homothety with positive scaling.

EXAMPLES:

sage: from flatsurf import translation_surfaces
sage: S = translation_surfaces.square_torus()
sage: 'Positive' in S.category().axioms()
True

class ParentMethods

Provides methods available to all positive dilation surfaces.

If you want to add functionality for such surfaces you most likely want to put it here.

is_dilation_surface(positive=False)

Return whether this surface is a dilation surface.

See similarity_surfaces.SimilaritySurfaces.ParentMethods.is_dilation_surface() for details.

class SubcategoryMethods

Positive()

Return the subcategory of surfaces glued by positive dilation.

EXAMPLES:

sage: from flatsurf.geometry.categories import DilationSurfaces
sage: C = DilationSurfaces()
sage: C.Positive()
Category of positive dilation surfaces

super_categories()

Return the categories that a dilation surfaces is also always a member of.

EXAMPLES:

sage: from flatsurf.geometry.categories import DilationSurfaces
sage: DilationSurfaces().super_categories()
[Category of rational similarity surfaces]