cone_surfaces

The category of cone surfaces.

A cone surface is a surface that can be built by gluing Euclidean polygons along their edges such that the matrix describing monodromy along a closed path is an isometry; that matrix is given by multiplying the individual matrices that describe how to transition between pairs of glued edges, see edge_matrix().

In sage-flatsurf, we restrict cone surfaces slightly by requiring that a cone surface is given by polygons such that each edge matrix is an isometry.

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:

We glue the sides of a square with a rotation of π/2. Since each gluing is just a rotation, this is a cone surface:

sage: from flatsurf import Polygon, MutableOrientedSimilaritySurface
sage: P = Polygon(vertices=[(0,0), (1,0), (1,1), (0,1)])
sage: S = MutableOrientedSimilaritySurface(QQ)
sage: S.add_polygon(P, label=0)
0
sage: S.glue((0, 0), (0, 1))
sage: S.glue((0, 2), (0, 3))
sage: S.set_immutable()

sage: C = S.category()

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

class flatsurf.geometry.categories.cone_surfaces.ConeSurfaces

The category of surfaces built by gluing (Euclidean) polygons with isometries on the edges.

See cone_surfaces and is_cone_surface() on how this differs slightly from the customary definition of a cone surface.

EXAMPLES:

sage: from flatsurf.geometry.categories import ConeSurfaces
sage: ConeSurfaces()
Category of cone surfaces

class FiniteType(base_category)

The category of cone surfaces built from finitely many polygons.

EXAMPLES:

sage: from flatsurf import Polygon, similarity_surfaces
sage: P = Polygon(edges=[(2, 0),(-1, 3),(-1, -3)])
sage: S = similarity_surfaces.self_glued_polygon(P)

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

class Oriented(base_category)

The category of oriented cone surfaces, i.e., orientable cone surfaces whose orientation can be chosen to be compatible with the embedding of its polygons in the real plane.

EXAMPLES:

sage: from flatsurf import Polygon, similarity_surfaces
sage: P = Polygon(edges=[(2, 0),(-1, 3),(-1, -3)])
sage: S = similarity_surfaces.self_glued_polygon(P)

sage: from flatsurf.geometry.categories import ConeSurfaces
sage: S in ConeSurfaces().Oriented()
True

class ParentMethods

Provides methods available to all oriented cone surfaces.

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

class WithoutBoundary(base_category)

The category of oriented cone surfaces without boundary.

EXAMPLES:

sage: from flatsurf import Polygon, similarity_surfaces
sage: P = Polygon(edges=[(2, 0),(-1, 3),(-1, -3)])
sage: S = similarity_surfaces.self_glued_polygon(P)

sage: from flatsurf.geometry.categories import ConeSurfaces
sage: S in ConeSurfaces().Oriented().WithoutBoundary()
True

class Connected(base_category)

The category of oriented connected cone surfaces without boundary.

EXAMPLES:

sage: from flatsurf import Polygon, similarity_surfaces
sage: P = Polygon(edges=[(2, 0),(-1, 3),(-1, -3)])
sage: S = similarity_surfaces.self_glued_polygon(P)

sage: from flatsurf.geometry.categories import ConeSurfaces
sage: S in ConeSurfaces().Oriented().Connected()
True

class ParentMethods

Provides methods available to all oriented connected cone surfaces without boundary.

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

class ParentMethods

Provides methods available to all oriented cone surfaces without boundary.

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

angles(numerical=False, return_adjacent_edges=False)

Return the set of angles around the vertices of the surface.

EXAMPLES:

sage: from flatsurf import polygons, similarity_surfaces
sage: T = polygons.triangle(3, 4, 5)
sage: S = similarity_surfaces.billiard(T)
sage: S.angles()
[1/3, 1/4, 5/12]
sage: S.angles(numerical=True)   # abs tol 1e-14
[0.333333333333333, 0.250000000000000, 0.416666666666667]

sage: S.angles(return_adjacent_edges=True)
[(1/3, [(0, 1), (1, 2)]), (1/4, [(0, 0), (1, 0)]), (5/12, [(1, 1), (0, 2)])]

class ParentMethods

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

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

area()

Return the area of this surface.

EXAMPLES:

sage: from flatsurf import Polygon, similarity_surfaces
sage: P = Polygon(edges=[(2, 0),(-1, 3),(-1, -3)])
sage: S = similarity_surfaces.self_glued_polygon(P)
sage: S.area()
3

class ParentMethods

Provides methods available to all cone surfaces.

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

is_cone_surface()

Return whether this surface is a cone surface, i.e., whether its edges are glued by isometries.

Note

This is a stronger requirement than the usual definition of a cone surface, see cone_surfaces for details.

EXAMPLES:

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

super_categories()

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

EXAMPLES:

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