polygons

The category of polyogons

This module provides shared functionality for all polygons in sage-flatsurf.

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 of a polygon is automatically determined.

EXAMPLES:

sage: from flatsurf.geometry.categories import Polygons
sage: C = Polygons(QQ)

sage: from flatsurf import polygons
sage: polygons.square() in C
True

class flatsurf.geometry.categories.polygons.Polygons(base, name=None)

The category of polygons defined over a base ring.

This comprises arbitrary base ring, e.g., this category contains Euclidean polygons and hyperbolic polygons.

EXAMPLES:

sage: from flatsurf.geometry.categories import Polygons
sage: Polygons(QQ)
Category of polygons over Rational Field

class Convex(base_category)

The axiom satisfied by convex polygons.

EXAMPLES:

sage: from flatsurf.geometry.categories import Polygons
sage: C = Polygons(QQ)
sage: C.Convex()
Category of convex polygons over Rational Field

class ParentMethods

Provides methods available to all convex polygons.

If you want to add functionality to all such polygons, you probably want to put it here.

is_convex(strict=False)

Return whether this is a convex polygon, which it is.

EXAMPLES:

sage: from flatsurf import polygons
sage: P = polygons.square()
sage: P.is_convex()
True

class ParentMethods

Provides methods available to all polygons.

If you want to add functionality to all polygons, independent of implementation, you probably want to put it here.

base_ring()

Return the ring over which this polygon is defined.

EXAMPLES:

sage: from flatsurf import polygons
sage: S = polygons.square()
sage: S.base_ring()
Rational Field

change_ring(ring)

Return a copy of this polygon which is defined over ring.

EXAMPLES:

sage: from flatsurf import polygons
sage: S = polygons.square()
sage: K.<sqrt2> = NumberField(x^2 - 2, embedding=AA(2)**(1/2))
sage: S.change_ring(K)
Polygon(vertices=[(0, 0), (1, 0), (1, 1), (0, 1)])

describe_polygon()

Return a textual description of this polygon for generating human-readable messages.

The description is returned as a triple (indeterminate article, singular, plural).

EXAMPLES:

sage: from flatsurf import polygons
sage: s = polygons.square()
sage: s.describe_polygon()
('a', 'square', 'squares')

is_convex(strict=False)

Return whether this is a convex polygon.

INPUT:

• strict – whether to check for strict convexity, i.e., a polygon with a π angle is not considered convex.

EXAMPLES:

sage: from flatsurf import polygons
sage: S = polygons.square()
sage: S.is_convex()
True
sage: S.is_convex(strict=True)
True

is_degenerate()

Return whether this polygon is considered degenerate.

EXAMPLES:

Polygons with zero area are considered degenerate:

sage: from flatsurf import Polygon
sage: p = Polygon(vertices=[(0, 0), (2, 0), (1, 0)], check=False)
sage: p.is_degenerate()
True

Polygons with marked vertices are considered degenerate:

sage: from flatsurf import Polygon
sage: p = Polygon(vertices=[(0, 0), (2, 0), (4, 0), (2, 2)])
sage: p.is_degenerate()
True

is_simple()

Return whether this polygon is not self-intersecting.

EXAMPLES:

sage: from flatsurf import polygons
sage: s = polygons.square()
sage: s.is_simple()
True

class Rational(base_category)

The axiom satisfied by polygons whose inner angles are rational multiples of π.

EXAMPLES:

sage: from flatsurf.geometry.categories import Polygons
sage: C = Polygons(QQ)
sage: C.Rational()
Category of rational polygons over Rational Field

class ParentMethods

Provides methods available to all rational polygons.

If you want to add functionality to all such polygons, you probably want to put it here.

is_rational()

Return whether all inner angles of this polygon are rational multiples of π, i.e., return True.

EXAMPLES:

sage: from flatsurf import polygons
sage: s = polygons.square()
sage: s.is_rational()
True

class Simple(base_category)

The axiom satisfied by polygons that are not self-intersecting.

EXAMPLES:

sage: from flatsurf.geometry.categories import Polygons
sage: C = Polygons(QQ)
sage: C.Simple()
Category of simple polygons over Rational Field

class ParentMethods

Provides methods available to all simple polygons.

If you want to add functionality to all such polygons, you probably want to put it here.

is_simple()

Return whether this polygon is not self-intersecting, i.e., return True.

EXAMPLES:

sage: from flatsurf import polygons
sage: s = polygons.square()
sage: s.is_simple()
True

class SubcategoryMethods

Convex()

Return the subcategory of convex polygons.

EXAMPLES:

sage: from flatsurf.geometry.categories import Polygons
sage: Polygons(QQ).Convex()
Category of convex polygons over Rational Field

Rational()

Return the subcategory of polygons with rational angles.

EXAMPLES:

sage: from flatsurf.geometry.categories import Polygons
sage: Polygons(QQ).Rational()
Category of rational polygons over Rational Field

Simple()

Return the subcategyr of simple polygons.

EXAMPLES:

sage: from flatsurf.geometry.categories import Polygons
sage: Polygons(QQ).Simple()
Category of simple polygons over Rational Field

base_ring()

Return the ring over which the polygons in this category are defined.

sage: from flatsurf.geometry.categories import Polygons sage: C = Polygons(QQ).Rational().Simple().Convex()

sage: C.base_ring() Rational Field

change_ring(ring)

Return this category but defined over the ring ring.

EXAMPLES:

sage: from flatsurf import polygons
sage: s = polygons.square()
sage: C = s.category()
sage: C
Category of convex simple euclidean rectangles over Rational Field
sage: C.change_ring(AA)
Category of convex simple euclidean rectangles over Algebraic Real Field

field()

Return the field over which these polygons are defined.

EXAMPLES:

sage: from flatsurf import Polygon
sage: P = Polygon(vertices=[(0,0),(1,0),(2,1),(-1,1)])
sage: P.category().field()
doctest:warning
...
UserWarning: field() has been deprecated and will be removed from a future version of sage-flatsurf; use base_ring() or base_ring().fraction_field() instead
Rational Field

super_categories()

Return the categories polygons automatically belong to.

EXAMPLES:

sage: from flatsurf.geometry.categories import Polygons
sage: C = Polygons(QQ)
sage: C.super_categories()
[Category of sets]