def ward(n):
r"""
Return the surface formed by gluing a regular 2n-gon to two regular n-gons.
These surfaces have Veech's lattice property due to work of Ward.
EXAMPLES::
sage: from flatsurf import *
sage: s=translation_surfaces.ward(3)
sage: TestSuite(s).run()
sage: s=translation_surfaces.ward(7)
sage: TestSuite(s).run()
"""
assert n>=3
from .polygon import polygons
from .surface import Surface_list
from .translation_surface import TranslationSurface
o = ZZ_2*polygons.regular_ngon(2*n)
p1 = polygons(*[o.edge((2*i+n)%(2*n)) for i in xrange(n)])
p2 = polygons(*[o.edge((2*i+n+1)%(2*n)) for i in xrange(n)])
s = Surface_list(base_ring=o.parent().field())
s.add_polygon(o)
s.add_polygon(p1)
s.add_polygon(p2)
s.change_polygon_gluings(1, [(0,2*i) for i in xrange(n)])
s.change_polygon_gluings(2, [(0,2*i+1) for i in xrange(n)])
s.make_immutable()
return TranslationSurface(s)
def veech_2n_gon(n):
r"""
The regular 2n-gon with opposite sides identified.
EXAMPLES::
sage: from flatsurf import *
sage: s=translation_surfaces.veech_2n_gon(5)
sage: s.polygon(0)
Polygon: (0, 0), (1, 0), (-1/2*a^2 + 5/2, 1/2*a), (-a^2 + 7/2, -1/2*a^3 + 2*a), (-1/2*a^2 + 5/2, -a^3 + 7/2*a), (1, -a^3 + 4*a), (0, -a^3 + 4*a), (1/2*a^2 - 3/2, -a^3 + 7/2*a), (a^2 - 5/2, -1/2*a^3 + 2*a), (1/2*a^2 - 3/2, 1/2*a)
sage: TestSuite(s).run()
"""
from .polygon import polygons
from .surface import Surface_list
from .translation_surface import TranslationSurface
p = polygons.regular_ngon(2*n)
s = Surface_list(base_ring=p.base_ring())
s.add_polygon(p,[ ( 0, (i+n)%(2*n) ) for i in xrange(2*n)] )
s.make_immutable()
return TranslationSurface(s)
def veech_double_n_gon(n):
r"""
A pair of regular n-gons with each edge of one identified to an edge of the other to make a translation surface.
EXAMPLES::
sage: from flatsurf import *
sage: s=translation_surfaces.veech_double_n_gon(5)
sage: TestSuite(s).run()
"""
from .polygon import polygons
from .surface import Surface_list
from .translation_surface import TranslationSurface
from sage.matrix.constructor import Matrix
p = polygons.regular_ngon(n)
s = Surface_list(base_ring=p.base_ring())
m = Matrix([[-1,0],[0,-1]])
s.add_polygon(p) # label=0
s.add_polygon(m*p, [(0,i) for i in xrange(n)])
s.make_immutable()
return TranslationSurface(s)
