def RandomTriangulation(n, embed=False, base_ring=QQ):
"""
Returns a random triangulation on n vertices.
A triangulation is a planar graph all of whose faces are
triangles (3-cycles).
The graph is built by independently generating `n` points
uniformly at random on the surface of a sphere, finding the
convex hull of those points, and then returning the 1-skeleton
of that polyhedron.
INPUT:
- ``n`` -- number of vertices (recommend `n \ge 3`)
- ``embed`` -- (optional, default ``False``) whether to use the
stereographic point projections to draw the graph.
- ``base_ring`` -- (optional, default ``QQ``) specifies the field over
which to do the intermediate computations. The default setting is slower,
but works for any input; one can instead use ``RDF``, but this occasionally
fails due to loss of precision, as mentioned on :trac:10276.
EXAMPLES::
sage: g = graphs.RandomTriangulation(10)
sage: g.is_planar()
True
sage: g.num_edges() == 3*g.order() - 6
True
TESTS::
sage: for i in range(10):
....: g = graphs.RandomTriangulation(30,embed=True)
....: assert g.is_planar() and g.size() == 3*g.order()-6
"""
from sage.misc.prandom import normalvariate
from sage.geometry.polyhedron.constructor import Polyhedron
from sage.rings.real_double import RDF
# this function creates a random unit vector in R^3
def rand_unit_vec():
vec = [normalvariate(0, 1) for k in range(3)]
mag = sum([x * x for x in vec])**0.5
return [x / mag for x in vec]
# generate n unit vectors at random
points = [rand_unit_vec() for k in range(n)]
# find their convex hull
P = Polyhedron(vertices=points, base_ring=base_ring)
# extract the 1-skeleton
g = P.vertex_graph()
g.rename('Planar triangulation on {} vertices'.format(n))
if embed:
from sage.geometry.polyhedron.plot import ProjectionFuncStereographic
from sage.modules.free_module_element import vector
proj = ProjectionFuncStereographic([0, 0, 1])
ppoints = [proj(vector(x)) for x in points]
g.set_pos({i: ppoints[i] for i in range(len(points))})
return g
def RandomTriangulation(n, embed=False, base_ring=QQ):
"""
Returns a random triangulation on n vertices.
A triangulation is a planar graph all of whose faces are
triangles (3-cycles).
The graph is built by independently generating `n` points
uniformly at random on the surface of a sphere, finding the
convex hull of those points, and then returning the 1-skeleton
of that polyhedron.
INPUT:
- ``n`` -- number of vertices (recommend `n \ge 3`)
- ``embed`` -- (optional, default ``False``) whether to use the
stereographic point projections to draw the graph.
- ``base_ring`` -- (optional, default ``QQ``) specifies the field over
which to do the intermediate computations. The default setting is slower,
but works for any input; one can instead use ``RDF``, but this occasionally
fails due to loss of precision, as mentioned on :trac:10276.
EXAMPLES::
sage: g = graphs.RandomTriangulation(10)
sage: g.is_planar()
True
sage: g.num_edges() == 3*g.order() - 6
True
TESTS::
sage: for i in range(10):
....: g = graphs.RandomTriangulation(30,embed=True)
....: assert g.is_planar() and g.size() == 3*g.order()-6
"""
from sage.misc.prandom import normalvariate
from sage.geometry.polyhedron.constructor import Polyhedron
from sage.rings.real_double import RDF
# this function creates a random unit vector in R^3
def rand_unit_vec():
vec = [normalvariate(0, 1) for k in range(3)]
mag = sum([x * x for x in vec]) ** 0.5
return [x / mag for x in vec]
# generate n unit vectors at random
points = [rand_unit_vec() for k in range(n)]
# find their convex hull
P = Polyhedron(vertices=points, base_ring=base_ring)
# extract the 1-skeleton
g = P.vertex_graph()
g.rename('Planar triangulation on {} vertices'.format(n))
if embed:
from sage.geometry.polyhedron.plot import ProjectionFuncStereographic
from sage.modules.free_module_element import vector
proj = ProjectionFuncStereographic([0, 0, 1])
ppoints = [proj(vector(x)) for x in points]
g.set_pos({i: ppoints[i] for i in range(len(points))})
return g
