def cone_triangulate(C, hyperplane=None):
r"""
Triangulation of rational cone contained in the positive quadrant.
EXAMPLES::
sage: from surface_dynamics.misc.linalg import cone_triangulate
sage: P = Polyhedron(rays=[(1,0,0),(0,1,0),(1,0,1),(0,1,1)])
sage: list(cone_triangulate(P)) # random
[[(0, 1, 1), (0, 1, 0), (1, 0, 0)], [(0, 1, 1), (1, 0, 1), (1, 0, 0)]]
sage: len(_)
2
sage: rays = [(0, 1, 0, -1, 0, 0),
....: (1, 0, -1, 0, 0, -1),
....: (0, 1, -1, 0, 0, -1),
....: (0, 0, 1, 0, 0, 0),
....: (0, 0, 0, 1, 0, -1),
....: (1, -1, 0, 0, 1, -1),
....: (0, 0, 0, 0, 1, -1),
....: (0, 0, 1, -1, 1, 0),
....: (0, 0, 1, -1, 0, 0),
....: (0, 0, 1, 0, -1, 0),
....: (0, 0, 0, 1, -1, -1),
....: (1, -1, 0, 0, 0, -1),
....: (0, 0, 0, 0, 0, -1)]
sage: P = Polyhedron(rays=rays)
sage: list(cone_triangulate(P, hyperplane=(1, 2, 3, -1, 0, -5))) # random
[[(0, 0, 0, 0, 0, -1),
(0, 0, 0, 0, 1, -1),
(0, 0, 0, 1, -1, -1),
(0, 0, 1, 0, 0, 0),
(0, 1, -1, 0, 0, -1),
(1, -1, 0, 0, 1, -1)],
...
(0, 0, 1, 0, 0, 0),
(0, 1, -1, 0, 0, -1),
(0, 1, 0, -1, 0, 0),
(1, -1, 0, 0, 1, -1),
(1, 0, -1, 0, 0, -1)]]
sage: len(_)
16
"""
rays = [r.vector() for r in C.rays()]
dim = len(rays[0])
if hyperplane is None:
hyperplane = [1] * dim
scalings = [sum(x * h for x, h in zip(r, hyperplane)) for r in rays]
assert all(s > 0 for s in scalings)
normalized_rays = [r / s for r, s in zip(rays, scalings)]
P = Polyhedron(vertices=normalized_rays)
for t in P.triangulate():
simplex = [P.Vrepresentation(i).vector() for i in t]
yield [(r / gcd(r)).change_ring(ZZ) for r in simplex]
