"""
The Schur cone, as described in the "Critical angles..." papers by
Seeger and Sossa.
"""

from sage.all import *

def is_pointed(P,Q):
    newP = Cone([ list(p) for p in P ])
    newQ = Cone([ list(q) for q in Q ])
    return newP.intersection(-newQ).is_trivial()

def ext_angles(P,Q):
    angles = []
    for p in P:
        for q in Q:
            p = vector(SR, p)
            q = vector(SR, q)
            p = p/p.norm()
            q = q/q.norm()
            angles.append(arccos(p.inner_product(q)))

    return angles

def schur(n):
    r"""
    Return the Schur cone in ``n`` dimensions.

    INPUT:

    - ``n`` -- the ambient dimension of the Schur cone and its ambient space.

    OUTPUT:

    A rational closed convex Schur cone of dimension ``n``.
    """

    hs = []
    for i in range(1,n):
        h_i = [0]*n
        h_i[i-1] = QQ(1)
        h_i[i] = -QQ(1)
        hs.append(vector(QQ,n,h_i))

    return Cone(hs)