- # Represent the cone in terms of a basis for W, i.e. with smaller
- # vectors.
- ws = [ W.coordinate_vector(w1) for (w1, _) in ray_pairs ]
-
- L = ToricLattice(W.dimension())
-
- return Cone(ws, lattice=L)
-
-
-
-def lineality(K):
- r"""
- Compute the lineality of this cone.
-
- The lineality of a cone is the dimension of the largest linear
- subspace contained in that cone.
-
- OUTPUT:
-
- A nonnegative integer; the dimension of the largest subspace
- contained within this cone.
-
- REFERENCES:
-
- .. [Rockafellar] R.T. Rockafellar. Convex Analysis. Princeton
- University Press, Princeton, 1970.
-
- EXAMPLES:
-
- The lineality of the nonnegative orthant is zero, since it clearly
- contains no lines::
-
- sage: K = Cone([(1,0,0), (0,1,0), (0,0,1)])
- sage: lineality(K)
- 0
-
- However, if we add another ray so that the entire `x`-axis belongs
- to the cone, then the resulting cone will have lineality one::
-
- sage: K = Cone([(1,0,0), (-1,0,0), (0,1,0), (0,0,1)])
- sage: lineality(K)
- 1
-
- If our cone is all of `\mathbb{R}^{2}`, then its lineality is equal
- to the dimension of the ambient space (i.e. two)::
-
- sage: K = Cone([(1,0), (-1,0), (0,1), (0,-1)])
- sage: lineality(K)
- 2
-
- Per the definition, the lineality of the trivial cone in a trivial
- space is zero::
-
- sage: K = Cone([], lattice=ToricLattice(0))
- sage: lineality(K)
- 0
-
- TESTS:
-
- The lineality of a cone should be an integer between zero and the
- dimension of the ambient space, inclusive::