Added the missing methods

Author: jplab

src/sage/geometry/fan.py

@@ -2439,6 +2439,58 @@ def Gale_transform(self):
         m = m.augment(matrix(ZZ, m.nrows(), 1, [1] * m.nrows()))
         return matrix(ZZ, m.integer_kernel().matrix())
 
+    def is_regular(self):
+        r"""
+        Check if ``self`` is regular.
+
+        A rational polyhedral fan is *regular* if it is the normal fan of a
+    polytope.
+
+        OUTPUT: ``True`` if ``self`` is complete and ``False`` otherwise
+
+        EXAMPLES:
+
+        This is the mother of all examples, which is not regular (see Section
+        7.1.1 in [DLRS2010]_)::
+
+            sage: epsilon = 0
+            sage: rays = [(4-epsilon,epsilon,0),(0,4-epsilon,epsilon),(epsilon,0,4-epsilon),(2,1,1),(1,2,1),(1,1,2),(-1,-1,-1)]
+            sage: S1 = [Cone([rays[i] for i in indices]) for indices in [(0,1,4),(0,3,4),(1,2,5),(1,4,5),(0,2,3),(2,3,5),(3,4,5),(6,0,1),(6,1,2),(6,2,0)]]
+            sage: mother = Fan(S1)
+            sage: mother.is_regular()
+            False
+
+        Doing a slight perturbation makes the same subdivision regular::
+
+            sage: epsilon = 1/2
+            sage: rays = [(4-epsilon,epsilon,0),(0,4-epsilon,epsilon),(epsilon,0,4-epsilon),(2,1,1),(1,2,1),(1,1,2),(-1,-1,-1)]
+            sage: S1 = [Cone([rays[i] for i in indices]) for indices in [(0,1,4),(0,3,4),(1,2,5),(1,4,5),(0,2,3),(2,3,5),(3,4,5),(6,0,1),(6,1,2),(6,2,0)]]
+            sage: mother = Fan(S1)
+            sage: mother.is_regular()
+            True
+
+        .. SEEALSO::
+
+            :meth:`is_projective`.
+        """
+        if not self.is_complete():
+            raise ValueError('the fan is not complete')
+        from sage.geometry.triangulation.point_configuration import PointConfiguration
+        from sage.geometry.polyhedron.constructor import Polyhedron
+        pc = PointConfiguration(self.rays())
+        v_pc = [vector(pc.point(i)) for i in range(pc.n_points())]
+        v_r = [vector(list(r)) for r in self.rays()]
+        cone_indices = [_.ambient_ray_indices() for _ in self.generating_cones()]
+        translator = [v_pc.index(v_r[i]) for i in range(pc.n_points())]
+        translated_cone_indices = [[translator[i] for i in ci] for ci in cone_indices]
+        dc_pc = pc.deformation_cone(translated_cone_indices)
+        lift = dc_pc.an_element()
+        ieqs = [[lift[i]] + list(v_pc[i]) for i in range(self.nrays())]
+        poly = Polyhedron(ieqs=ieqs)
+        return self.is_equivalent(poly.normal_fan())
+
+    is_projective = is_regular
+
     def generating_cone(self, n):
         r"""
         Return the ``n``-th generating cone of ``self``.

src/sage/geometry/polyhedron/base5.py

@@ -669,7 +669,10 @@ def deformation_cone(self):
         that the resulting polytope has a normal fan which is a coarsening of
         the normal fan of ``self``.
 
-        EXAMPLES::
+        EXAMPLES:
+
+        Let's examine the deformation cone of the square with one truncated
+        vertex::
 
             sage: tc = Polyhedron([(1, -1), (1/3, 1), (1, 1/3), (-1, 1), (-1, -1)])
             sage: dc = tc.deformation_cone()
@@ -685,6 +688,19 @@ def deformation_cone(self):
              A ray in the direction (0, 1, 1),
              A ray in the direction (1, 0, 2))
 
+        Now, let's compute the deformation cone of the pyramid over a square
+        and verify that it is not full dimensional::
+
+            sage: py = Polyhedron([(0, -1, -1), (0, -1, 1), (0, 1, -1), (0, 1, 1), (1, 0, 0)])
+            sage: dc_py = py.deformation_cone(); dc_py
+            A 4-dimensional polyhedron in QQ^5 defined as the convex hull of 1 vertex, 1 ray, 3 lines
+            sage: [_.b() for _ in py.Hrepresentation()]
+            [0, 1, 1, 1, 1]
+            sage: r = dc_py.rays()[0]
+            sage: l1,l2,l3 = dc_py.lines()
+            sage: r.vector()-l1.vector()/2-l2.vector()-l3.vector()/2
+            (0, 1, 1, 1, 1)
+
         .. SEEALSO::
 
             :meth:`~sage.schemes.toric.variety.Kaehler_cone`
@@ -699,7 +715,7 @@ def deformation_cone(self):
         A = A.transpose()
         A_ker = A.right_kernel_matrix(basis='computed')
         gale = tuple(A_ker.columns())
-        collection = [_.ambient_H_indices() for _ in self.faces(0)]
+        collection = [f.ambient_H_indices() for f in self.faces(0)]
         n = len(gale)
         K = None
         for cone_indices in collection:

src/sage/geometry/triangulation/point_configuration.py

@@ -2121,7 +2121,7 @@ def Gale_transform(self, points=None, homogenize=True):
             except TypeError:
                 pass
         if homogenize:
-            m = matrix([ (1,) + p.affine() for p in points])
+            m = matrix([(1,) + p.affine() for p in points])
         else:
             m = matrix([p.affine() for p in points])
         return m.left_kernel().matrix()
@@ -2167,6 +2167,55 @@ def deformation_cone(self, collection):
              A ray in the direction (0, 1, 0),
              A ray in the direction (1, 0, -1))
 
+        Let's verify the mother of all examples explained in Section 7.1.1 of
+        [DLRS2010]_::
+
+            sage: epsilon = 0
+            sage: mother = PointConfiguration([(4-epsilon,epsilon,0),(0,4-epsilon,epsilon),(epsilon,0,4-epsilon),(2,1,1),(1,2,1),(1,1,2)])
+            sage: mother.points()
+            (P(4, 0, 0), P(0, 4, 0), P(0, 0, 4), P(2, 1, 1), P(1, 2, 1), P(1, 1, 2))
+            sage: S1 = [(0,1,4),(0,3,4),(1,2,5),(1,4,5),(0,2,3),(2,3,5)]
+            sage: S2 = [(0,1,3),(1,3,4),(1,2,4),(2,4,5),(0,2,5),(0,3,5)]
+
+        Both subdivisions `S1` and `S2` are not regular::
+
+            sage: mother_dc1 = mother.deformation_cone(S1)
+            sage: mother_dc1
+            A 4-dimensional polyhedron in QQ^6 defined as the convex hull of 1 vertex, 1 ray, 3 lines
+            sage: mother_dc2 = mother.deformation_cone(S2)
+            sage: mother_dc2
+            A 4-dimensional polyhedron in QQ^6 defined as the convex hull of 1 vertex, 1 ray, 3 lines
+
+        Notice that they have a ray which provides a degenerate lifting which
+        only provides a coarsening of the subdivision from the lower hull (it
+        has 5 facets, and should have 8)::
+
+            sage: result = Polyhedron([vector(list(mother.points()[_])+[mother_dc1.rays()[0][_]]) for _ in range(len(mother.points()))])
+            sage: result.f_vector()
+            (1, 6, 9, 5, 1)
+
+        But if we use epsilon to perturb the configuration, suddenly
+        `S1` becomes regular::
+
+            sage: epsilon = 1/2
+            sage: mother = PointConfiguration([(4-epsilon,epsilon,0), 
+                                               (0,4-epsilon,epsilon),
+                                               (epsilon,0,4-epsilon),
+                                               (2,1,1),
+                                               (1,2,1),
+                                               (1,1,2)])
+            sage: mother.points()
+            (P(7/2, 1/2, 0),
+             P(0, 7/2, 1/2),
+             P(1/2, 0, 7/2),
+             P(2, 1, 1),
+             P(1, 2, 1),
+             P(1, 1, 2))
+            sage: mother_dc1 = mother.deformation_cone(S1);mother_dc1
+            A 6-dimensional polyhedron in QQ^6 defined as the convex hull of 1 vertex, 3 rays, 3 lines
+            sage: mother_dc2 = mother.deformation_cone(S2);mother_dc2
+            A 3-dimensional polyhedron in QQ^6 defined as the convex hull of 1 vertex and 3 lines
+
         .. SEEALSO::                                                                                                                                                                                        
                    
             :meth:`~sage.schemes.toric.variety.Kaehler_cone`