Trac #22310: Use PPL for facet normals of full-dimensional polytopes

Before:
{{{
sage: timeit("LatticePolytope(lattice_polytope.cross_polytope(3).vertice
s()).facet_normals()")
5 loops, best of 3: 43.2 ms per loop
}}}
After:
{{{
sage: timeit("LatticePolytope(lattice_polytope.cross_polytope(3).vertice
s()).facet_normals()")
125 loops, best of 3: 6.81 ms per loop
}}}
PPL will of course work for non-full-dimensional polytopes as well,
however the treatment of this case is spread around several places and
its removal will be treated separately. Once this is done the speed up
will be even more significant.

Next in the chain of lattice polytope improvements is #22391

URL: https://trac.sagemath.org/22310
Reported by: novoselt
Ticket author(s): Andrey Novoseltsev
Reviewer(s): Travis Scrimshaw

src/sage/geometry/fan.py

@@ -76,50 +76,50 @@
 The last output is not very illuminating. Let's try to improve it::
 
     sage: for cone in fan1: print(cone.rays())
-    M(1, 0,  0),
-    M(0, 1,  0),
-    M(0, 0, -1)
-    in 3-d lattice M
-    M( 0, 1,  0),
-    M(-1, 0,  0),
-    M( 0, 0, -1)
+    M( 0, 1, 0),
+    M( 0, 0, 1),
+    M(-1, 0, 0)
     in 3-d lattice M
-    M(1,  0,  0),
-    M(0, -1,  0),
-    M(0,  0, -1)
+    M( 0,  0, 1),
+    M(-1,  0, 0),
+    M( 0, -1, 0)
     in 3-d lattice M
     M(-1,  0,  0),
     M( 0, -1,  0),
     M( 0,  0, -1)
     in 3-d lattice M
+    M( 0, 1,  0),
+    M(-1, 0,  0),
+    M( 0, 0, -1)
+    in 3-d lattice M
+    M(1, 0,  0),
+    M(0, 1,  0),
+    M(0, 0, -1)
+    in 3-d lattice M
     M(1, 0, 0),
     M(0, 1, 0),
     M(0, 0, 1)
     in 3-d lattice M
-    M( 0, 1, 0),
-    M( 0, 0, 1),
-    M(-1, 0, 0)
-    in 3-d lattice M
     M(1,  0, 0),
     M(0,  0, 1),
     M(0, -1, 0)
     in 3-d lattice M
-    M( 0,  0, 1),
-    M(-1,  0, 0),
-    M( 0, -1, 0)
+    M(1,  0,  0),
+    M(0, -1,  0),
+    M(0,  0, -1)
     in 3-d lattice M
 
 You can also do ::
 
     sage: for cone in fan1: print(cone.ambient_ray_indices())
-    (0, 1, 5)
-    (1, 3, 5)
-    (0, 4, 5)
+    (1, 2, 3)
+    (2, 3, 4)
     (3, 4, 5)
+    (1, 3, 5)
+    (0, 1, 5)
     (0, 1, 2)
-    (1, 2, 3)
     (0, 2, 4)
-    (2, 3, 4)
+    (0, 4, 5)
 
 to see indices of rays of the fan corresponding to each cone.
 
@@ -618,17 +618,17 @@ def FaceFan(polytope, lattice=None):
         M( 0, -1)
         in 2-d lattice M
         sage: for cone in P1xP1: print(cone.rays())
-        M(1,  0),
-        M(0, -1)
-        in 2-d lattice M
         M(-1,  0),
         M( 0, -1)
         in 2-d lattice M
+        M( 0, 1),
+        M(-1, 0)
+        in 2-d lattice M
         M(1, 0),
         M(0, 1)
         in 2-d lattice M
-        M( 0, 1),
-        M(-1, 0)
+        M(1,  0),
+        M(0, -1)
         in 2-d lattice M
 
     TESTS::
@@ -720,12 +720,12 @@ def NormalFan(polytope, lattice=None):
         sage: P1xP1.rays()
         N( 1,  0),
         N( 0,  1),
-        N( 0, -1),
-        N(-1,  0)
+        N(-1,  0),
+        N( 0, -1)
         in 2-d lattice N
         sage: for cone in P1xP1: print(cone.rays())
-        N( 0, -1),
-        N(-1,  0)
+        N(-1,  0),
+        N( 0, -1)
         in 2-d lattice N
         N(1,  0),
         N(0, -1)
@@ -2797,11 +2797,11 @@ def is_smooth(self, codim=None):
             sage: fan.is_smooth(codim=1)
             True
             sage: fan.generating_cone(0).rays()
-            N(-1,  1),
-            N(-1, -1)
+            N(-1, -1),
+            N(-1,  1)
             in 2-d lattice N
             sage: fan.generating_cone(0).rays().matrix().det()
-            2
+            -2
         """
         if codim is None or codim < 0:
             codim = 0

src/sage/geometry/fan_morphism.py

@@ -47,18 +47,18 @@
     Domain fan: Rational polyhedral fan in 2-d lattice N
     Codomain fan: Rational polyhedral fan in 2-d lattice N
     sage: fm.domain_fan().rays()
-    N(-1,  1),
     N( 1,  1),
-    N(-1, -1),
     N( 1, -1),
+    N(-1, -1),
+    N(-1,  1),
     N( 0, -1),
     N( 0,  1)
     in 2-d lattice N
     sage: normal.rays()
-    N(-1,  1),
     N( 1,  1),
+    N( 1, -1),
     N(-1, -1),
-    N( 1, -1)
+    N(-1,  1)
     in 2-d lattice N
 
 As you see, it was necessary to insert two new rays (to prevent "upper" and
@@ -358,14 +358,14 @@ def _RISGIS(self):
             sage: fm = FanMorphism(identity_matrix(2),
             ....:         normal, face, subdivide=True)
             sage: fm._RISGIS()
-            (frozenset({3}),
-             frozenset({2}),
-             frozenset({1}),
+            (frozenset({2}),
+             frozenset({3}),
              frozenset({0}),
-             frozenset({1, 3}),
+             frozenset({1}),
              frozenset({0, 1}),
-             frozenset({0, 2}),
-             frozenset({2, 3}))
+             frozenset({0, 3}),
+             frozenset({2, 3}),
+             frozenset({1, 2}))
         """
         if "_RISGIS_" not in self.__dict__:
             try: