Trac #30605: improve cone containment tests

If `K` is a polyhedral cone, the test `x in K` only supports `x` with
rational coordinates. It is desirable in many cases to perform the same
test with irrational numbers as well.

This will simplify the implementation in ticket #29169.

URL: https://trac.sagemath.org/30605
Reported by: mjo
Ticket author(s): Michael Orlitzky
Reviewer(s): Jonathan Kliem

build/pkgs/configure/checksums.ini

@@ -1,4 +1,4 @@
 tarball=configure-VERSION.tar.gz
-sha1=384646584233b2b43d67518772f298064ef46887
-md5=82a3a0b22d5bb54dde5eba74356e5c73
-cksum=462963871
+sha1=0844b6fc043ed89d8cc0d1ffd32ab38d3e2d355a
+md5=22f22483c1da51c1cef22833e3ec82ea
+cksum=1843480734

build/pkgs/configure/package-version.txt

@@ -1 +1 @@
-7be0559bbb823f318412c21e720bf91710ffac12
+e90b2d71aab2289a07c3e04c41fd536001faa1d2

src/sage/geometry/cone.py

@@ -209,7 +209,7 @@
 from sage.matrix.all import column_matrix, matrix, MatrixSpace
 from sage.misc.all import cached_method, flatten, latex
 from sage.modules.all import span, vector, VectorSpace
-from sage.rings.all import QQ, RR, ZZ
+from sage.rings.all import QQ, ZZ
 from sage.structure.all import SageObject, parent
 from sage.structure.richcmp import richcmp_method, richcmp
 from ppl import (C_Polyhedron, Generator_System, Constraint_System,
@@ -526,47 +526,85 @@ def _ambient_space_point(body, data):
 
     OUTPUT:
 
-    - integral, rational or numeric point of the ambient space of ``body``
-      if ``data`` were successfully interpreted in such a way, otherwise a
-      ``TypeError`` exception is raised
+    An integral, rational, real algebraic, or numeric point of the
+    ambient space of ``body`` is returned if ``data`` were
+    successfully interpreted in such a way. A ``TypeError`` is raised
+    otherwise.
 
     TESTS::
 
         sage: from sage.geometry.cone import _ambient_space_point
         sage: c = Cone([(1,0), (0,1)])
         sage: _ambient_space_point(c, [1,1])
         N(1, 1)
+        sage: _ambient_space_point(c, vector(ZZ,[1,1]))
+        N(1, 1)
         sage: _ambient_space_point(c, c.dual_lattice()([1,1]))
         Traceback (most recent call last):
         ...
         TypeError: the point M(1, 1) and
          2-d cone in 2-d lattice N have incompatible lattices
         sage: _ambient_space_point(c, [1,1/3])
         (1, 1/3)
+        sage: _ambient_space_point(c, vector(QQ,[1,1/3]))
+        (1, 1/3)
         sage: _ambient_space_point(c, [1/2,1/sqrt(3)])
-        (0.500000000000000, 0.577350269189626)
+        (1/2, 0.5773502691896258?)
+        sage: _ambient_space_point(c, vector(AA,[1/2,1/sqrt(3)]))
+        (1/2, 0.5773502691896258?)
         sage: _ambient_space_point(c, [1,1,3])
         Traceback (most recent call last):
         ...
         TypeError: [1, 1, 3] does not represent a valid point
-         in the ambient space of 2-d cone in 2-d lattice N
+        in the ambient space of 2-d cone in 2-d lattice N
+        sage: _ambient_space_point(c, vector(ZZ,[1,1,3]))
+        Traceback (most recent call last):
+        ...
+        TypeError: (1, 1, 3) does not represent a valid point
+        in the ambient space of 2-d cone in 2-d lattice N
+
+    Ensure that transcendental elements can, at the very least, be
+    represented numerically::
+
+        sage: from sage.geometry.cone import _ambient_space_point
+        sage: c = Cone([(1,0), (0,1)])
+        sage: _ambient_space_point(c, [1, pi])
+        (1.00000000000000, 3.14159265358979)
+        sage: _ambient_space_point(c, vector(SR,[1, pi]))
+        (1.00000000000000, 3.14159265358979)
+
     """
+    from sage.rings.all import AA, RR
+
     L = body.lattice()
-    try: # to make a lattice element...
-        return L(data)
-    except TypeError:
-        # Special treatment for toric lattice elements
-        if is_ToricLattice(parent(data)):
-            raise TypeError("the point %s and %s have incompatible "
-                            "lattices" % (data, body))
-    try: # ... or an exact point...
-        return L.base_extend(QQ)(data)
-    except TypeError:
-        pass
-    try: # ... or at least a numeric one
-        return L.base_extend(RR)(data)
-    except TypeError:
-        pass
+
+    def try_base_extend(ring):
+        # Factor out the "try this ring..." code that's repeated four
+        # times.
+        try:
+            return L.base_extend(ring)(data)
+        except TypeError:
+            pass
+        except ValueError as ex:
+            if str(ex).startswith("Cannot coerce"):
+                pass
+
+    # Special treatment for toric lattice elements
+    p = try_base_extend(ZZ)
+    if p is not None:
+        return p
+    if is_ToricLattice(parent(data)):
+        raise TypeError("the point %s and %s have incompatible "
+                        "lattices" % (data, body))
+
+    # If we don't have a lattice element, try successively
+    # less-desirable ambient spaces until (as a last resort) we
+    # attempt a numerical representation.
+    for ring in [QQ, AA, RR]:
+        p = try_base_extend(ring)
+        if p is not None:
+            return p
+
     # Raise TypeError with our own message
     raise TypeError("%s does not represent a valid point in the ambient "
                     "space of %s" % (data, body))
@@ -1546,34 +1584,67 @@ def _contains(self, point, region='whole cone'):
         This function is called by :meth:`__contains__` and :meth:`contains`
         to ensure the same call depth for warning messages.
 
+        By default, a point on the boundary of the cone is considered
+        part of the cone. If you want to test whether the
+        **interior** of the cone contains the point, you need to pass
+        the optional argument ``'interior'``.  If you want to test
+        whether the **relative interior** of the cone contains the
+        point, you need to pass the optional argument
+        ``'relative_interior'``.
+
+        .. WARNING::
+
+            The boundary of a closed convex cone is determined by a
+            set of inequalities. If your ``point`` has entries in an
+            inexact ring, it will sometimes be impossible to say (with
+            confidence) if that point lies on the boundary of the cone
+            or slightly inside it.
+
         INPUT:
 
-        - ``point`` -- anything. An attempt will be made to convert it into a
-          single element of the ambient space of ``self``. If it fails,
-          ``False`` is returned;
+        - ``point`` -- anything; an attempt will be made to convert it
+          into an element compatible with the ambient space of ``self``.
 
-        - ``region`` -- string. Can be either 'whole cone' (default),
-          'interior', or 'relative interior'. By default, a point on
-          the boundary of the cone is considered part of the cone. If
-          you want to test whether the **interior** of the cone
-          contains the point, you need to pass the optional argument
-          ``'interior'``.  If you want to test whether the **relative
-          interior** of the cone contains the point, you need to pass
-          the optional argument ``'relative_interior'``.
+        - ``region`` -- a string (default: 'whole cone'); can be
+          either 'whole cone', 'interior', or 'relative interior'.
 
         OUTPUT:
 
-        - ``True`` if ``point`` is contained in the specified ``region`` of
-          ``self``, ``False`` otherwise.
+        ``True`` is returned if ``point`` is contained in the
+        specified ``region`` of ``self``. ``False`` is returned
+        otherwise, in particular when ``point`` is incompatible with
+        the ambient space.
 
-        Raises a ``ValueError`` if ``region`` is not one of the
+        A ``ValueError`` is raised if ``region`` is not one of the
         three allowed values.
 
         TESTS::
 
             sage: c = Cone([(1,0), (0,1)])
             sage: c._contains((1,1))
             True
+
+        We can test vectors with irrational components::
+
+            sage: c = Cone([(1,0), (0,1)])
+            sage: c._contains((1,sqrt(2)))
+            True
+            sage: c._contains(vector(SR, [1,pi]))
+            True
+
+        Ensure that complex vectors are not contained in a real cone::
+
+            sage: c = Cone([(1,0), (0,1)])
+            sage: c._contains((1,I))
+            False
+            sage: c._contains(vector(QQbar,[1,I]))
+            False
+
+        And we refuse to coerce elements of another lattice into ours::
+
+            sage: c = Cone([(1,0), (0,1)])
+            sage: c._contains(c.dual().ray(0))
+            False
         """
         try:
             point = _ambient_space_point(self, point)