implement ambient volume using normaliz

src/sage/geometry/polyhedron/backend_normaliz.py

@@ -1319,14 +1319,17 @@ def _volume_normaliz(self, measure='euclidean'):
 
         INPUT:
 
-        - ``measure`` -- (default: 'euclidean') the measure to take. 'euclidean'
-          correspond to ``EuclideanVolume`` in normaliz and 'induced_lattice'
-          correspond to ``Volume`` in normaliz.
+        - ``measure`` -- string. The measure to use. Allowed values are:
+
+          * ``'euclidean'`` (default): corresponds to ``'EuclideanVolume`` in normaliz
+          * ``'induced_lattice'``: corresponds to ``'Volume'`` in normaliz
+          * ``'ambient'``: Lebesgue measure of ambient space (volume)
 
         OUTPUT:
 
-        A float value (when ``measure`` is 'euclidean') or a rational number
-        (when ``measure`` is 'induced_lattice').
+        A float value (when ``measure`` is 'euclidean'),
+        a rational number (when ``measure`` is 'induced_lattice'),
+        a rational number or symbolic number otherwise (dependent on base ring).
 
         .. NOTE::
 
@@ -1342,7 +1345,7 @@ def _volume_normaliz(self, measure='euclidean'):
             sage: s._volume_normaliz()                         # optional - pynormaliz
             0.3333333333333333
 
-        The other possibility is to compute the scaled volume where a unimodual
+        One other possibility is to compute the scaled volume where a unimodual
         simplex has volume 1::
 
             sage: s._volume_normaliz(measure='induced_lattice')  # optional - pynormaliz
@@ -1354,13 +1357,42 @@ def _volume_normaliz(self, measure='euclidean'):
             sage: cube._volume_normaliz(measure='induced_lattice')  # optional - pynormaliz
             6
 
+        Or one can can calculate the ambient volume, which is the above multiplied by the
+        volume of the unimodal simplex (or zero if not full-dimensional)::
+
+            sage: cube._volume_normaliz(measure='ambient')  # optional - pynormaliz
+            1
+            sage: s._volume_normaliz(measure='ambient')     # optional - pynormaliz
+            0
+
         TESTS:
 
         Check that :trac:`28872` is fixed::
 
             sage: P = polytopes.dodecahedron(backend='normaliz')  # optional - pynormaliz
             sage: P.volume(measure='induced_lattice')             # optional - pynormaliz
             -1056*sqrt5 + 2400
+
+        Some sanity checks that the ambient volume works correctly::
+
+            sage: (2*cube)._volume_normaliz(measure='ambient')    # optional - pynormaliz
+            8
+            sage: (1/2*cube)._volume_normaliz(measure='ambient')  # optional - pynormaliz
+            1/8
+            sage: s._volume_normaliz(measure='ambient')           # optional - pynormaliz
+            0
+
+            sage: P = polytopes.regular_polygon(3, backend='normaliz')          # optional - pynormaliz
+            sage: P._volume_normaliz('ambient') == P.volume(engine='internal')  # optional - pynormaliz
+            True
+
+            sage: P = polytopes.dodecahedron(backend='normaliz')                # optional - pynormaliz
+            sage: P._volume_normaliz('ambient') == P.volume(engine='internal')  # optional - pynormaliz
+            True
+
+            sage: P = Polyhedron(rays=[[1]], backend='normaliz')  # optional - pynormaliz
+            sage: P.volume()                                      # optional - pynormaliz
+            +Infinity
         """
         cone = self._normaliz_cone
         assert cone
@@ -1371,6 +1403,20 @@ def _volume_normaliz(self, measure='euclidean'):
                 return self._nmz_result(cone, 'Volume')
             else:
                 return self._nmz_result(cone, 'RenfVolume')
+        elif measure == 'ambient':
+            if self.dim() < self.ambient_dim():
+                return self.base_ring().zero()
+            if not self.is_compact():
+                from sage.rings.infinity import infinity
+                return infinity
+
+            from sage.functions.other import factorial
+            volume = self._volume_normaliz('induced_lattice')/factorial(self.dim())
+
+            return volume
+
+        else:
+            raise TypeError("the measure should be `ambient`, `euclidean`, or `induced_lattice`")
 
     def _triangulate_normaliz(self):
         r"""

src/sage/geometry/polyhedron/base.py

@@ -7088,6 +7088,9 @@ def volume(self, measure='ambient', engine='auto', **kwds):
                 except FeatureNotPresentError:
                     raise RuntimeError("the induced rational measure can only be computed with the optional packages `latte_int`, or `pynormaliz`")
 
+        if engine == 'auto' and measure == 'ambient' and self.backend() == 'normaliz':
+            engine = 'normaliz'
+
         if measure == 'ambient':
             if self.dim() < self.ambient_dim():
                 return self.base_ring().zero()
@@ -7102,7 +7105,7 @@ def volume(self, measure='ambient', engine='auto', **kwds):
             elif engine == 'latte':
                 return self._volume_latte(**kwds)
             elif engine == 'normaliz':
-                return self._volume_normaliz(measure='euclidean')
+                return self._volume_normaliz(measure='ambient')
 
             triangulation = self.triangulate(engine=engine, **kwds)
             pc = triangulation.point_configuration()

src/sage/geometry/polyhedron/library.py

@@ -247,7 +247,7 @@ def regular_polygon(self, n, exact=True, base_ring=None, backend=None):
             sage: octagon.n_vertices()                                        # optional - pynormaliz
             8
             sage: octagon.volume()                                            # optional - pynormaliz
-            2.828427124746190?
+            2*a
         """
         n = ZZ(n)
         if n <= 2: