Added tips and some more methods

src/doc/en/thematic_tutorials/geometry.rst

@@ -19,5 +19,6 @@ So far, the tutorials cover mostly things related to polyhedral computations.
    geometry/new_from_old
    geometry/related_objects
    geometry/is_this_polyhedron
+   geometry/tips
    geometry/visualization
    geometry/polytope_tikz

src/doc/en/thematic_tutorials/geometry/related_objects.rst

@@ -119,6 +119,44 @@ several methods:
 ... geometric objects and properties
 ==============================================================
 
+Base ring
+--------------------------------------------------------------
+
+The first important object related to a polyhedron is its base ring.
+
+::
+
+    sage: P1.base_ring()
+    Integer Ring
+
+.. end of output
+
+Bounded edges
+--------------------------------------------------------------
+
+The bounded edges are edges between two vertices of the polyhedron. This
+function returns an iterator.
+
+::
+
+    sage: list(P1.bounded_edges())
+    [(A vertex at (0, 1), A vertex at (1, 0))]
+
+.. end of output
+
+Bounding box
+--------------------------------------------------------------
+
+The bounding box returns the two corners (the ones with minimal and maximal 
+coordinates) of a box with integer coordinates that contains the polyhedron.
+
+::
+
+    sage: Cube.bounding_box()
+    ((-1, -1, -1), (1, 1, 1))
+
+.. end of output
+
 Center and Representative point
 --------------------------------------------------------------
 
@@ -433,6 +471,36 @@ There are two ways to get it.
 
 .. end of output
 
+Automorphic groups
+--------------------------------------------------------------
+
+The first one gives the automorphism group of the vertex graph of the polyhedron.
+
+::
+
+    sage: S.combinatorial_automorphism_group()
+    Permutation Group with generators [(3,4), (2,3), (1,2)]
+
+    sage: Octa = polytopes.octahedron()
+    sage: Octa.combinatorial_automorphism_group()
+    Permutation Group with generators [(3,4), (2,3)(4,5), (1,2)(5,6)]
+
+.. end of output
+
+The second automorphism group is the restricted automorphism group which
+contains the affine transformations that preserve the :math:`V`-representation.
+
+::
+
+    sage: P2 = Polyhedron(vertices = [[1, 0], [0, 1]], rays = [[1, 1], [0, 1]])
+    sage: P2.combinatorial_automorphism_group()
+    Permutation Group with generators [(2,3)]
+    sage: P2.restricted_automorphism_group()
+    Permutation Group with generators [()]
+
+.. end of output
+
+??
 
 Incidence matrix
 --------------------------------------------------------------

src/doc/en/thematic_tutorials/geometry/tips.rst

@@ -0,0 +1,66 @@
+.. -*- coding: utf-8 -*-
+
+.. linkall
+
+.. _tips:
+
+==============================================================
+Tips, Tricks and Good Things to Know
+==============================================================
+
+.. MODULEAUTHOR:: Jean-Philippe Labbé <[email protected]>
+
+
+Sage input function
+==============================================================
+
+If you are working with a polyhedron that was difficult to construct
+and you would like to get back the proper Sage input code to reproduce this
+object, you can!
+
+::
+
+    sage: Cube = polytopes.cube()
+    sage: TCube = Cube.truncation()
+    sage: sage_input(TCube)
+    Polyhedron(base_ring=QQ, vertices=[(-QQ(1), -QQ(1), -1/3), (-QQ(1), -QQ(1),
+    1/3), (-QQ(1), -1/3, -QQ(1)), (-QQ(1), -1/3, QQ(1)), (-QQ(1), 1/3, -QQ(1)),
+    (-QQ(1), 1/3, QQ(1)), (-QQ(1), QQ(1), -1/3), (-QQ(1), QQ(1), 1/3), (-1/3,
+    -QQ(1), -QQ(1)), (-1/3, -QQ(1), QQ(1)), (-1/3, QQ(1), -QQ(1)), (-1/3,
+    QQ(1), QQ(1)), (1/3, -QQ(1), -QQ(1)), (1/3, -QQ(1), QQ(1)), (1/3, QQ(1),
+    -QQ(1)), (1/3, QQ(1), QQ(1)), (QQ(1), -QQ(1), -1/3), (QQ(1), -QQ(1), 1/3),
+    (QQ(1), -1/3, -QQ(1)), (QQ(1), -1/3, QQ(1)), (QQ(1), 1/3, -QQ(1)), (QQ(1),
+    1/3, QQ(1)), (QQ(1), QQ(1), -1/3), (QQ(1), QQ(1), 1/3)])
+
+.. end of output
+
+
+:code:`repr_pretty_Hrepresentation`
+==============================================================
+
+If you would like to visualize the :math:`H`-representation nicely and even get
+the latex presentation, there is a method for that!
+
+::
+
+    sage: Nice_repr = TCube.repr_pretty_Hrepresentation(separator='\n')
+    sage: print Nice_repr
+    1 >= x0
+    1 >= x1
+    3*x1 + 7 >= 3*x0 + 3*x2
+    x0 + 1 >= 0
+    x1 + 1 >= 0
+    3*x0 + 7 >= 3*x1 + 3*x2
+    3*x0 + 3*x1 + 7 >= 3*x2
+    3*x0 + 3*x2 + 7 >= 3*x1
+    3*x0 + 3*x1 + 3*x2 + 7 >= 0
+    x2 + 1 >= 0
+    1 >= x2
+    3*x1 + 3*x2 + 7 >= 3*x0
+    3*x2 + 7 >= 3*x0 + 3*x1
+    7 >= 3*x0 + 3*x1 + 3*x2
+
+    sage: Latex_repr = LatexExpr(TCube.repr_pretty_Hrepresentation(separator=",\\\\", latex=True))
+    sage: view(Latex_repr)  # not tested
+
+.. end of output