Implement join of polytopes

[jplab]

From ​https://www.csun.edu/~ctoth/Handbook/chap15.pdf:

The join of two polytopes P (of dimension d) and Q (of dimension d') is the (d+d'+1)-polytope obtained as the convex hull of P\cup Q after
embedding P and Q in a space where their affine hulls are skew.

Depends on #22572

CC: @videlec @mo271 @mkoeppe @fchapoton

Component: geometry

Keywords: days93, polytope, IMA-PolyGeom

Author: Jean-Philippe Labbé

Branch/Commit: 0c9793f

Reviewer: Vincent Delecroix, Moritz Firsching

Issue created by migration from https://trac.sagemath.org/ticket/24848

[jplab]

Commit: a14d2c7

[jplab]

Branch: u/jipilab/24848

[jplab]

New commits:

c807893	first version of the join
a14d2c7	pep8

[jplab]

Description changed:

--- 
+++ 
@@ -2,3 +2,5 @@
 
 The join of two polytopes `P` (of dimension d) and `Q` (of dimension d') is the (d+d'+1)-polytope obtained as the convex hull of `P\cup Q` after
 embedding P and Q in a space where their affine hulls are skew.
+
+The usual product is obtained when the intersection of their affine hull is the origin.

[jplab]

Description changed:

--- 
+++ 
@@ -2,5 +2,3 @@
 
 The join of two polytopes `P` (of dimension d) and `Q` (of dimension d') is the (d+d'+1)-polytope obtained as the convex hull of `P\cup Q` after
 embedding P and Q in a space where their affine hulls are skew.
-
-The usual product is obtained when the intersection of their affine hull is the origin.

[videlec]

Author: Vincent Delecroix

[videlec]

comment:3

According to the developer guide:

• the docstring should start with a one line sentence. Then a line break, then a longer description.

• better than - ``other`` -- a :class:`Polyhedron_base`., I would just use - ``other`` -- a polyhedron.

• there is no need to fill an OUTPUT section as it is already described before.

And I would like to see the definition of a "skew subspace".

This is not the proper thing to do

        try:
            new_ring = self.parent()._coerce_base_ring(other)
        except TypeError:
            raise TypeError("no common canonical parent for objects with parents: " + str(self.parent())
                     + " and " + str(other.parent()))

You should try to coerce the objects to a common parent as in

sage: C = polytopes.cube()
sage: I = polytopes.icosahedron()
sage: cm = get_coercion_model()
sage: cm.common_parent(C, I)
Polyhedra in (Number Field in sqrt5 with defining polynomial x^2 - 5)^3

And even simpler, you could use the decorator @coerce_binop from sage.structure.element.

Moreover, when you construct the resulting polytope you should just use directly the common parent.

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

347f99f

Made docstring corrections