Implement a proper equality check for polyhedron representation objects

[kliem]

sage: P = (1/2)*polytopes.cube()                                                                                                                                                    
sage: Q = (1/2)*polytopes.cube(backend='field')                                                                                                                                     
sage: for p in P.inequalities(): 
....:     assert p in Q.inequalities() 
....:                                                                                                                                                                               
---------------------------------------------------------------------------
AssertionError                            Traceback (most recent call last)
<ipython-input-16-d0a8f3fbfdbb> in <module>
      1 for p in P.inequalities():
----> 2     assert p in Q.inequalities()
      3 

AssertionError: 

This isn't acceptable. Those inequalites are the same, only the defining vector differs by a positive scalar.

We also define a method is_facet_defining_inequality that checks whether self could replace an inequality of the polyhedron other. This is useful, when other is not full-dimensional and inequalities aren't unique (only after affine_hull_projection).

CC: @jplab @LaisRast

Component: geometry

Keywords: scalars, polyhedra

Author: Jonathan Kliem

Branch/Commit: 9ee3335

Reviewer: Jean-Philippe Labbé

Issue created by migration from https://trac.sagemath.org/ticket/30954

[kliem]

Commit: 4c1f295

[kliem]

New commits:

4c1f295

scale rays, lines, inequalities and equations for comparison

[kliem]

Branch: public/30954

[sagetrac-git]

Changed commit from 4c1f295 to b1eac2c

[sagetrac-git]

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

b1eac2c

implement is_facet_defining_inequality

[kliem]

Description changed:

--- 
+++ 
@@ -15,3 +15,5 @@
 AssertionError: 
 ```
 This isn't acceptable. Those inequalites are the same, only the defining vector differs by a positive scalar.
+
+We also define a method `is_facet_defining_inequality` that checks whether `self` could replace an inequality of the polyhedron `other`. This is useful, when `other` is not full-dimensional and inequalities aren't unique (only after `affine_hull_projection`).

[jplab]

Reviewer: Jean-Philippe Labbé

[jplab]

comment:4

Minor things:

-        cross_slack_matrix = self_matrix*hom_Vrep
+        cross_slack_matrix = self_matrix * hom_Vrep

-        if any(self.A()*line.vector() for line in other.lines()):
+        if any(self.A() * line.vector() for line in other.lines()):

In the TEST block, I would remove the QQ(*) to save some space... (could you also indent/line split?)

Why is there a lonely :: ?:

+            False
+
+        ::
+
+            sage: P = Polyhedron(vertices=[[0,0,0],[0,1,0]], lines=[[1,0,0]])

[kliem]

comment:5

Replying to @jplab:

Why is there a lonely :: ?:

+            False
+
+        ::
+
+            sage: P = Polyhedron(vertices=[[0,0,0],[0,1,0]], lines=[[1,0,0]])

People use it to group examples, when they are too lazy to write down a nice sentence for each example.