add orthogonal/orthonormal feature to affine_hull of polyhedra

[mo271]

As mentioned in #16045 comment:22 it would be desirable to get the affine hull of a polyhderon, while preserving the polyhedron isometrically (instead of applying an affine transformation, that might distort lenght, volumes, angels, etc.).

I propose to add an option "algorithm" for the .affine_hull method for polyhedra.
This can then be used to correctly calculate volume of non full-diemsional polyhedra, see #16045.

CC: @videlec

Component: geometry

Keywords: polyhedron

Author: Moritz Firsching

Branch: 2ca1a84

Reviewer: Matthias Koeppe, Vincent Delecroix

Issue created by migration from https://trac.sagemath.org/ticket/22804

[mo271]

Branch: u/moritz/affine_hull

[videlec]

comment:2

Two quick comments

1. The keyword algorithm is very misleading. "Algorithm" refers to a method to compute something not the result itself. Moreover, both constructions are projections. You should find something more coherent! And also describe in the documentation what is the result.

2. I like better the more explicit

P.vertices()[0].vector() == P.ambient_space().zero()

---

New commits:

91f239b

adding isometry feature to affine_hull

[videlec]

Commit: 91f239b

[sagetrac-git]

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

43dd759

change name of parameter to 'force_isometry'

[sagetrac-git]

Changed commit from 91f239b to 43dd759

[mo271]

comment:4

Thanks for your comments, Vincent! I hope the new commit addresses all the proposed changes.

I was not quite happy with the choice of "algorithm" for a name of the parameter. Now I named the parameter "force_isometry".

[videlec]

comment:5

I don't understand why this step

    #if the self is full-dimensional, return it immediately
    if self.ambient_dim() == self.dim():
        return self

is inside the if force_isometry.

[mo271]

comment:6

Replying to @videlec:

I don't understand why this step

    #if the self is full-dimensional, return it immediately
    if self.ambient_dim() == self.dim():
        return self

is inside the if force_isometry.

The intent was not to disturb the original method too much.

Howeer , I will gladly move it above the if force_isometry. (I don't know why the old method didn't include this check..)

[sagetrac-git]

Changed commit from 43dd759 to 7afd9aa

[sagetrac-git]

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

7afd9aa

move dimenion check to top of method

[mkoeppe]

comment:8

This looks like a nice interface, but let me just say that for volume computations it's overkill to compute this (which will lead to a double description computation in the slow "field" backend), when all one needs is to compute a scaling factor for the volume...

Another comment: I don't think this is a suitable test:

if self.base_ring() in [ZZ,QQ]

because you could as well have some number field that is too small.

sage: D = polytopes.dodecahedron()
sage: F = D.faces(2)[0].as_polyhedron()
sage: F.affine_hull(force_isometry=True)
TypeError: QR decomposition unable to compute square roots in Number Field in sqrt5 with defining polynomial x^2 - 5