Make outer normal fans readily available

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For a polytope P, both NormalFan(P) and P.normal_fan() return the inner normal fan having the inner facet normals as rays. The outer normal fan, using the outer facet normals, is well-known, but not as easy to find in Sage.

As an example, create the polytope on p.193 of Ziegler's "Lectures on Polytopes" with its normal fan:

sage: V=[(-8,-4),(-8,6),(-4,-8),(0,-9),(0,6),(4,-8)]
sage: P=Polyhedron(V)
sage: NormalFan(P)

The result is the inner normal fan of P, which is the outer normal fan of -P.

This ticket adds to Polyhedron.normal_fan an argument direction set to either 'inner' or 'outer' to determine which of the directions to use. The default will stay 'inner' in order to match the default of NormalFan. The description of NormalFan is extended by a hint to use NormalFan(-P) for the outer normal fan.

Additionally, this ticket adds in sage.geometry.polyhedron.representation a method Inequality.outer_normal.
This allows to obtain the outer normal vector without having to think about the appropriate sign of Inequality.A() whenever dealing with the facet inequalities of a polyhedron.

CC: @jplab

Component: geometry

Keywords: NormalFan, polytope, days100

Author: Julian Ritter

Branch/Commit: 440689c

Reviewer: Jean-Philippe Labbé, Frédéric Chapoton

Issue created by migration from https://trac.sagemath.org/ticket/27993

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Branch: u/nailuj/normalfan_outer

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New commits:

57d5508	added direction argument to NormalFan
484bf56	added outer_normal method

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Commit: 484bf56

[embray]

comment:3

Sage 8.8 is to be released fairly soon so unless this is a blocker this should be moved to the next release.

[novoselt]

comment:4

I don't care much about adding direction argument, but I have extremely strong objection to making it the default. There was a reason why inner normals were used for many, many years (like since 2006) - it is consistent with the literature. I understand that some books/articles use the other way, but when neither way is exceptionally uncommon the default choice is pretty much random personal preference when you first create these classes and methods. Once they were created and used for a long time, there is absolutely no reason to change the default whatever it is.

[jplab]

comment:5

Replying to @novoselt:

I don't care much about adding direction argument, but I have extremely strong objection to making it the default. There was a reason why inner normals were used for many, many years (like since 2006) - it is consistent with the literature. I understand that some books/articles use the other way, but when neither way is exceptionally uncommon the default choice is pretty much random personal preference when you first create these classes and methods. Once they were created and used for a long time, there is absolutely no reason to change the default whatever it is.

+1

I would leave the default as it is and add the direction argument to allow to get the variant.

[novoselt]

comment:6

Actually, I think now it would be better to allow quick "direction flip" either on the polytope or the fan, i.e. make them support - operator, instead of making constructors more complicated.

[jplab]

comment:7

Replying to @novoselt:

Actually, I think now it would be better to allow quick "direction flip" either on the polytope or the fan, i.e. make them support - operator, instead of making constructors more complicated.

This seems to already be possible for Polyhedron objects:

sage: p = polytopes.regular_polygon(5)
sage: p.vertices()
(A vertex at (0.5877852522924731?, -0.8090169943749474?),
 A vertex at (0.9510565162951536?, 0.3090169943749474?),
 A vertex at (0, 1),
 A vertex at (-0.5877852522924731?, -0.8090169943749474?),
 A vertex at (-0.9510565162951536?, 0.3090169943749474?))
sage: (-p).vertices()
(A vertex at (-0.5877852522924731?, 0.8090169943749474?),
 A vertex at (-0.9510565162951536?, -0.3090169943749474?),
 A vertex at (0, -1),
 A vertex at (0.587785252292474?, 0.8090169943749474?),
 A vertex at (0.9510565162951536?, -0.3090169943749474?))

Or you have something else in mind?

Yes, I would use NormalFan(-P) or NormalFan(P) depending on whether one wants the inner or outer. This seems like the easiest thing to do. One does not need to go into the Polyhedron constructor...