Make outer normal fans readily available

[sagetrac-nailuj]

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

[sagetrac-nailuj]

Branch: u/nailuj/normalfan_outer

[sagetrac-nailuj]

New commits:

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

[sagetrac-nailuj]

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...

[novoselt]

comment:8

That's exactly what I meant. I think those who care about directions are likely to read the documentation and discover that these are inner normals. After that, if this is not the desired choice for the user, NormalFan(-P) seems easier and cleaner to use than NormalFan(P, direction="outer").

[jplab]

comment:9

Replying to @novoselt:

That's exactly what I meant. I think those who care about directions are likely to read the documentation and discover that these are inner normals. After that, if this is not the desired choice for the user, NormalFan(-P) seems easier and cleaner to use than NormalFan(P, direction="outer").

I believe that the proposed change would be in the method .normal_fan() of Polyhedron objects, and not at the level of NormalFan, for the reason you mention.

That said, I would change the description of the ticket as follows:

I propose to add to normal_fan an argument direction set to either 'outer' or 'inner' to determine which of the directions to use. The default will stay inner in order to match the default of NormalFan.

Above I would keep the current default.

Additionally, I propose to add 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.

This is good to me.

Further, I would suggest to add

Use NormalFan(-P) to get the outer normal fan.

in the documentation of NormalFan.

[sagetrac-git]

Changed commit from 484bf56 to b41def5

[sagetrac-git]

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

d8763cf	Revert "added direction argument to NormalFan"
7bfda6e	added hint to NormalFan(-P)
0c653c9	added direction argument to Polyhedron.normal_fan
b41def5	Merge branch 'develop' into normalfan_outer

[sagetrac-git]

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

fd3e899

Fixed typo

[sagetrac-git]

Changed commit from b41def5 to fd3e899

[sagetrac-nailuj]

comment:12

Replying to @jplab:

Replying to @novoselt:

That's exactly what I meant. I think those who care about directions are likely to read the documentation and discover that these are inner normals. After that, if this is not the desired choice for the user, NormalFan(-P) seems easier and cleaner to use than NormalFan(P, direction="outer").

I believe that the proposed change would be in the method .normal_fan() of Polyhedron objects, and not at the level of NormalFan, for the reason you mention.

Yes. I stumbled upon this inner-outer issue through Polyhedron.normal_fan(). This is why I added jipilab in CC in the first place, and this is where I should have proposed to make 'outer' the default – my bad. I'm fine with 'inner' as default in both files. My main goal was easier access to outer normal fans, which is now available and explained in the docstrings.

That said, I would change the description of the ticket as follows:

I propose to add to normal_fan an argument direction set to either 'outer' or 'inner' to determine which of the directions to use. The default will stay inner in order to match the default of NormalFan.

Done.

Further, I would suggest to add

Use NormalFan(-P) to get the outer normal fan.

in the documentation of NormalFan.

Done.

[jplab]

Reviewer: Jean-Philippe Labbé

[jplab]

Changed keywords from NormalFan, polytope to NormalFan, polytope, days100

[jplab]

comment:13

• The optional parameters should receive an appropriate doctest. For example:

  sage: p = polytopes.regular_polygon(4,base_ring=QQ).pyramid()
  sage: inner_nf = p.normal_fan()
  sage: inner_nf.rays()
  N( 1,  0,  0),
  N(-1,  1,  1),
  N(-1,  1, -1),
  N(-1, -1, -1),
  N(-1, -1,  1)
  in 3-d lattice N
  
  sage: outer_nf = p.normal_fan(direction='outer')
  sage: outer_nf.rays()
  N(-1,  0,  0),
  N( 1,  1, -1),
  N( 1,  1,  1),
  N( 1, -1,  1),
  N( 1, -1, -1)
  in 3-d lattice N
  

- Further, it would be nice to disallow such things:

sage: outer_nf = p.normal_fan(direction='blabla')

   by checking that direction is one of the two allowed things (see the volume function for a similar handling).

- Remove the space before the semi-colon in the docstring of `normal_fan`.

[sagetrac-git]

Changed commit from fd3e899 to 308dada

[sagetrac-git]

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

257d367	Added check that the direction is inner or outer
5ae5e68	pyflakes fix
691046a	fixed typo
308dada	added examples

[sagetrac-nailuj]

comment:15

Replying to @jplab:

The optional parameters should receive an appropriate doctest. For example:

sage: p = polytopes.regular_polygon(4,base_ring=QQ).pyramid()
sage: inner_nf = p.normal_fan()
sage: inner_nf.rays()
N( 1,  0,  0),
N(-1,  1,  1),
N(-1,  1, -1),
N(-1, -1, -1),
N(-1, -1,  1)
in 3-d lattice N

sage: outer_nf = p.normal_fan(direction='outer')
sage: outer_nf.rays()
N(-1,  0,  0),
N( 1,  1, -1),
N( 1,  1,  1),
N( 1, -1,  1),
N( 1, -1, -1)
in 3-d lattice N

I added a 2-d example testing inner, outer, and an invalid direction.

Further, it would be nice to disallow such things:

sage: outer_nf = p.normal_fan(direction='blabla')

by checking that direction is one of the two allowed things (see the volume function for a similar handling).

Done.

Remove the space before the semi-colon in the docstring of normal_fan.

Done.

[jplab]

comment:16

Great!

Could you do one change: the backtick in the error message should be a single quote.

I would also fix the title of the ticket and the description to reflect the actual changes involved.

[sagetrac-git]

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

9d20a1d

replaced backtick by single quote

[sagetrac-git]

Changed commit from 308dada to 9d20a1d

[sagetrac-nailuj]

comment:18

Replying to @jplab:

Could you do one change: the backtick in the error message should be a single quote.

Done.

I would also fix the title of the ticket and the description to reflect the actual changes involved.

Done.

[jplab]

comment:19

The doctest still has backticks.

[sagetrac-git]

Changed commit from 9d20a1d to 440689c

[sagetrac-git]

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

440689c

replaced backticks in doctest

[fchapoton]

comment:21

ok, c'est correct

[fchapoton]

Changed reviewer from Jean-Philippe Labbé to Jean-Philippe Labbé, Frédéric Chapoton

[vbraun]

Changed branch from u/nailuj/normalfan_outer to 440689c