Add .is_inscribed() method to polytopes

[jipilab]

A full-dimensional compact polytope is inscribed if there exists a point in space which is equidistant to all its vertices.

This function checks if this point exists and returns it if asked for.

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​ecaf3ee

pep8 conventions

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​2a831bd

Added a comment on the method used.

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​ef42b4a

corrected indentation

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​d3dce24

Added examples with NotImplemented

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​8e6df07

Updated the next to python3 standard

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​21ec0e7

Changed keyword to certificate

[tmonteil]

Here are few first small comments after first reading:

• you forgot to replace certify with certificate in the code of the function.
• inscribable is about the possibility to find a combinatorially equivalent polytope that can be inscribed in a sphere. So, i guess the current method should be called is_inscribed instead (if you agree, do not forget to also modify the ticket title).
• i do not see any reason why the polytope should be full-dimensional. In case it is not, there might be various centers, but you could return the one that belongs to the affine hull of the polytope.
• you could be lazier by not constructing the list other_vertices, but an iterator instead (the all call might end far before testing all points).
• since you only do comparisons with the circumradius, you should not compute it, but use its square (computing the square root can be very costly in some rings).
• row_data shoud be raw_data.
• Why not returning a boolean when certificate is set to False ? The problem here is that bool((False,None)) is True, so that this will cause issues if the method is used within if statements.

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​820910a

First edits after review

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​a563094

Second small edits

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​361914c

Make tests pass

[tmonteil]

Some more comments:

• The documentation should start with a one-sentence description, see ​http://doc.sagemath.org/html/en/developer/coding_basics.html#documentation-strings
• in describing the input, you should say that the default for certificate is False (look at other parts of Sage source code, as well as ​http://doc.sagemath.org/html/en/developer/coding_basics.html#documentation-strings)
• the OUTPUT section should separate cases depending on the value of certificate.
• tests like sage: q.is_inscribed(True) could be misleading for the user. Where does the True comes from. You can write this since certificate is the first option, but it is not a required parameter, so you should prefer the more explicit sage: q.is_inscribed(certificate=True).
• simplex_vertices works because you assume that the polytope is full-dimensional, you should add a short comment to explain that just before that command, so that we understand what happens (and it will ease the work of someone who might try to extend the method to non-full-dimensional polytopes).
• since you do not use other_vertices anymore, you should remove its definition
• you have:

  if not self.is_compact() or not self.is_full_dimensional():
      raise NotImplementedError("This function is implemented for full-dimensional polytopes only.")
  

If the polyhedron is full-dimensional but not compact, the user will get a misleading message. So make two different tests/statements, this adds informations to the user.

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

​c886ddb	Second edits after review
​649a3b4	pep8 and pyflakes conventions

[jipilab]

All test pass on 7.6.b4.

[tscrim]

Two minor things. For the INPUT:, you should format it as:

        - ``certificate`` -- (default: ``False``) boolean; specifies whether to
          return the circumcenter, if found

Also, we don't technically consider error messages to be full sentences (at least the "shorter" ones like here) following Python's convention. So it should start with a lower case and not end with a period.

[jipilab]

Thanks for the tip! Will do!

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

​4d9b5c5

Added a indentation for doc to build

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

​d9e5415

Correct an indentation in input block

[chapoton]

ok, let it be