#10132: Differential Geometry via Sage
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Reporter: mikarm | Owner: mhampton
Type: enhancement | Status: new
Priority: major | Milestone:
Component: geometry | Keywords: differential geometry,
parametrized surface
Author: Mikhail Malakhaltsev | Upstream: N/A
Reviewer: | Merged:
Work_issues: |
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Comment(by jvkersch):
I had a great time playing with this code during the weekend and I am
really looking forward to the time where I'll be able to take a book on
curves and surfaces and work through the examples using Sage. I didn't
really look at the mathematics or the usability (both seem fine for now),
but I went ahead and made this into a Sage class (following the
instructions at
[http://www.sagemath.org/doc/developer/coding_in_python.html#creating-a
-new-directory this page]). Please see the attached patch for details.
What it does is the following:
1. Create a new directory `riemann`, containing the parametrized surface
class that Mikhail uploaded in the zip file, together with the vector
functions class.
2. Updates `setup.py` and `all.py` to make Sage aware of these additions.
I deliberately did not want to change any code if I could help it, but
here are the modifications that I made:
1. Changed the capitalization to `ParametrizedSurface3D` to conform to
the Sage standard;
2. Removed the references to `RR` and `CC` since these already have
standard meanings in Sage;
3. I added a class `GenericManifold` as per the suggestions of Andrey. I
think this is a good idea (even if we don't implement anything generic
now), as it will keep us focused on the need to tie this in with more
general differential geometric classes.
4. Throughout the code, I added various import statements where it was
necessary. I ended up changing some of the constructions in the functions
`principal_curvatures`, ` geodesics_numerical` and
`parallel_translation_numerical` as these relied on some constructions
that weren't pure Python. As you will see, the code that I added is quite
hideous and just serves to make things work -- definitely needs to be
streamlined.
With these changes, my version of Sage (4.6.alpha3 without any other
patches) built cleanly and I was able to replicate all of the
constructions in the Sage worksheet. So hopefully the uploaded patch can
provide a good starting point for further work.
Here are a few remarks that I came across while tweaking the code.
1. Sometimes the terminology "vector" is used where "vector field" would
be more appropriate.
2. It seems as if this class would benefit from having a small template
for a generic tensor class. Right now, many of the member functions
return tensors as lists (which is fine), but there are also member
functions which take an optional argument to return only one component of
that tensor. This code then needs to check whether the optional argument
is valid, adding some complexity to the code. Maybe we could define a
simple tensor class which just has `__getitem__` and `__setitem__` and
then let this class deal with components, input checking, etc.
3. I think that some of the functions in the `vector_functions` module
have Sage alternatives, which should be used instead.
4. We need to think on providing an interface to `parametric_plot3d`, so
that these kind of surfaces can easily be plotted.
As Mikhail remarked in his original post, the code hasn't been optimized
for speed. I thought this could be addressed by the following (could be a
longer term goal for this patch).
1. Caching the result of computing the fundamental forms, the Christoffel
symbols, etc. This would make subsequent calls to the components of these
tensors much faster, at the expense of some memory.
2. The numerical routines should be optimized for float operations using
`fast_float` or similar.
This is just my 2 cents. I'm looking forward to hearing your comments and
I would definitely like to help in implementing some of these (or other)
suggestions.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/10132#comment:6>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
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