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[WIP] Adding ParametricIntegral class #19539

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merged 8 commits into from Jun 25, 2020
Merged

[WIP] Adding ParametricIntegral class #19539

merged 8 commits into from Jun 25, 2020

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friyaz
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@friyaz friyaz commented Jun 11, 2020

References to other Issues or PRs

#19320

Brief description of what is fixed or changed

An object of ParametricIntegral represents integral of a scalar or vector field over a Parametric Region.

Other comments

_bounds_case function is taken from @prasoon2211 PR with some modification. I have added him as a coauthor for the corresponding commit.

Release Notes

  • vector
    • added class to represent integral of scalar/vector field over a parametric surface.

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sympy-bot commented Jun 11, 2020

Hi, I am the SymPy bot (v160). I'm here to help you write a release notes entry. Please read the guide on how to write release notes.

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  • vector
    • added class to represent integral of scalar/vector field over a parametric surface. (#19539 by @friyaz)

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#19320 

#### Brief description of what is fixed or changed
An object of ParametricIntegral represents integral of a scalar or vector field over a Parametric Region.

#### Other comments
`_bounds_case` function is taken from @prasoon2211  [PR](https://github.com/sympy/sympy/pull/2208/files) with some modification. I have added him as a coauthor for the corresponding commit. 
#### Release Notes

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* vector
    * added class to represent integral of scalar/vector field over a parametric surface.
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The release notes on the wiki have been updated.

@friyaz friyaz marked this pull request as draft Jun 11, 2020
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    • sympy/vector/integrals.py
    • sympy/vector/tests/test_integrals.py

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if len(coord_sys) > 1:
raise ValueError

coord_sys = next(iter(coord_sys))
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@friyaz friyaz Jun 13, 2020

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Since no coordinate system is associated with the parametric region, we use the base scalars of the field. But if the field does not have any basescalar or vector, we cannot derive coordinate system. This breaks the logic

Example:

>>> p = ParametricRegion(t, (3*t - 2, t + 1), {t: (1, 2)}).
>>> I = ParametricIntegral(C.x + C.y, p)
>>> I.eval()
5*sqrt(10)
>>>
>>> I = ParametricIntegral(1, p) ##We cannot extract coordinate system from 1
StopIteration

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@sylee957 sylee957 Jun 14, 2020

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My opinion is that we should equip base scalars and base vector (fields) to the ParametricRegion if we want to use its intrinsic scalars and vector field components.
I don't think that it makes much sense to have vector integral like ParametricIntegral(C.x*C.i, p) where C.x is linked to the base scalars of the parametric region and C.i is linked to embedding cartesian coordinate system.

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@friyaz friyaz Jun 14, 2020

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Agreed. But if we allow the coordsys parameter to be optional and None by default in ParametricRegion class, a similar situation can arise.

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@friyaz friyaz Jun 15, 2020

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@Upabjojr, your thoughts on this?

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@Upabjojr Upabjojr Jun 15, 2020

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Probably it's better to separate vector and parametric integrals. At least for now.

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@friyaz friyaz Jun 23, 2020

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The point of having subclasses is that the result of the vector integration may be a vector

I do not see how this is possible. Can you give an example?

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@friyaz friyaz Jun 24, 2020

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I think this is possible if the user want to perform cross product of vector field with normal vector instead of the usual dot product.

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@Upabjojr Upabjojr Jun 24, 2020

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Well, integrating velocity over time is a simple example. Though it's probably already handled by the current Integral class. But you see, this case could be a case where you may want to subclass Integral in order to supply it with the vector methods.

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@Upabjojr Upabjojr Jun 24, 2020

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But indeed, probably we don't need this case, it's pretty rare that some users may need to use it.

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@friyaz friyaz Jun 25, 2020

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Although it is a kind of vector integral, it is not integrating a vector over a region. As you said, the integrate function is able to handle such kind of integrals (integral over variables). But if we wish to handle such a case, we can allow passing a tuple of variables in place of the parametric region.

@friyaz friyaz requested a review from Upabjojr Jun 15, 2020
(v, lower_v, upper_v))
else:
result = integrate(parametricfield*normal_vector.magnitude(), (u, lower_u, upper_u), (v, lower_v, upper_v))

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@friyaz friyaz Jun 15, 2020

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Changing the order of parameters cuases Integral to fail or get slow.

>>> field = sqrt(C.x**2 + C.y**2)
>>> cone1 = ParametricRegion((u,v), ((2-2*u/3)*cos(v), (2-2*u/3)*sin(v), u), {u: (0, 3), v: (0, 2*pi)})
>>> cone2 = ParametricRegion((u,v), ((2-2*u/3)*cos(v), (2-2*u/3)*sin(v), u), {u: (0, 3), v: (0, 2*pi)})
>>> ParametricIntegral(field, cone1)
# It gets stuck
>>> ParametricIntegral(field, cone2)
8*sqrt(13)*pi/3

I think this is because normal_vector and normal_vector.magnitude becomes a large expression and Integral is not able to work on it. Using simplify on normal_vector.magnitude solves the problem.

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@Upabjojr Upabjojr Jun 19, 2020

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simplify is very expensive in computational terms. What about some more specific functions (e.g. expand)?

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@friyaz friyaz Jun 19, 2020

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@Upabjojr It gets slow if I use expand.

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@friyaz friyaz Jun 19, 2020

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I think Integrate function should be able to handle or perform such simplification instead of simplifying them here.

# Case 1 : Both bounds are constants : can perform integration in any order
# Case 2 : Bounds for u are in terms of v : integrate wrt to u first
# Case 3 : Bounds for v are in terms of u : integrate wrt to v first

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@friyaz friyaz Jun 15, 2020

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Determining order for integration is necessary when bounds of parameters are interdependent.
Suppose we need to calculate area of a triangle:

>>> integrate(1, (x, 0, 2), (y, 10 - 5*x))
20 - 10*x
>>> integrate(1, (y, 0, 10 - 5*x), (x, 0, 2))
10 # correct answer

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@Upabjojr Upabjojr Jun 15, 2020

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Topological sort?

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@friyaz friyaz Jun 16, 2020

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Yes, we need to perform a topological sort.
This function generates the graph and performs topological sort. It works on any number of parameters.

    @classmethod             
    def _topological_sort_parameters(cls, parameters, limits):
    
        V = list(parameters)
        E = list()
        
        for p in parameters:
            lower_p = limits[p][0]
            upper_p = limits[p][1]
            
            lower_p = lower_p.atoms()
            upper_p = upper_p.atoms()
            for q in parameters:
                if p == q:
                    continue
                if lower_p.issuperset(set([q])) or upper_p.issuperset(set([q])):
                    E.append((p, q))   
        return topological_sort((V, E), key=default_sort_key)   

The only problem with this approach is when bounds are independent of each other, we can perform integration in any order. So if one order fails to calculate the integral, we can reverse the order and perform again. With my previous approach using cases, we can determine whether bounds are independent of each other but it works only for two parameters.

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@Upabjojr Upabjojr Jun 17, 2020

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What about calling the equation solver? The equation solver should be able to find a parametrization somehow.


lower, upper = parametricregion.limits[parameter][0], parametricregion.limits[parameter][1]

if isinstance(parametricfield, Vector):
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@Upabjojr Upabjojr Jun 17, 2020

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Maybe this case should be moved to another class, e.g. ParametricVectorField?

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Upabjojr commented Jun 24, 2020

Tests are broken.

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friyaz commented Jun 25, 2020

@Upabjojr I have fixed them.
Also, I have added vector_integrate function. I will add the test cases for it in the next PR if it is decided to keep this function.

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codecov bot commented Jun 25, 2020

Codecov Report

Merging #19539 into master will decrease coverage by 0.018%.
The diff coverage is 84.810%.

@@              Coverage Diff              @@
##            master    #19539       +/-   ##
=============================================
- Coverage   75.675%   75.656%   -0.019%     
=============================================
  Files          654       658        +4     
  Lines       169943    170778      +835     
  Branches     40064     40289      +225     
=============================================
+ Hits        128605    129205      +600     
- Misses       35721     35917      +196     
- Partials      5617      5656       +39     

@friyaz friyaz marked this pull request as ready for review Jun 25, 2020
@Upabjojr Upabjojr merged commit 200788d into sympy:master Jun 25, 2020
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Upabjojr commented Jun 25, 2020

It works for me. Thanks.

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4 participants