Allow vector_integrate to handle ImplicitRegion objects

[friyaz]

References to other Issues or PRs

#19320

Brief description of what is fixed or changed

The purpose of this PR is to modify the vector_integrate method to directly integrate over ImplicitRegion objects.

Example:

>>> circle = ImplicitRegion((x, y), x**2 + y**2 - 4)
>>> parametric_region_list(circle)
[ParametricRegion((2*sin(2*t), -2*cos(2*t)), (t, 0, 2*pi))]
>>> vector_integrate(1, circle)
8*pi

Other comments

Release Notes

• vector
  • vector_integrate can integrate over ImplicitRegion objects.

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• vector
  • vector_integrate can integrate over ImplicitRegion objects. (#19883 by @friyaz)

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#### References to other Issues or PRs
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#19320

#### Brief description of what is fixed or changed
The purpose of this PR is to modify the `vector_integrate` method to directly integrate over `ImplicitRegion` objects.

Example:
```python
>>> circle = ImplicitRegion((x, y), x**2 + y**2 - 4)
>>> parametric_region_list(circle)
[ParametricRegion((2*sin(2*t), -2*cos(2*t)), (t, 0, 2*pi))]
>>> vector_integrate(1, circle)
8*pi
```
#### Other comments

#### Release Notes
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<!-- BEGIN RELEASE NOTES -->
* vector
    * `vector_integrate` can integrate over `ImplicitRegion` objects.
<!-- END RELEASE NOTES -->

Update

The release notes on the wiki have been updated.

[friyaz]

In #19807, the rational_parametrization method was implemented to obtain a rational parametric representation of an implicit region. In most cases, the integrate method fails to integrate over the region if it is defined in terms of its rational parametric representation.

>>> circle = ImplicitRegion((x, y), x**2 + (y - 2)**2 - 4) 
>>> definition = circle.rational_parametrization()
>>> definiton
(4*t/(t**2 + 1), 4*t**2/(t**2 + 1))
>>>
>>> p = ParametricRegion(definiton, (t, 0, tan(2*pi)))
>>> vector_integrate(1, p)
4*Integral(sqrt(t**4/(t**8 + 4*t**6 + 6*t**4 + 4*t**2 + 1) + 2*t**2/(t**8 + 4*t**6 + 6*t**4 + 4*t**2 + 1) + 1/(t**8 + 4*t**6 + 6*t**4 + 4*t**2 + 1)), (t, 0, tan(2*pi)))

If t is replaced by tan(t), the parametric representation gets simplified (2sin(2 * t), 4sin(t)**2) and integrate is able to get the result.

>>> p2 = ParametricRegion((2*sin(2*t), 4*sin(t)**2), (t, 0, 2*pi))
>>> vector_integrate(1, p2)
8*pi

In the current implementation, every parameter is replaced by its tan. I think there should be flag like trig_rep=True to enable/ disable replacing with tan and use rational parametrization instead.

But replacing with tan does not always work. Consider the simple case of a sphere:

>>> sphere = ImplicitRegion((x, y, z), x**2 + y**2 + z**2 - 16)
>>> sphere.rational_parametrization()
(-4 + 8/(s**2 + t**2 + 1), 8*s/(s**2 + t**2 + 1), 8*t/(s**2 + t**2 + 1))
>>> parametric_region_list(sphere)
[ParametricRegion((-4 + 8/(tan(s)**2 + cos(t)**(-2)), 8*tan(s)/(tan(s)**2 + cos(t)**(-2)), 8*tan(t)/(tan(s)**2 + cos(t)**(-2))), (t, 0, 2*pi), (s, 0, 2*pi))]
>>> vector_integrate(1, sphere)
# Gets stuck

[friyaz]

How do we determine the bounds of the parameter? For closed regions like circle or ellipse, the default bounds for a parameter t can be (tan(t), 0, 2*pi) to represent the entire region. But for regions like parabola, I am not sure about the default bounds.

>>> parabola = ImplicitRegion((x,y), y**2 - 4*x)
>>> parametric_region_list(parabola)
[ParametricRegion((4/tan(t)**2, 4/tan(t)), (t, 0, 2*pi))]
>>> vector_integrate(1, parabola)
# Gets stuck

[codecov]

Codecov Report

Merging #19883 into master will decrease coverage by 3.321%.
The diff coverage is 100.000%.

@@              Coverage Diff              @@
##            master    #19883       +/-   ##
=============================================
- Coverage   75.704%   72.382%   -3.322%     
=============================================
  Files          667       670        +3     
  Lines       172829    173686      +857     
  Branches     40737     40976      +239     
=============================================
- Hits        130839    125718     -5121     
- Misses       36261     42039     +5778     
- Partials      5729      5929      +200     

[Upabjojr]

But replacing with tan does not always work. Consider the simple case of a sphere:

These are bugs of the integration algorithm. I wouldn't care if SymPy is unable to evaluate the integral, that's a bug in sympy.integrals... you should only care about the correctness of sympy.vector.

There is a lot of work to do in sympy.integrals, so don't worry if SymPy is unable to evaluate your integral.

[Upabjojr]

How do we determine the bounds of the parameter? For closed regions like circle or ellipse, the default bounds for a parameter t can be (tan(t), 0, 2*pi) to represent the entire region. But for regions like parabola, I am not sure about the default bounds.

Have you tried the equation solver on the coordinates of the reference point?

[friyaz]

Have you tried the equation solver on the coordinates of the reference point?

Can you expand on this? Solving for the parameter will give either the lower/upper bound. We need to find one more point.

[friyaz]

These are bugs of the integration algorithm. I wouldn't care if SymPy is unable to evaluate the integral, that's a bug in sympy.integrals... you should only care about the correctness of sympy.vector.

So should we parameters with their tan or leave it as it is? I think we should use a flag subs_tan default to true to do this.

[Upabjojr]

Ok, let's keep the trigsimp it as it is for now.

[Upabjojr]

@friyaz I would say... let's make this work somehow and then merge it. I think at this point of GSoC it would be better to focus on writing some detailed documentation with examples of how to apply the newly created code to physics and/or engineering.

[friyaz]

@Upabjojr I agree. My classes have started so I have not been able to give enough time to this.

[Upabjojr]

@Upabjojr I agree. My classes have started so I have not been able to give enough time to this.

Not it's important to write the documentation, otherwise people will not realize that your code has been added. I mean, without documentation it's likely that the code might be forgotten in the future.

[friyaz]

Not it's important to write the documentation, otherwise people will not realize that your code has been added. I mean, without documentation it's likely that the code might be forgotten in the future.

Yes, I was referring to the documentation itself. I will try to come up with a draft by sunday.

[czgdp1807]

@friyaz Any news on this?

[friyaz]

Currently, I am working on adding some examples for the rest of my GSoC project. I think this PR is almost ready. I will make this ready once my work on examples is complete.

[friyaz]

I have decided not to add trigsimp flag. It is useless as of now. integrate is unable to work on the rational parametric form generated from rational_parametrization.

I think we can merge this PR. With this PR, vector_integrate can work on simple implicit regions like circle. I have not added many tests as vector_integrate method is too slow for most cases I will open an issue to list all the problems which occur due integrate method.

[Upabjojr]

I have decided not to add trigsimp flag. It is useless as of now.

OK, let's focus on having everything merged before the end of GSoC.