Adding classes to represent implicitly defined regions #19681

friyaz · 2020-07-01T13:06:49Z

References to other Issues or PRs

#19320

Brief description of what is fixed or changed

The vector module has support to define regions using parametric representation and integrate over them.
In some cases, It is easier to work to define regions using their implicit equation. This Pull Request aims to add classes to represent implicitly defined regions.

Other comments

This PR is a draft and the structure of the PR is not fixed.

Release Notes

vector
- Added support to create implictly defined regions.

friyaz · 2020-07-01T13:09:59Z

Algorithm to determine the parametric representation of a Conic:

Fix a point p on the conic. Consider the pencil of lines through p. Formulate the line equations.
Substitute for y in the conic equation, solve for x(t).
Use the fine equations to determine y(t).

sympy-bot · 2020-07-01T13:10:20Z

✅

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.

Your release notes are in good order.

Here is what the release notes will look like:

vector
- Added support to create implictly defined regions. (#19681 by @friyaz)

This will be added to https://github.com/sympy/sympy/wiki/Release-Notes-for-1.7.

Note: This comment will be updated with the latest check if you edit the pull request. You need to reload the page to see it.

Click here to see the pull request description that was parsed.

<!-- Your title above should be a short description of what
was changed. Do not include the issue number in the title. -->

#### References to other Issues or PRs
<!-- If this pull request fixes an issue, write "Fixes #NNNN" in that exact
format, e.g. "Fixes #1234" (see
https://tinyurl.com/auto-closing for more information). Also, please
write a comment on that issue linking back to this pull request once it is
open. -->
#19320 

#### Brief description of what is fixed or changed
The vector module has support to define regions using parametric representation and integrate over them.
In some cases, It is easier to work to define regions using their implicit equation. This Pull Request aims to add classes to represent implicitly defined regions.

#### Other comments
This PR is a draft and the structure of the PR is not fixed. 
#### Release Notes

<!-- Write the release notes for this release below. See
https://github.com/sympy/sympy/wiki/Writing-Release-Notes for more information
on how to write release notes. The bot will check your release notes
automatically to see if they are formatted correctly. -->

<!-- BEGIN RELEASE NOTES -->
* vector
    * Added support to create implictly defined regions.
<!-- END RELEASE NOTES -->

Update

The release notes on the wiki have been updated.

sympy-bot · 2020-07-01T13:10:22Z

🟠

Hi, I am the SymPy bot (v160). I've noticed that some of your commits add or delete files. Since this is sometimes done unintentionally, I wanted to alert you about it.

This is an experimental feature of SymPy Bot. If you have any feedback on it, please comment at sympy/sympy-bot#75.

The following commits add new files:

4e5ddc0:
- sympy/vector/implicitregion.py
1ba0aca:
- sympy/vector/tests/test_implicitregion.py

If these files were added/deleted on purpose, you can ignore this message.

friyaz · 2020-07-01T13:12:26Z

sympy/vector/implicitregion.py

+
+    # Finding  a point on the Curve.
+    # Needs to find a better way than random looping.
+    for i in range(100):


I need to find a better way to find a point on the conic.

have you thought about some deterministic-iterative algorithm?

If you cannot solve the equation... maybe you are at a singular point, right? Can you determine singular points from the gradient of the implicit equation? Maybe thinking of example equations where this case happens, may help think about a solution.

I am working on it. I think I will implement a general algorithm for monoids which will work for all conics.

Found on an algorithm for determining a point on the conic:
https://www3.risc.jku.at/publications/download/risc_1355/Rational%20Points%20on%20Conics.pdf.

@Upabjojr the algorithm is non-trivial. Is it worth it to implement? Even parametrizing conics seems to be a tough task,

Another option is to make the user input a point on the curve apart from implicit equation.

An example:

C = ImplicitCurve(Eq(x**2 + y**2 - 4)) C.parametrization(Point(0, 2)) #Takes a point on a curve (2*cos(theta), 2*sin(theta))

@Upabjojr Any thoughts?

Suggesting the point is fine with me, especially if it makes the code simpler to implement.

We can work with this and maybe later implement the algorithm for finding a rational point on the curve.

Of course... always start with a simple algorithm. We can open as many pull requests as we like, so no need to do it now.

Thanks. I will complete this work then. This point will be optional. If not given, the algorithm will iterate over some point. If it is unable to find a point on the curve, it raises NotImplementedError.

friyaz · 2020-07-01T13:15:52Z

A better way is to use an algorithm to parameterize a monoid. A monoid is an algebraic curve of degree n that has a point of multiplicity n -1. All conics and singular conics are monoids. But this requires determining the (n -1) fold point. I currently do not know how to determine this.

friyaz · 2020-07-03T18:13:08Z

ping @Upabjojr

sympy/vector/implicitregion.py

…ioon defined imperically to ParametricRegion object

friyaz · 2020-07-13T21:28:44Z

@Upabjojr My concern is that the parametric representation determined by these algorithms are mostly in terms of large and complex expression. 'ParametricIntegralis not able to integrate over such regions due tointegrate` function being slow.
For example:

>>> ellipse = ImplicitRegion(x**2/4 + y**2/16 - 1, x, y)
>>> parametric_region_list(ellipse)
[ParametricRegion((8*tan(theta)/(tan(theta)**2 + 4), -4 + 8*tan(theta)**2/(tan(theta)**2 + 4)), (theta, 0, pi))]
>>> vector_integrate(2, ellipse)
Too slow

There is little benefit of adding functions to determine parametric representation of implicit region if integrate is not able to work on it.

friyaz · 2020-07-14T17:56:20Z

Sage has a function for rational parametrization of Curves. See rational_parametrization from https://github.com/sagemath/sage/blob/develop/src/sage/schemes/curves/affine_curve.py

But it requires the implementation of many other related classes.

Upabjojr · 2020-07-14T19:11:16Z

Sage has a function for rational parametrization of Curves. See rational_parametrization from https://github.com/sagemath/sage/blob/develop/src/sage/schemes/curves/affine_curve.py

But it requires the implementation of many other related classes.

Sage is GPL'd. We cannot reuse their code. Their approach to this problem is a lot more theoretical... they define lots of algebraic structures. Is this really the purpose of sympy.vector? Complicated algebraic structures will make the code useful only to mathematicians or very theoretical people, cutting off many SymPy users.

Upabjojr · 2020-07-14T19:12:59Z

There is little benefit of adding functions to determine parametric representation of implicit region if integrate is not able to work on it.

That is a bug of integrate... maybe someone will fix that in the future? My understanding is that integrate tries a list of fast algorithms, if all fail, it falls back to a very slow backup algorithm.

friyaz · 2020-07-14T19:35:52Z

Sage is GPL'd. We cannot reuse their code. Their approach to this problem is a lot more theoretical... they define lots of algebraic structures. Is this really the purpose of sympy.vector? Complicated algebraic structures will make the code useful only to mathematicians or very theoretical people, cutting off many SymPy users.

Probably something like this can become part of the geometry module. I had to email authors of a few research papers to understand some of these concepts. I highly doubt Mathematica integrates over implicit regions this way. It can handle many complex curves and surfaces including inequalities that most likely are not easily parametrizable.

That is a bug of integrate... maybe someone will fix that in the future? My understanding is that integrate tries a list of fast algorithms, if all fail, it falls back to a very slow backup algorithm.

I agree. I did not thought it in this way.

Upabjojr · 2020-07-14T20:04:36Z

Probably something like this can become part of the geometry module. I had to email authors of a few research papers to understand some of these concepts. I highly doubt Mathematica integrates over implicit regions this way. It can handle many complex curves and surfaces including inequalities that most likely are not easily parametrizable.

That's a nice research. In case you decide not to implement this algorithm, what about writing a page on SymPy's wiki?

friyaz · 2020-07-14T20:09:11Z

That's a nice research. In case you decide not to implement this algorithm, what about writing a page on SymPy's wiki?

OK, For either case, I will try to summarize what I have learned on parametrizing rational curves/surfaces.

friyaz · 2020-07-15T19:19:51Z

@Upabjojr I have implemented the algorithm for the parametrization of monoids. We can now determine the rational parametrization of all algebraic curves and surfaces of degree > 2 if a parametrization exists.
This algorithm also works for some cases of degree 2 curves/surfaces. To handle every case, proper implementation of regular_point is needed.

It is important to note that the algorithm returns rational parametrization so in some cases the result may not be convenient.
For example, In the case of a circle:

>>> c = ImplicitRegion((x, y), x**2 + y**2 - 16)
>>> c.rational_parametrization()
(-4 + 8/(t**2 + 1), 8*t/(t**2 + 1))

codecov · 2020-07-16T18:52:59Z

Codecov Report

Merging #19681 into master will decrease coverage by 0.004%.
The diff coverage is 86.776%.

@@              Coverage Diff              @@
##            master    #19681       +/-   ##
=============================================
- Coverage   75.709%   75.704%   -0.005%     
=============================================
  Files          659       664        +5     
  Lines       171401    172364      +963     
  Branches     40437     40652      +215     
=============================================
+ Hits        129766    130487      +721     
- Misses       35982     36164      +182     
- Partials      5653      5713       +60

Upabjojr · 2020-07-16T21:32:19Z

@Upabjojr I have implemented the algorithm for the parametrization of monoids. We can now determine the rational parametrization of all algebraic curves and surfaces of degree > 2 if a parametrization exists.
This algorithm also works for some cases of degree 2 curves/surfaces. To handle every case, proper implementation of regular_point is needed.

It is important to note that the algorithm returns rational parametrization so in some cases the result may not be convenient.
For example, In the case of a circle:
>>> c = ImplicitRegion((x, y), x**2 + y**2 - 16)
>>> c.rational_parametrization()
(-4 + 8/(t**2 + 1), 8*t/(t**2 + 1))

That's pretty nice.

friyaz · 2020-07-16T21:43:48Z

I will try to implement the algorithm for finding a regular point on a Conic in a separate PR. Maybe we can then get rid of reg_point parameter in rational_parametrization.

Upabjojr · 2020-07-16T21:51:38Z

sympy/vector/implicitregion.py

+
+        return m
+
+    def rational_parametrization(self, parameter1='t', parameter2='s', reg_point=None):


Is this supposed to work for 2D only? What about parameters = ('t', 's') instead?

It will work for curves and surfaces given that they are monoid.

The report which I followed had algorithms only for curves and surfaces. No Volume regions or higher dimensional region.

What about parameters = ('t', 's') instead?

This looks better. I will fix it.

Is this supposed to work for 2D only? What about parameters = ('t', 's') instead?

It does not work for 3-D curves. It works for 3-D surfaces but does not support higher-dimensional surfaces..

friyaz commented Jul 1, 2020

View reviewed changes

friyaz marked this pull request as draft July 1, 2020 13:16

friyaz added GSoC vector labels Jul 1, 2020

friyaz requested a review from Upabjojr July 1, 2020 20:09

Upabjojr reviewed Jul 4, 2020

View reviewed changes

sympy/vector/implicitregion.py Outdated Show resolved Hide resolved

friyaz requested a review from Upabjojr July 8, 2020 21:26

Upabjojr reviewed Jul 8, 2020

View reviewed changes

sympy/vector/implicitregion.py Outdated Show resolved Hide resolved

Upabjojr reviewed Jul 8, 2020

View reviewed changes

sympy/vector/implicitregion.py Outdated Show resolved Hide resolved

friyaz added 4 commits July 10, 2020 02:04

(Experiment) Adding ImplictConicSection class to convert a conic sect…

4e5ddc0

…ioon defined imperically to ParametricRegion object

Use solveset instead of solve in implicitregion.py

1411f02

Determine implicit point

6631b04

Use class for Implicit Region insteaad of function

736b3ae

friyaz force-pushed the implicitregion branch from 0485ae9 to 736b3ae Compare July 9, 2020 20:59

friyaz requested a review from Upabjojr July 9, 2020 21:00

Implemented Monoids parametrization algorithm

fd0cd31

Added tests and docstring for ImplicitRegion class

1ba0aca

friyaz marked this pull request as ready for review July 16, 2020 16:44

friyaz requested review from Upabjojr and removed request for Upabjojr July 16, 2020 16:49

Upabjojr reviewed Jul 16, 2020

View reviewed changes

Pass parameters as tuple rather than individual keywords

cae0b36

friyaz force-pushed the implicitregion branch from e39c360 to cae0b36 Compare July 17, 2020 13:39

friyaz changed the title ~~[WIP] Adding classes to represent implicitly defined regions~~ Adding classes to represent implicitly defined regions Jul 17, 2020

friyaz requested a review from Upabjojr July 17, 2020 15:57

Upabjojr merged commit 1b8a63b into sympy:master Jul 18, 2020

Provide feedback

Saved searches

Use saved searches to filter your results more quickly

Adding classes to represent implicitly defined regions #19681

Adding classes to represent implicitly defined regions #19681

friyaz commented Jul 1, 2020 •

edited

Loading

friyaz commented Jul 1, 2020

sympy-bot commented Jul 1, 2020 •

edited

Loading

sympy-bot commented Jul 1, 2020 •

edited

Loading

friyaz Jul 1, 2020

Upabjojr Jul 4, 2020

Upabjojr Jul 4, 2020

friyaz Jul 6, 2020

friyaz Jul 13, 2020

friyaz Jul 14, 2020

Upabjojr Jul 14, 2020

friyaz Jul 14, 2020

Upabjojr Jul 14, 2020

friyaz Jul 14, 2020

friyaz commented Jul 1, 2020

friyaz commented Jul 3, 2020

friyaz commented Jul 13, 2020

friyaz commented Jul 14, 2020

Upabjojr commented Jul 14, 2020

Upabjojr commented Jul 14, 2020

friyaz commented Jul 14, 2020 •

edited

Loading

Upabjojr commented Jul 14, 2020

friyaz commented Jul 14, 2020

friyaz commented Jul 15, 2020

codecov bot commented Jul 16, 2020 •

edited

Loading

Upabjojr commented Jul 16, 2020

friyaz commented Jul 16, 2020

Upabjojr Jul 16, 2020

friyaz Jul 16, 2020

friyaz Jul 16, 2020 •

edited

Loading

friyaz Jul 18, 2020


		return m

		def rational_parametrization(self, parameter1='t', parameter2='s', reg_point=None):

Adding classes to represent implicitly defined regions #19681

Adding classes to represent implicitly defined regions #19681

Conversation

friyaz commented Jul 1, 2020 • edited Loading

References to other Issues or PRs

Brief description of what is fixed or changed

Other comments

Release Notes

friyaz commented Jul 1, 2020

sympy-bot commented Jul 1, 2020 • edited Loading

sympy-bot commented Jul 1, 2020 • edited Loading

🟠

Choose a reason for hiding this comment

Choose a reason for hiding this comment

Choose a reason for hiding this comment

Choose a reason for hiding this comment

Choose a reason for hiding this comment

Choose a reason for hiding this comment

Choose a reason for hiding this comment

Choose a reason for hiding this comment

Choose a reason for hiding this comment

Choose a reason for hiding this comment

friyaz commented Jul 1, 2020

friyaz commented Jul 3, 2020

friyaz commented Jul 13, 2020

friyaz commented Jul 14, 2020

Upabjojr commented Jul 14, 2020

Upabjojr commented Jul 14, 2020

friyaz commented Jul 14, 2020 • edited Loading

Upabjojr commented Jul 14, 2020

friyaz commented Jul 14, 2020

friyaz commented Jul 15, 2020

codecov bot commented Jul 16, 2020 • edited Loading

Codecov Report

Upabjojr commented Jul 16, 2020

friyaz commented Jul 16, 2020

Choose a reason for hiding this comment

Choose a reason for hiding this comment

friyaz Jul 16, 2020 • edited Loading

Choose a reason for hiding this comment

Choose a reason for hiding this comment

friyaz commented Jul 1, 2020 •

edited

Loading

sympy-bot commented Jul 1, 2020 •

edited

Loading

sympy-bot commented Jul 1, 2020 •

edited

Loading

friyaz commented Jul 14, 2020 •

edited

Loading

codecov bot commented Jul 16, 2020 •

edited

Loading

friyaz Jul 16, 2020 •

edited

Loading