Week 9

This week, I spent most of my time reading about algorithms to parametrize algebraic curves and surfaces.

Parametrization of algebraic curves and surfaces

In many cases, it is easiar to define a region using an implicit equation over its parametric form. For example, a sphere can be defined with the implict eqution x² + y² + z² = 4. Its parametric equation although easy to determine are tedious to write.

To integrate over implicitly defined regions, it is necessary to determine its parametric representation. I found the report on conversion methods between parametric and implicit curves from Christopher M. Hoffmann very useful. It lists several algorithms for curves and surfaces of different nature.

Every plane parametric curve can be expressed as an implicit curve. Some, but not all implicit curves can be expressed as parametric curves. Similarly, we can state of algebraic surfaces. Every plane parametric surface can be expressed as an implicit surface. Some, but not all implicit surfaces can be expressed as parametric surfaces.

Conic sections

One of the algorithm to parametrize conics is given below:

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

The difficult part is to find a point on the conic. For the first implementation, I just iterated over a few points and selected the point which satisfied the equation. I need to implement an algorithm that can find a point on the curve.

The parametric equations determined by the algoithm are large and complex expressions. The vector_integrate function in most cases is taking too much time to calculate the integral. I tried fixing this using simplification functions in SymPy like trigsimp and expand_trig. The problem still persist.

>>> 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)
### It gets stuck

The algorithm does not work for curves of higher algebraic degree like cubics.

Monoids

A monoid is an algebraic curve of degree n that has a point of multiplicity n -1. All conics and singular cubics are monoids. Parametrizing a monoid is easy if the special point is known.

To parametrize a monoid, we need to determine a singular point of n - 1 multiplicity. The singular points are those points on the curve where both partial derivatives vanish. A singular point (xo, yo) of a Curve C is said to be of multiplicity n if all the partial derivatives off to order n - 1 vanish there. This paper describes an algorithm to calculate such a point but the algorithm is not trivial.

Next week’s goal

My goal is to complete the work on parametrizing conics. If I can find a simple algorithm to determine the singular point of the required multiplicity, I will start working on it.

My goal for this week was to add support of integration over objects of geometry module.

Integrating over objects of geometry module

In my GSoC proposal, I mentioned implementing classes to represent commonly used regions. It will allow a user to easily define regions without bothering about its parametric representation. SymPy’s geometry module has classes to represent commonly used geometric entities like line, circle, and parabola. Francesco told me it would be better to add support to use these classes instead. The integral takes place over the boundary of the geometric entity, not over the area or volume enclosed.

My first approach was to add a function parametric_region to the classes of geometry module. Francesco suggested not to modify the geometry module as this would make it dependent on vector module. We decided to implement a function parametric_region to return a ParametricRegion of objects of geometry module.

I learned about the singledispatch decorater of python. It is used to create overloaded functions.
Polygons cannot be represented as a single ParametricRegion object. Each side of a polygon has its separate parametric representation. To resolve this, we decided to return a list of ParametricRegion objects.

My next step was to modify vector_integrate to support objects of geometry module. This was easy and involved calling the parametric_region_list function and integrating each ParametricRegion object.

The PR for the work has been merged. Note that we still do not have any way to create regions like disc and cone without their parametric representation. To support such regions, I think we need to add new classes to the geometry module.

Next week’s goal

I plan to work on implementing a class to represent implicitly defined regions.

The first phase of GSoC is over. We can now integrate the scalar or vector field over a parametrically defined region.

The PR for ParametricIntegral class has been merged. In my last post, I mentioned about a weird error. It turns out that I was importing pi as a symbol instead of as a number. Due to this, the expressions were returned unevaluated causing tests to fail.

The route ahead

I had a detailed discussion with Francesco about the route ahead. My plan has drifted from my proposal because we have decided not to implement new classes for special regions like circles and spheres. The geometry module already has classes to represent some of these regions. We have decided to use these classes.

Francesco reminded me that we have to complete the documentation of these changes. I was planning to complete the documentation in the last phase. But I will try completing it soon as I may forget some details.

We also discussed adding support to perform integral transformation using Stoke’s and Green’s theorem. In many cases, such transformation can be useful but adding them may be outside the scope of this project.

I have convinced Francesco on adding classes to represent implicitly defined regions. It might be the most difficult part of the project but hopefully, it will be useful to SymPy users.

Next week’s goal

My next week’s goal is to make a PR for adding suppor to integrate over objects of the geometry module.

Key highlights of this week’s work are:

• Fixed incorrect limit evaluation related to LambertW function

  • This was a minor bug fix. We added the _singularities feature to the LambertW function so that its limit gets evaluated using the meromorphic check already present in the limit codebase.
    
    

• Fixed errors in assumptions when rewriting RisingFactorial / FallingFactorial as gamma or factorial

  • This was a long pending issue. The rewrite to gamma or factorial methods of RisingFactorial and FallingFactorial did not handle all the possible cases, which caused errors in some evaluations. Thus, we decided to come up with a proper rewrite using Piecewise which accordingly returned the correct rewrite depending on the assumptions on the variables. Handling such rewrites using Piecewise is never easy, and thus there were a lot of failing testcases. After spending a lot of time debugging and fixing each failing testcase, we were finally able to merge this.

This marks the end of Phase-2 of the program. I learnt a lot during these two months and gained many important things from my mentors.

This blog describes the 7th week of the program and the 3rd week of Phase 2. Some of the key highlights on the discussions and the implementations during this week are:

Key highlights of this week’s work are:

• Improved the limit evaluations of Power objects

  • This PR improves the limit evaluations of Power objects. We first check if the limit expression is a Power object and then accordingly evaluate the limit depending on different cases. First of all, we express b**e in the form of exp(e*log(b)). After this, we check if e*log(b) is meromorphic and accordingly evaluate the final result. This check helps us to handle the trivial cases in the beginning itself.

    Now, if e*log(b) is not meromorphic, then we separately evaluate the limit of the base and the exponent. This helps us to determine the indeterminant form of the limit expression if present. As we know, there are 3 indeterminate forms corresponding to power objects: 0**0, oo**0, and 1**oo, which need to be handled carefully. If there is no indeterminate form present, then no further evaluations are required. Otherwise, we handle all the three cases separately and correctly evaluate the final result.

    We also added some code to improve the evaluation of limits having Abs() expressions. For every Abs() term present in the limit expression, we replace it simply by its argument or the negative of its argument, depending on whether the value of the limit of the argument is greater than zero or less than zero for the given limit variable.

    Finally, we were able to merge this after resolving some failing testcases.
    
    

• Fixed limit evaluations involving error functions

  • The incorrect limit evaluations of error functions were mainly because the tractable rewrite was wrong and did not handle all the possible cases. For a proper rewrite, it was required that the limit variable be passed to the corresponding rewrite method. This is because, to define a correct rewrite we had to evaluate the limit of the argument of the error function, for the passed limit variable. Thus, we added a default argument limitvar to all the tractable rewrite methods and resolved this issue. While debugging, we also noticed that the _eval_as_leading_term method of error function was wrong, hence it was also fixed.

    Finally, we were able to merge this after resolving some failing testcases.

This blog describes the 6th week of the official program and the 2nd week of Phase 2. By the end of this week, Compound Distributions framework is ready as targeted and I would now focus on the Joint Distributions in the upcoming weeks of this Phase.

This blog post describes my progress in Week 5 of Google Summer of Code 2020!

I ended up Week 4 by adding unit tests and a rough draft for Series and Parallel classes. Now in this week, to complete the implementation, we decided to add another...

Key highlights of this week’s work are:

• Fixed RecursionError and Timeout in limit evaluations

  • The Recursion Errors in limit evaluations were mainly due to the fact that the indeterminant form of 1**oo was not handled accurately in the mrv() function of the Gruntz algorithm. So, some minor changes were required to fix those.

    The major issue was to handle those cases which were timing out. On deep digging, we identified that the cancel() function of polytools.py was the reason. Thus, we decided to completely transform the cancel() function to speed up its algorithm. Now after this major modification, many testcases were failing as the cancel() function plays an important role in simplifying evaluations and is thus used at many places across the codebase. Therefore, a lot of time was spent in debugging and rectifying these testcases.

    Finally we were able to merge this, enhancing the limit evaluation capabilities of SymPy.

This blogs describes the week 5, the beginning week of the Phase 2. Phase 2 will be mostly focused on Compound Distributions which were stalled from 2018, and additions to Joint Distributions.

With this, the fourth week and phase 1 of GSoC 2020 is over. Here I will give you a brief summary of my progress this week.

The initial days were spent mostly on modifying unit tests for Series and Parallel classes which I added in the...

I spent this week working on the implementation of the ParametricIntegral class.

Modifying API of ParametricRegion

When I was writing the test cases, I realized that the API of the ParametricRegion could be improved. Instead of passing limits as a dictionary, tuples can be used. So I modified the API of the ParametricRegion class. The new API is closer to the API of the integral class and more intuitive. I made a separate PR for this change to make it easy for reviewers.

Example of the new API:

ParametricRegion( ( x+y, x*y ), (x, 0, 2), (y, 0, 2))

Handling scalar fields with no base scalars

As discussed in previous posts, we decided to not associate a coordinate system with the parametric region. Instead, we assume that the parametricregion is defined in the coordinate system of the field of which the integral is being calculated. We calculate the position vector and normal vector of the parametric region using the base scalars and vectors of the fields. This works fine for most cases. But when the field does not contain any base scalar or vector in its expression, we cannot extract the coordinate system from the field.

ParametricIntegral(150, circle)
# We cannot determine the coordinate system from the field. 
# To calculate the line integral, we need to find the derivative of the position vector.
# This is not possible until we know the base vector of the field

To handle this situation, I assign a coordinate system C to the region. This does not affect the result in any way as the result is independent of it. It just allows the existing algorithm to work in this case.

Separate class for vector and scalar fields

Francesco suggested making separate classes based on the nature of the field: vector and scalar. I am open to this idea. But I think it will be more easy and intuitive for the users if they can use the same class to calculate the integral. I do not think they are different enough from a user’s perspective to have a separate interface.

Maybe we can have a function vectorintegrate which returns the object of ParametricVectorIntegral or ParametricScalarIntegral depending on the nature of the field. This can work for other types of integrals too. Suppose we implement a class ImplicitIntegral to calculate the integral over an implicit region. The vectorintegrate function can then return an object of ImplicitIntegral object by identifying the region is defined implicitly. I think this will be great. I will have more discussion with Francesco on this aspect.

Topological sort of parameters

When evaluating double integral, the result some times depend upon the order in which the integral is evaluated. If the bounds of one parameter u depend on another parameter v, we should integrate first with respect to u and then v.

For example, consider the problem of evaluating the area of the triangle.

T = ParametricRegion((x, y), (x, 0, 2), (y, 10 - 5*x))

The area of the triangle is 10 units and should be independent of the order parameters are specified at the time of object initialization. But the double integral depends on the order of integration.

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

So parameters must be passed to integrate in the correct order. To overcome this issue, we topologically sort the parameters. SymPy already had a function to perform topologically sort in its utilities module. I implemented a function that generates the graph and passes it to the topological_sort function. This made my work easy.

Long computation time of Integrals

Some integrals are taking too long to compute. When base scalars in the field are replaced by their parametric equivalents, the expression of the field becomes large. Further, the integrand is the dot product of the field with a vector or product of the field and the magnitude of the vector. The integrate function takes about 20-30 seconds to calculate the integral. I think this behavior is due to the expression of integrand growing structurally despite it being simple.

For example,

>>>solidsphere = ParametricRegion((r*sin(phi)*cos(theta), r*sin(phi)*sin(theta), r*cos(phi)),\
                                (r, 0, 2), (theta, 0, 2*pi), (phi, 0, pi))
>>> ParametricIntegral(C.x**2 + C.y**2, solidsphere)

In this case, the parametric field when replaced with parametersr become r**2*sin(phi)**2*sin(theta)**2 + r**2*sin(phi)**2*cos(theta)**2 although it can be easily simplified to r**2*sin(phi).

SymPy has a function called simplify. simplify attempts to apply various methods to simplify an expression. When the integrand is simplified using it before passing to integrate, the result is returned almost immediately. Francesco rightly pointed out that simplify is computationally expensive and we can try to some specific simplification. I will look into it.

Failing test cases

Some test cases are failing because of integrate function returning results in different forms. The results are mathematically equivalent but different in terms of structure. I found this strange. I do not think this has to do with hashing. I still have not figured out this problem.

Next week’s goal

Hopefully, I will complete the work on the ParamatricIntegral and get it merged. We can then start discussing about representing implicit regions.

Part of this week was spent in getting the PR for ParametricRegion merged. I made some fixes to get the failing tests to pass. I also added the docstring.

ParametricIntegral

I started working on the ParametricIntegral class. ParametricIntegral class returns the integral of a scalar or vector field over a region defined parametrically. It should be able to calculate line, surface, or volume integral depending on the nature of the parametric region. To identify the nature of the region, the dimensions property needs to be added in the ParametricRegion class.

I started with writing the test cases to avoid my previous mistake of working on the implementation before deciding the API. The API of the ParametricIntegral class:

ParametricIntegral(field, parametricregion)

Some examples:

>>> curve = ParametricRegion((3*t - 2, t + 1), {t: (1, 2)})
>>> ParametricIntegral(C.x, curve)
5*sqrt(10)/2
>>> semisphere = ParametricRegion((theta, pi), (2*sin(phi)*cos(theta), 2*sin(phi)*sin(theta), 2*cos(phi)),\
                            {theta: (0, 2*pi), phi: (0, pi/2)})
>>> ParametricIntegral(C.z, semisphere)
8*pi

I initially thought to make it a subclass of Integral as the ParametricIntegral is a kind of integral. But there was no reuse of any code from Integral in the ParametricIntegral so decided against it.
A rough sketch of the algorithm to calculate integral is described below:

1. Replace the base scalars in the field with their definition in terms of parameters and obtain the parametric representation of the field.
2. Form the position vector of the region using definition tuple and base vectors determined from the field.
3. Depending on the type of region, calculate the derivative of the position vector, or find the normal vector or calculate dV using Jacobian.
4. Calculate the integrand using dot product or multiplication with the parametric field.
5. Use the integrate function to calculate the integral.
6. If the integrate function can evaluate the integral, return it otherwise return an object of the ParametricRegion class.

We also had discussion on whether swapping the upper and lower limit has any affect on the result of a vector integral. For multiple integrals, the sign of the result changes when the upper and lower bound of a parameter are interchanged. I do not think we came to any conclusion. But this won’t matter for integrals over parametric regions as the bounds are directly defined by the user.

Next week’s goal

I will start working on the implementation of the ParametricIntegral class. I am also trying to get more involved in SymPy outside of my GSoC project. I hope I can become a contributing member of the community.