I had a few discussions after Christopher Smith's comments, which I believe could be useful to share:
- The Area Method uses constructive geometry, so it might be beneficial to provide a tool that can draw actual diagrams, in addition to solving the geometry problems (for example, the ability to draw random points, lines, circles). - There were discussions between myself and others regarding the choice between the Wu Method and the Area Method. However, this ultimately led to the conclusion that the Area Method should be used due to the poor performance of sparse multivariate polynomials at that time. (Intuitively, we considered using separate x and y symbols for every point in the geometry.) - Maybe things have improved by 2024, but I'm not 100% sure. Nonetheless, the performance of sparse multivariate polynomials could clearly contribute to improvements in SymPy as well. - I also note that the performance of the 'cancel' function and algebraic number operations could be improved. I believe that many of the datasets from the book on the Area Method could also be useful for providing test cases for these issues. Some geometry problems took dozens of seconds to compute or were not feasible to implement due to performance bottlenecks with algebraic numbers. - I note that the debate between using geometry invariants (like the Area Method) and polynomials (like the Wu Method or Gröbner basis) has not yet been concluded. The problem is that geometric identities lead to very inefficient representations of polynomials (think of writing a matrix determinant for every triangle, which is obviously inefficient). This has motivated the use of a symbolic system that represents the determinant expression directly and efficiently. - I believe that research on this topic was advanced by people like Hongbo Lee (Invariant Algebras and Geometry Reasoning), but I haven't fully read his work yet. It could offer improvements. - I also note that the Area Method could potentially be implemented with Galgebra due to its relevance, but I previously skipped this because Galgebra was more focused on geometric analysis (such as computing derivatives), which was not relevant to the topic. However, I think suggestions from the Galgebra community could be valuable here. - I also had access to a full version of the Area Method (including Pythagorean identities) from my previous employer. While it may not be possible to share the code, it's fairly easy to figure out from the original paper I cited. On Saturday, August 24, 2024 at 1:30:18 AM UTC+2 smi...@gmail.com wrote: > Area method: https://github.com/sympy/sympy/issues/22160 > > Angle method: https://github.com/sympy/sympy/issues/22644 > > /c > > On Friday, August 23, 2024 at 6:24:36 PM UTC-5 Oscar wrote: > >> On Fri, 23 Aug 2024 at 23:29, Sangyub Lee <syle...@gmail.com> wrote: >> > >> > To be more productive, SymPy itself can enhance its geometry >> capabilities by incorporating Wu's method or a deductive database approach, >> > which are useful in addressing geometric challenges. >> > I’ve also shared the implementation of the Area method with SymPy, >> which can be searched. >> >> I know that you have said that this can be searched but can you share >> a link to what you are referring to? >> >> I think that would be useful to others reading this. >> >> -- >> Oscar >> > -- You received this message because you are subscribed to the Google Groups "sympy" group. To unsubscribe from this group and stop receiving emails from it, send an email to sympy+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/sympy/0d3d94ac-b3f9-48b1-bbea-512ce4ce1f22n%40googlegroups.com.
