"Symbolic-Numeric Computations for Computer Graphics" is one of the projects of "KSS Projects for Communication", which is started since 2014. Please see also KSS Projects for Communication.
Please contact Kosaku Nagasaka if you have a question on this project.
We would like to have better discussions on Computer Graphics from a bit new viewpoints of Mathematics and Computer Science (especially symbolic-numeric computations). In this project, we build some preliminary implementations to have discussions in KSS seminar series with various specialists (especially with researchers). For instance, to organize discussions on fluid simulations from the viewpoint of symbolic algebraic computation, numerical computation with guaranteed accuracy, symbolic-numeric computations and so on, we may need to have common notations, languages (technical terms), implementations (basic framework) and so on. This is our aim. Therefore, this project is not so special. We just try to reexamine common technologies (standards) from a broader perspective. In fact, reviewing the results of the Studio Phones Fellowship Program is one of such trials.
Since Yusuke Kiriu (Studio Phones) in his film making, has been working with music, acoustics, clothing, image processing, furniture, architecture and 3D computer graphics, he has achieved some fundamental research results on computer which contributed in part to many researchers as the Studio Phones Fellowship Program. For example, his implementation covered some basics of hyperbolic geometry and topology (Mathematics), which includes symbolic computations can generate some corresponding data structure to design several algorithms of topological invariant for the given knot diagram drawn on your computer screen (e.g. drawing link diagram on Mathematica by hand). In this project, based on his fundamental research results, with a bit new viewpoints of Mathematics and Computer Science, we try to build new implementations being able to contribute to our long discussions in KSS seminar series.
Seeking an implementation to contribute to finding some simple idea or deepening the theory may need us to change the specification and confirm its proper functionality. For the present, we may reorganize the needs (notations and implementations for our discussions) and do the PDCA cycle in KSS seminar series. During this period, we will ask some of the participants (other than the team members) to examine our perspective from their diversified viewpoints regardless our specialities (other than symbolic-numeric computations, for example).
In this project, we are to reexamine and reorganize the fundamental research results (studied by Yusuke Kiriu) to build a new implementation to accomplish our goal (enriching the discussions in KSS). For instance, reviewing the results from symbolic-numeric computations and symbolic algebraic computation (other than simple symbolic operations) is one of the important investigations. In fact, the known study is based on general theories in symbolic computation and artificial intelligence, so that it was very impressive for the mathematicians at the Fellowship Program. However, it seemed to be difficult for them to expand or modify the implementation further. This is one of the most notable issues. Therefore, it has been difficult to deepen discussion between several sorts of specialists beyond their speciality, even with the implementation.
Our design of this project is to resolve the issues by reorganizing the known work at the level of the software architecture in computer graphics with the viewpoints of symbolic-numeric computations. It may be difficult to achieve the desired implementation but our development process will be meaningful to enrich the discussions in KSS seminar series. Moreover, as possible as we can, we would like to progres, especially from the academical viewpoint other than the practical viewpoint which was the most important one in the known work, and build it independently of the known work of Yusuke Kiriu. This is our long term goal.
For the present, we may not distribute the resulting preliminary implementations to the public since our goal is to contribute to deepening discussion between several sorts of specialists and making specialists involved in their own networking contacts regardless of their speciality. Basically, we practice it in Studio Phones and use it in KSS seminar series. However, in the future, we wish to distribute it (by BSD license, for example) if it becomes being fairly content with its maturity and performance.
This project in part is grounded on the contributions of the Studio Phones Fellowship Program. Moreover, themes achieved in the Program agree with the purpose of KSS and are useful and meaningful for the potential participants of KSS. We itemize the themes in the Program as follows.
My purpose in this fellowship program is to study the recent works of Yusuke Kiriu and Mikami Hirasawa on topology from a point of view of computer graphics, primarily focusing on the visual simulation of clouds.
The themes of my research are the following topics;
Bunraku, also known as Ningyō jōruri, is the traditional Japanese puppet theater founded in Osaka in 1684. In this 14 years, I worked at National Bunraku Theatre to maintain these puppets, and I retired. My main purpose is to discuss the possibilities of character animation in the context of computer graphics.
My purpose in this fellowship program is to assist the recent works of Yusuke Kiriu and Mikami Hirasawa on formal language and formalizations in mathematics, particularly from the view point of geometric structures and combinatorial structures of 3-manifolds.
I study a formalization of mathematics by means of logical methods. Key objects in my research area are mathematical theories, as naive set theories and truth theories, with circularity in non-classical logics. The purposes during this fellowship program are to make a contribution to a studio phones's research project by giving advices from logical point of view, to help an its researcher to formalize non-formalized ideas in terms of logic and category theory, and to give a new insight for them with Yusuke Kiriu.
I study a commutative ring of positive characteristic. Key objects in my research area are a singularity called the F-purity, an invariant called the F-pure threshold, and the test ideal, which are defined for a ring via its Frobenius morphism. The purpose during this fellowship program is to represent them in terms of formal language and category theory, and to give a new insight for them with Yusuke Kiriu.
My purpose in this fellowship program is to assist the recent works of Yusuke Kiriu and Mikami Hirasawa on some formulations in low dimensional topology and computer graphics, particularly from the viewpoint of Morse Theory and the Theory of Singularities of Differentiable Maps.
The theme of my research is to assist the recent works of Yusuke Kiriu and Mikami Hirasawa for some formulations on low dimensional topology, particularly from the viewpoint of Heegaard splittings and Morse functions on 3-manifolds.
Let K be a knotted circle in the 3-space. In the context of low-dimensional topology, it is known that there exists a 2-sided surface whose boundary coincides with K. Using such surfaces, one can calculate various topological invariants of knots, such as Alexander polynomials, determinants, and signatures. Essential algorithms which deal with such objects are desired. For clear understanding of them, we study their algorithm via formal language and category theory.
Following researches about hyperbolic geometry done with Yusuke Kiriu, our interest spreads across several fields of mathematics. Under this fellowship program Yusuke Kiriu and I will put together our current interests into articles.
I defined a new polynomial invariant for a given link diagram in three-dimensional space when I was a student of doctor course in Graduate School of Mathematics, Nagoya University. This link invariant is Poincare polynomial associated to a homology which is a generalization of Khovanov-Rozansky link homology using matrix factorizations. I study a structure of an algorithm to compute this polynomial link invariant in the context of computer language, especially, further possibilities of this algorithm in the context of formal language.