# Registered Data

## [00882] Geometric Shape Generation II: Design

**Session Date & Time**:- 00882 (1/3) : 5B (Aug.25, 10:40-12:20)
- 00882 (2/3) : 5C (Aug.25, 13:20-15:00)
- 00882 (3/3) : 5D (Aug.25, 15:30-17:10)

**Type**: Proposal of Minisymposium**Abstract**: This mini-symposium is based on the JSIAM activity group “Geometric Shape Generation”, and aims at exhibiting the latest research in this activity group and relevant researchers, especially putting its focus on the design. We discuss the mathematical aspects of design and analysis of shapes under various settings, and special curves and surfaces useful for generating desirable shapes on CAD.**Organizer(s)**: Yoshiki Jikumaru, Kenji Kajiwara, Kenjiro T. Miura, Masaaki Umehara**Classification**:__53A04__,__53A05__,__53A10__,__53A35__,__53A70__**Speakers Info**:- Masashi Yasumoto (Tokushima University)
- Hiroyuki Kitahata (Chiba University)
- Eriko Shinkawa (Tohoku University)
- Dan Wang (Shizuoka University)
- Kenjiro Takai Miura (Department of Mechanical Engineering, Shizuoka University)
- Syed Ahmad Aidil Adha Said Mad Zain (Universiti Sains Malaysia)
**Kenji Kajiwara**(Institute of Mathematics for Industry, Kyushu University)- Sampei Hirose (Shibaura Institute of Technology)
- R U Gobithaasan (University Malaysia Terengganu)
- Rushan Ziatdinov ( Keimyung University)

**Talks in Minisymposium**:**[01406] Bifurcation of the trajectory shape in self-propelled motions****Author(s)**:**Hiroyuki Kitahata**(Chiba University)

**Abstract**: We consider the motion of a self-propelled particle which is driven by the surface tension gradient originating from the concentration of the chemicals released from the particle itself. First, we discuss the trajectory shape of the particle confined in a circular region. Next, we discuss the bifurcation of the motion for the motion observed in the system with a self-propelled particle and a passive particle.

**[01561] The uniqueness theorem on the shape of free-form curves****Author(s)**:**Kenjiro Takai Miura**(Shizuoka University)- R.U. Gobithaasan (University Malaysia Terengganu)
- Md Yushalify Misro (University Science Malaysia)
- Tadatoshi Sekine (Shizuoka University)
- Shin Usuki (Shizuoka University)

**Abstract**: We will discuss about the shape uniqueness theorem for curves defined by three or more control points and show several examples of applications ofthe theorem.

**[01581] Generation of κ-Space Curve****Author(s)**:**DAN WANG**(Shizuoka University)- Tadatoshi Sekine (Shizuoka University)
- Shin Usuki (Shizuoka University)
- Kenjiro Takai Miura (Department of Mechanical Engineering, Shizuoka University)

**Abstract**: The κ-curve is a recently published interpolating spline which consists of quadratic Bezier segments passing through input points at the loci of local curvature extrema. But their interpolation can only deal with planar curves. Therefore, in this research We propose a method that enables to extend this representation to deal with space curves in a new scheme called κ-space curves

**[01613] On the relationship between mimetic discretization and discrete exterior calculus****Author(s)**:**Sampei Hirose**(Shibaura Institute of Technology)

**Abstract**: The mimetic discretization is a general framework for the discretization of differential operators using differential forms, including finite element and finite volume methods. On the other hand, the discrete exterior calculus is a method for dealing with differential forms on discrete spaces and is used in computer graphics and other applications. In this talk, the relationship between the mimetic discretization and the discrete exterior calculus developed by Anil Hirani and others will be discussed.

**[01640] Quantifying the shape of data using Topological Data Analysis****Author(s)**:**R U Gobithaasan**(University Malaysia Terengganu)- Kenjiro Takai Miura (Shizuoka University)
- Pawel Dlotko (Dioscuri Centre in Topological Data Analysis, Mathematical Institute, Polish Academy.)

**Abstract**: Topological Data Analysis (TDA) encodes the global structure and overall connectivity of high dimensional dataset, hence revealing the linearity, distribution, clusters and groups abnormality. TDA has two methodologies namely Persistent homology which produces Persistence Diagram and, TDA Mapper which is a graph representing the structure of the data. In this talk, we will discuss both the methodologies with numerical examples for efficient implementation.

**[01713] Biangular coordinates: moving forward****Author(s)**:**Rushan Ziatdinov**(Keimyung University)

**Abstract**: A bipolar coordinate system, known as biangular coordinates, uses two angles rather than one to describe the location of a point in a plane. Most common curves' equations in biangular coordinates are still unknown, and little research has been done on the features of this coordinate system and its potential uses. This work aims to advance our understanding of biangular coordinates.

**[01764] Generation of Aesthetic Shape by Integrable Geometry****Author(s)**:**Kenji Kajiwara**(Institute of Mathematics for Industry, Kyushu University)

**Abstract**: We consider log-aesthetic curves (LAC), a family of planar curves developed in industrial design, as curves that car designers regard as “aesthetic.” We present a new mathematical framework of LAC on the theory of integrable systems and similarity geometry. Using this framework, LAC is shown to be a similarity geometry analogue of Euler’s Elastica. Based on this, we present generalizations of LAC to space curves and surfaces, which may be useful for generating aesthetic shapes.

**[01771] Construction of weaving structures by standard realizations with repulsive interactions****Author(s)**:**Eriko Shinkawa**- Motoko Kotani (Tohoku University)
- Hisashi Naito (Nagoya University)

**Abstract**: We consider weaving structures. Let T be two sets of threads in 2-dimensional Euclidean space, where all the threads in each set are parallel and assign up/down information at their intersections. To find a suitable configuration of T in 3-dimensional Euclidean space, we take the energy that is given by the standard realization with repulsive interactions introduced by A. Dechant et al. We discuss the existence of energy minimizing configurations, which are called weaving structures.

**[01775] Construction of κ-Curve Using Fractional Bézier Curve****Author(s)**:**Syed Ahmad Aidil Adha Said Mad Zain**(Universiti Sains Malaysia)- Md Yushalify Misro (Universiti Sains Malaysia)
- Kenjiro Takai Miura (Department of Mechanical Engineering, Shizuoka University)

**Abstract**: The $\kappa$-curve is one of the famous curves that has been applied as a curvature pen tool in Adobe Illustrator® and Photoshop®. The $\kappa$-curve has an excellent property where the local maxima of curvature have occurred at the control points. This will prevent the formation of cusps and loops. In this work, the construction of the $\kappa$-curve will be shown by using the fractional Bézier curve with the help of fractional continuity.

**[01784] Construcion of discrete zero mean curvature surfaces in Euclidean and Lorentz-Minkowski spaces****Author(s)**:**Masashi Yasumoto**(Tokushima University)

**Abstract**: In this talk we first introduce discrete timelike minimal surfaces in Lorentz-Minkowski 3-space. Compared with other dicrete zero mean surfaces, discrete timelike minimal surfaces possess richer mathematical structures. Starting from discrete timelike minimal surfaces, we construct discrete zero mean curvature surfaces in Lorentz-Minkowski 3-space that can have both spacelike and timelike parts. As an application, we construct discrete holomorphhic functions and new discrete minimal surfaces in Euclidean space.