Construction of Hyperfigure Theory and Its Applications

超图理论的构建及其应用

• 批准号: 13450039
• 负责人: SUGIHARA Kokichi
• 金额: $ 9.28万
• 依托单位: The University of Tokyo
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-13450039/
• 关键词: Minkowski sum; slope-monotone curve; invertibility; hyperfigure; polygon; surface of revolution; convex figure; 超図解の解釈; 回転曲面

We found that the world of figures can be extended in such a way that the Minkowski operation always admits its inverse, and name the new objects "hyperfigures". The goal of this project was to study the mathematical structures of hyperfigures and establish a basis of hyperfigure theory. Aiming at this goal, we obtained the following results.(1) The object world is extended from the world of slope-monotone curves so that any polygons are included. This was achieved by the property that a slope-monotone curve can be the boundary of the union of a finite number of convex figures and that the polygonal lines are obtained as the limits of those curves.(2) Physical interpretation was given to hyperfigures. In the conventional Minkowski algebra, sometimes the result of the operation becomes just an empty set. In our new theory, such a result can be represent by a nontrivial hyperfigure, and can be interpreted as, for example, the ability of a numerical control machine or the ability of a water-spread vehicle.(3) An efficient algorithm was obtained for the Minkowski sum of three-dimensional solids. We considered surfaces of revolution obtained by rotating two-dimensional slope-monotone closed curves, and designed an algorithm for finding the correspondence of points with the same outward normal, and thus computing the Minkowski sum. This algorithm was also applied to the problem of finding the shortest distance of two moving solids.

我们发现，世界的数字可以扩展到这样一种方式，闵可夫斯基操作总是承认其逆，并命名为新的对象“hyperfigures”。该项目的目标是研究超数的数学结构，并建立超数理论的基础。针对这一目标，我们取得了以下成果。(1)对象世界是从斜率单调曲线的世界中扩展出来的，因此任何多边形都包括在内。这是通过以下性质实现的：斜率单调曲线可以是有限个凸图形的并的边界，并且多边形线作为这些曲线的极限而获得。(2)对超形进行了物理解释。在传统的闵可夫斯基代数中，有时运算的结果会变成一个空集。在我们的新理论中，这样的结果可以用一个非平凡的超形来表示，并且可以解释为，例如，数控机床的能力或水传播车辆的能力。(3)给出了三维立体Minkowski和的一个有效算法。考虑了旋转二维斜率单调闭曲线得到的旋转曲面，设计了一种算法，求出具有相同外法线的点的对应关系，从而计算Minkowski和。该算法也被应用到两个移动的固体的最短距离的问题。

项目成果

Kei KOBAYASHI, Kokichi SUGIHARA: "Crystal Voronoi diagram and its applications"Future Generation Computer Systems. 18. 681-692 (2002)

Kei KOBAYASHI、Kokichi SUGIHARA：“水晶 Voronoi 图及其应用”未来一代计算机系统。

Takeshi KANDA, and Kokichi SUGIHARA: "Hybrid arithmetic for acceleration of geometric algorithms-suitable error estimation for each of sign decisions"6th Korea-Japan Joint Workshop on Algorithms and Computation (Pusan, June 28-29). 37-44 (2001)

Takeshi KANDA 和 Kokichi SUGIHARA：“几何算法加速的混合算术 - 每个符号决策的适当误差估计”第六届韩日算法与计算联合研讨会（釜山，6 月 28-29 日）。

Kokichi Sugihara: "Exact Computation of 4-D Convex Hulls With Perturbation and Acceleration"6th Korea-Japan Joint Workshop on Algorithms and Computation (Pusan, June 28-29). 51-58 (2001)

Kokichi Sugihara：“带有扰动和加速的 4-D 凸包的精确计算”第六届韩日算法与计算联合研讨会（釜山，6 月 28-29 日）。

坂井秀行, 杉原厚吉: "図形の中心軸の安定した生成法"電子情報通信学会論文誌. J85-D-II・11. 1637-1644 (2002)

Hideyuki Sakai，Atsuyoshi Sugihara：“图形中心轴的稳定生成方法”电子信息通信工程师学会会刊J85-D-II·11（2002）。

Daniel FOGARAS, and Kokichi SUGIHARA: "Topology-oriented robust algorithm for constructing lint arrangements"6th Korea-Japan Joint Workshop on Algorithms and Computation (Pusan, June 28-29). 21-58 (2001)

Daniel FOGARAS 和 Kokichi SUGIHARA：“用于构建 lint 排列的拓扑导向的鲁棒算法”第六届韩日算法与计算联合研讨会（釜山，6 月 28-29 日）。