Development of software to and researches in hyperbolic geometry and kleinnian group

在这个项目中，我们为双曲几何和克莱因群的研究人员开发了一个研究工具OPTi，并以各种方式对其进行了改进。该项目取得的成果一直通过OPTi主页向公众开放（RUL：http：//vivaldi.ics.nara-wu.ac.jp/juwada/OPTi/index.html）。我们在许多会议和研讨会上发表了与OPTi相关的演讲，包括1999年墨西哥克利福德代数第五届国际会议，1999年沃里克大学群论和几何计算研讨会，2001年沃里克大学Kleinian群和双曲三维流形研讨会。我们还组织了2000年在奈良女子大学举办的“穿孔环面和2桥结“研讨会和2000年在奈良女子大学举办的“几何3-orbifolds研究”研讨会，并听取了许多研究人员对OPTi的评论和1项要求。因此，OPTi变得比第一个版本更有用，也更容易使用。同时，2001年，我们与山下、菅川和小森合作，将OPTi中使用的福特区域绘制到Bers切片的算法应用于Bers切片，并在世界上首次获得了奇异组件的图片。其中使用的算法后来被实现到OPTi 3.30版中，以实时绘制各种切片图片的变形。

In this project, we have developed a research tool OPTi for researchers in hyperbolic geometry and Kleinnian groups, and improved it in various ways. The results obtained in the project have always been open to the public through the OPTi Home Page (RUL: http://vivaldi.ics.nara-wu.ac.jp/〜wada/OPTi/index.html). We gave talks related to OPTi in many conferences and workshops including The 5th International Conference in Clifford Algebra, Mexico, 1999, Workshop on Computations in Group Theory and Geometry, University of Warwick, 1999, Workshop on Kleinian Groups and Hyperbolic 3-Manifolds, University of Warwick, 2001. We also organized workshops "Punctured tori and 2-bridge knots ", Nara Women's University, 2000 and "Studies on Geometric 3-orbifolds", Nara Women's Univeristy, 2000, and heard comments and 1 requests about OPTi from many researchers. As a result, OPTi has become much more useful and is much easier to use than the first version.Meanwhile, in 2001, in cooperation with Yamashita, Sugawa and Komori, we applied the algorithm for drawing the Ford regions used in OPTi to the Bers slice, and obtained pictures of the exotic components for the first time in the world. The algorithm used therein was later implemented into OPTi Version 3.30 to give real-time drawing of the deformations of various kinds of slice pictures.