施陶特综合射影几何学思想的历史探究

发布时间：2018-03-05 06:03

本文选题：射影几何学　切入点：综合法　出处：《河北师范大学》2017年硕士论文　论文类型：学位论文

【摘要】：综合法用纯粹的几何方法解决几何问题,在解析几何学诞生之前是几何学家们使用的唯一方法.综合法在18世纪处于劣势地位,在19世纪初期得到复兴.综合法的复兴带来了综合射影几何学的繁荣发展,代表人物有庞斯列、沙勒、斯坦纳、施陶特等人.其中,施陶特使射影几何学脱离度量,其综合射影几何学思想在射影几何学中占有举足轻重的地位.本文在大量搜集、整理、分析文献的基础上,对施陶特的综合射影几何学思想进行了历史探究,包括其思想的背景、核心内容以及后人对它的完善.所取得的研究成果如下:1.阐明了施陶特的综合射影几何学思想产生的背景.射影几何学起源于古希腊时期,在17世纪由德沙格等人发展,成熟于19世纪.主要介绍了19世纪庞斯列的奠基性工作及沙勒与斯坦纳关于交比的相关研究成果.2.探究了施陶特的综合射影几何学思想.施陶特的综合射影几何学思想集中体现在其两部著作《位置几何学》与《续论位置几何学》中.施陶特于1847年出版《位置几何学》,首次不依赖度量概念定义图形间的射影关系,并在此基础上借助对合与配极得到自己独特的圆锥曲线定义.施陶特于1856-1860年出版三卷《续论位置几何学》,引入“投”重新定义交比,并通过对合定义虚元素.3.分析了后人对施陶特的综合射影几何学思想的完善.施陶特由无关度量的射影定义得到施陶特定理,但是中间过程存在理论缺陷,分别由克莱因和皮耶里从连续性和公理化两个角度进行完善.
[Abstract]:The method of synthesis was the only method used by geometrists before the birth of analytic geometry, which was at a disadvantage in 18th century. In the beginning of 19th century, it was revived. The renaissance of the synthetic method brought about the prosperity of comprehensive projective geometry, with representatives of Ponsley, Shaler, Steiner, Stauter, etc. Among them, Special Envoy Shi Tao's projective geometry broke away from the metric. The idea of Synthetical projective Geometry plays an important role in projective Geometry. Based on the extensive collection, arrangement and analysis of literature, this paper makes a historical inquiry into the idea of Synthetical projective Geometry of Stauter. The research results are as follows: 1. The background of Stauter's thought of Synthetical projective Geometry. Projective Geometry originated in ancient Greece. In 17th century, developed by de Schag et al., It matured on 19th century. It mainly introduces the foundational work of Ponce Sligh in 19th century and the relative research results of Schaler and Steiner on the intersection ratio. 2. It probes into the comprehensive projective geometry thought of Stauter. The synthetic projective geometry of Stauter. The thought of learning is embodied in his two works, "position Geometry" and "continued position Geometry". In 1847, Stauter published "positional Geometry", and for the first time did not rely on the concept of metric to define the projective relationship between figures. On the basis of this, we get our own unique definition of conic curve by means of involution and collocation. Stauter published three volumes of Geometry of position in 1856 ~ 1860, and introduced "cast" to redefine the intersection. And through the definition of involutive virtual element .3.At the end of this paper, the author analyzes the perfection of the idea of Synthetical projective geometry of Stauter. Stauter obtains the Stautert theorem by the projective definition of independent metric, but there are some theoretical defects in the intermediate process. By Klein and Pieri from the continuity and axiom two angles to improve.
【学位授予单位】：河北师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O185

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