本文選題：射影幾何學(xué)　切入點(diǎn)：綜合法　出處：《河北師范大學(xué)》2017年碩士論文　論文類型：學(xué)位論文

【摘要】：綜合法用純粹的幾何方法解決幾何問題,在解析幾何學(xué)誕生之前是幾何學(xué)家們使用的唯一方法.綜合法在18世紀(jì)處于劣勢(shì)地位,在19世紀(jì)初期得到復(fù)興.綜合法的復(fù)興帶來了綜合射影幾何學(xué)的繁榮發(fā)展,代表人物有龐斯列、沙勒、斯坦納、施陶特等人.其中,施陶特使射影幾何學(xué)脫離度量,其綜合射影幾何學(xué)思想在射影幾何學(xué)中占有舉足輕重的地位.本文在大量搜集、整理、分析文獻(xiàn)的基礎(chǔ)上,對(duì)施陶特的綜合射影幾何學(xué)思想進(jìn)行了歷史探究,包括其思想的背景、核心內(nèi)容以及后人對(duì)它的完善.所取得的研究成果如下:1.闡明了施陶特的綜合射影幾何學(xué)思想產(chǎn)生的背景.射影幾何學(xué)起源于古希臘時(shí)期,在17世紀(jì)由德沙格等人發(fā)展,成熟于19世紀(jì).主要介紹了19世紀(jì)龐斯列的奠基性工作及沙勒與斯坦納關(guān)于交比的相關(guān)研究成果.2.探究了施陶特的綜合射影幾何學(xué)思想.施陶特的綜合射影幾何學(xué)思想集中體現(xiàn)在其兩部著作《位置幾何學(xué)》與《續(xù)論位置幾何學(xué)》中.施陶特于1847年出版《位置幾何學(xué)》,首次不依賴度量概念定義圖形間的射影關(guān)系,并在此基礎(chǔ)上借助對(duì)合與配極得到自己獨(dú)特的圓錐曲線定義.施陶特于1856-1860年出版三卷《續(xù)論位置幾何學(xué)》,引入“投”重新定義交比,并通過對(duì)合定義虛元素.3.分析了后人對(duì)施陶特的綜合射影幾何學(xué)思想的完善.施陶特由無關(guān)度量的射影定義得到施陶特定理,但是中間過程存在理論缺陷,分別由克萊因和皮耶里從連續(xù)性和公理化兩個(gè)角度進(jìn)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.
【學(xué)位授予單位】：河北師范大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O185

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