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【摘要】：可展曲面是Gauss曲率等于零的直紋面。它具有很多重要性質(zhì),例如它可以沒有拉伸和撕裂地展開到平面上；它是單參數(shù)平面族的包絡(luò)等等。這些性質(zhì)使可展曲面在曲面造型中具有非常重要的應(yīng)用價值。比如在實(shí)體外形的設(shè)計(jì)中,若實(shí)體外表面是可展曲面,則可以在平面上進(jìn)行設(shè)計(jì)；在計(jì)算機(jī)圖形學(xué)的紋理映射中,一張平面圖片可以沒有形變地貼到可展曲面上等等。所以,根據(jù)工程實(shí)際要求如何構(gòu)造所需的可展曲面,已成為當(dāng)前想要解決的一個重要問題。 因此,本文主要針對基于已知的曲線、曲面幾何條件如何構(gòu)造可展曲面及相關(guān)問題,進(jìn)行了如下幾方面的深入研究和探討： (1)完善了過曲面曲線構(gòu)造其可展切曲面的一般性理論和方法,得出了可展切曲面的表達(dá)形式,對可展切曲面進(jìn)行了分類,通過建立兩曲面間的映射關(guān)系,實(shí)現(xiàn)了它們間整體與局部的映射分析,較準(zhǔn)確地把握曲面上幾何要素的變形情況,并通過實(shí)例對理論和方法進(jìn)行了驗(yàn)證。 (2)提出了構(gòu)造回轉(zhuǎn)曲面的可展切曲面及它們間映射分析的理論與方法,建立了回轉(zhuǎn)曲面可展切柱面和可展切錐面的數(shù)學(xué)模型以及曲面間的映射關(guān)系,根據(jù)回轉(zhuǎn)曲面及其可展切曲面間微分長度比的理論分析,推出了映射中極值映射曲線和等距映射曲線的微分方程,通過整體和局部的變形分析,可以準(zhǔn)確地掌握回轉(zhuǎn)曲面與其可展切曲面間映射中的變形情況。 (3)以曲面片的可展切曲面研究為基礎(chǔ),得出了過曲面、曲線幾何要素構(gòu)造可展曲面的理論和方法,包括過一條曲線構(gòu)造與另一曲面相切的可展曲面和過兩曲線構(gòu)造的可展曲面,得出了可展曲面的解析表達(dá)形式。 (4)作為上述理論和方法的應(yīng)用,給出了可展曲面構(gòu)造與分析在曲面映射、不可展曲面近似展開和構(gòu)件表面可展化設(shè)計(jì)等方面的應(yīng)用舉例。
[Abstract]:A developable surface is a straight surface with Gauss curvature equal to zero. It has many important properties, such as it can expand to the plane without stretching and tearing, it is the envelope of the single parameter plane family and so on. These properties make developable surfaces have very important application value in surface modeling. For example, in the design of solid shape, if the external surface of the solid is an developable surface, it can be designed on the plane; in the texture mapping of computer graphics, a plane image can be attached to the developable surface without deformation, and so on. Therefore, how to construct the developable surface according to the practical engineering requirements has become an important problem to be solved. Therefore, this paper focuses on how to construct developable surfaces based on known geometric conditions of curves and surfaces and related problems. The following aspects are studied and discussed: (1) the general theory and method of constructing the developable tangent surface of hyperbolic curve are improved, the expression of developable tangent surface is obtained, and the developable tangent surface is classified. By establishing the mapping relationship between the two surfaces, the global and local mapping analysis between them is realized, and the deformation of geometric elements on the surface is accurately grasped. The theory and method are verified by examples. (2) the theory and method of constructing developable tangent surfaces and mapping analysis between them are presented. The mathematical models of developable tangent cylinder and developable tangent cone of rotary surface and the mapping relationship between them are established. The differential length ratio between the rotational surface and its developable tangent surface is analyzed theoretically. The differential equations of extreme mapping curve and equidistant mapping curve in mapping are derived, and the global and local deformation analysis are carried out. It is possible to accurately grasp the deformation of the mapping between the rotational surface and its developable tangent surface. (3) based on the research of the developable tangent surface of the surface slice, the theory and method of constructing the developable surface with the geometric elements of the curve are obtained. The analytic expression of developable surface is obtained by constructing the developable surface which is tangent to another curve and the developable surface constructed by two curves. (4) as the application of the above theory and method, The applications of construction and analysis of developable surfaces in surface mapping, approximate expansion of non-developable surfaces and developable design of component surfaces are given.
【學(xué)位授予單位】：東北大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2012
【分類號】：TH122

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本文編號：2151596