基于曲线曲面的可展曲面构造与分析

发布时间：2018-07-28 21:14

【摘要】：可展曲面是Gauss曲率等于零的直纹面。它具有很多重要性质,例如它可以没有拉伸和撕裂地展开到平面上；它是单参数平面族的包络等等。这些性质使可展曲面在曲面造型中具有非常重要的应用价值。比如在实体外形的设计中,若实体外表面是可展曲面,则可以在平面上进行设计；在计算机图形学的纹理映射中,一张平面图片可以没有形变地贴到可展曲面上等等。所以,根据工程实际要求如何构造所需的可展曲面,已成为当前想要解决的一个重要问题。因此,本文主要针对基于已知的曲线、曲面几何条件如何构造可展曲面及相关问题,进行了如下几方面的深入研究和探讨： (1)完善了过曲面曲线构造其可展切曲面的一般性理论和方法,得出了可展切曲面的表达形式,对可展切曲面进行了分类,通过建立两曲面间的映射关系,实现了它们间整体与局部的映射分析,较准确地把握曲面上几何要素的变形情况,并通过实例对理论和方法进行了验证。 (2)提出了构造回转曲面的可展切曲面及它们间映射分析的理论与方法,建立了回转曲面可展切柱面和可展切锥面的数学模型以及曲面间的映射关系,根据回转曲面及其可展切曲面间微分长度比的理论分析,推出了映射中极值映射曲线和等距映射曲线的微分方程,通过整体和局部的变形分析,可以准确地掌握回转曲面与其可展切曲面间映射中的变形情况。 (3)以曲面片的可展切曲面研究为基础,得出了过曲面、曲线几何要素构造可展曲面的理论和方法,包括过一条曲线构造与另一曲面相切的可展曲面和过两曲线构造的可展曲面,得出了可展曲面的解析表达形式。 (4)作为上述理论和方法的应用,给出了可展曲面构造与分析在曲面映射、不可展曲面近似展开和构件表面可展化设计等方面的应用举例。
[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.
【学位授予单位】：东北大学
【学位级别】：硕士
【学位授予年份】：2012
【分类号】：TH122

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