考虑虚数几何图像的阿波罗问题近似解的统一构造模型的视觉图形设计

Д. Волошинов, D. Voloshinov
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引用次数: 9

摘要

与平面上任意三个给定圆相切的圆的构造问题阿波罗尼乌斯问题是经典几何中研究得很好的问题之一。本文的材料旨在建立一个统一的理论来解决阿波罗问题,考虑到它不仅是真实的,而且是不可见的复值图像。本文证明了阿波罗尼乌斯问题所依据的基本几何结构不仅适用于实数,而且适用于复值数据,从而有可能消除目前存在的许多例外情况。本文揭示了阿波罗尼乌斯问题的基本性质及其与投影和二次几何变换的密切联系。已经证明,阿波罗尼乌斯问题及其类似问题有一个单一的解决方法,而不是普遍认为这些问题只能通过单独的特定方法来解决。由于几何模型视觉设计系统Simplex中设置了大量的计算试验,笔者提出的几何实验的概念使我们发现了许多以前不知道的和本文讨论过的共性因素。本文考虑的是三维空间中类似Apollonian问题的求解实例,但所提算法的运算具有普适性,可以同样适用于求解任意维空间中的类似问题。所得结果证明了构造建模和多维描述几何方法在解决复杂数学问题中的应用能力,并确定了构造几何建模自动化系统的发展趋势。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Visual-Graphic Design of a Unitary Constructive Model to Solve Analogues For Apollonius Problem Taking into Account Imaginary Geometric Images
The Apollonius problem on construction of circles, tangent to three arbitrary given circles of a plane, is one of classical geometry’s well-studied problems. The presented paper’s materials are directed at development a unified theory for Apollonius problem solving, taking into account it’s not only real, but also invisible complex-valued images. In the paper it has been demonstrated, that fundamental geometric structures, on which Apollonius problem is based on, are applicable not only to real, but also to complex-valued data, that makes possible to eliminate many exceptions, currently existing in it. In this paper Apollonius problem’s fundamental nature and its strong correlation with projective and quadratic geometric transformations has been disclosed. It has been proved that Apollonius problem and its analogues have a single solution method, in contrast to the prevailing idea that these problems can be solved only by separate particular methods. A concept of geometric experiment proposed by the author has allowed find out many previously unknown and discussed in this paper common factors, due to the set of many computational tests in the system Simplex for visual design of geometric models. In this paper is considered an example for solving an analogue of Apollonian problem for three-dimensional space, but proposed algorithm’s operation is universal, and it can be equally applied to solving similar problems in spaces of arbitrary dimensions. Obtained results demonstrate capabilities of methods for constructive modeling and multidimensional descriptive geometry in application to solving of complex mathematical problems, and determine the trends in development for automation systems of constructive geometric modeling.
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