Graphic Reconstruction Algorithms of the Second-Order Curve, Given by the Imaginary Elements

Viktor Korotkiy; A. Girsh

doi:doi:10.12737/22840

Graphic Reconstruction Algorithms of the Second-Order Curve, Given by the Imaginary Elements

Published в Geometry & Graphics · Volume 4, Issue 4, 2016 · Pages 19–30 · Rubrics: Scientific problems of geometry

DOI 10.12737/22840

Received: 24.11.2016 Accepted: 19.12.2016 Published: 19.12.2016 Language of publication: RUS

Authors

Viktor Korotkiy ¹ · A. Girsh ²

1 South Ural State University (Dep. “Engineering and Computer Graphics”, Professor)
Chelyabinsk, Chelyabinsk, Russian Federation

2 Kassel'skiy universitet
Kassel', Germany

ABSTRACT

Second-order curves are used as shape-generating elements in the design of technical devices and architectural structures. In such a case, a need for reconstruction task solution may emerge. The reconstruction is called the definition of the main axes and asymptotes of the second-order curve by its incomplete image containing n points and m tangents (n + m = 5). In CAD graphical systems there is no possibility for construction of the second order curve, given by real and imaginary points and tangents. Therefore, the second-order curve reconstruction cannot be made with the standard set of computer graphics tools. In this paper are proposed geometrically accurate algorithms for reconstruction of the secondorder curve, given by a mixed set of real and imaginary elements. A specialized software package has been developed for constructive realization of these algorithms. Imaginary geometric images are pair-conjugated, so there are only seven possible combinations of given data with imaginary elements participation: five points, two of which are imaginary ones; five points, four of which are imaginary ones; three real points, two imaginary tangents; a real point, four imaginary tangents; a real point, two imaginary points, two imaginary tangents; a real point, two imaginary points, two real tangents; two real points, two imaginary points, a real tangent. For reconstruction problem solution is used the main property of polar matching: if P and p are the pole and polar relative to the conic g, the harmonic homology with center P and axis p transforms the curve g in itself. The method of solution based on projective transformation of required conic into a circle. It has been shown that in some cases for reconstruction problem solution its necessary to apply the quadratic involution conversion, resting on plane by a conic beam. The developed technique and software package expand the capabilities of the computer geometric simulation for processes occurring with the second-order curves participation.

KEYWORDS

harmonic homology elliptic involution polarity conic beam autopolar triangle quadratic involution

Information Text References

Authors:

1. South Ural State University (Dep. “Engineering and Computer Graphics”, Professor)

Chelyabinsk, Chelyabinsk, Russian Federation

2. Kassel'skiy universitet

Kassel', Germany

Type:

Article

Pages:

from 19 to 30

Status:

Published

Language:

Russian

Keywords:

harmonic homology, elliptic involution, polarity, conic beam, autopolar triangle, quadratic involution

Text

Кривые второго порядка (КВП, конические сечения, коники) применяются как для изображения различных кинематических процессов (траектории, орбиты и т.п.), так и в качестве формообразующих элементов при проектировании технических устройств и архитектурных сооружений. При этом может возникать потребность в решении задачи реконструкции. Реконструкцией КВП называют определение метрики (главных осей, асимптот) кривой второго порядка по ее неполному изображению, содержащему n точек и m касательных (n + m = 5).

Известны графические алгоритмы определения метрики КВП для случая, когда заданные элементы (точки и касательные) действительны [4; 17; 18]. Для случая, когда некоторые точки и касательные к искомой КВП — мнимые (комплексно сопряженные), алгоритмы определения метрики КВП отсутствуют.

Цель работы — составить геометрически точный алгоритм реконструкции кривой второго порядка, заданной смешанным набором действительных и мнимых элементов (точек и касательных). Геометрически точным алгоритмом называют последовательность построений с использованием только двух графических примитивов — прямой линии и окружности [1].

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