Rostov-na-Donu, Russian Federation
Ruled surfaces have long been known and are widely used in construction, architecture, design and engineering. And if from the technical point of view the developable surfaces are more attractive, then architecture and design successfully experiment with non-developable ones. In this paper are considered non-developable ruled surfaces with three generators, two of which are curvilinear ones. According to classification, such surfaces are called twice oblique cylindroids. In this paper has been proposed an approach for obtaining of twice oblique cylindroids by immersing a curve in a line congruence of hyperbolic type. Real directrixes of such congruence are a straight line and a curve. It has been proposed to use helical lines (cylindrical and conical ones) as a curvilinear directrix, and a helical line’s axis as the straight one. Then the congruence’s rectilinear ray will simultaneously intersect the helical line and its axis. Congruence parameters are the line’s pitch and the guide cylinder or cone’s radius. The choice of the curvilinear directrix is justified by the fact that the helical lines have found a wide application in engineering and architecture. Accordingly, the helical lines based surfaces can have a great potential. In this paper have been presented parametric equations of the considered congruences. The congruence equations have been considered from the point of view related to introducing a new curvilinear coordinate system. The obtained system’s coordinate surfaces and coordinate lines have been also studied in the paper. To extract the surface, it is necessary to immerse the curve in the congruence. To synthesize the equations has been used a constructive-parametric method based on the substitution of the immersed line’s parametric equations in the congruence equations according to a special algorithm. In the paper have been presented 5 examples for the synthesis of ruled surfaces equations such as the twice oblique cylindroid and their visualization. The method is universal and algorithmic, and therefore easily adaptable for the automated construction of surfaces with variable parameters of both the congruence and the immersed line.
ruled surface, lined congruence, twice oblique cylindroids, helical line, parametric equations.
Линейчатые поверхности давно привлекают внимание геометров, архитекторов, машиностроителей и дизайнеров. Наиболее изученными и применяемыми из неразвертываемых поверхностей являются геликоиды, иперболический параболоид, однополостной гиперболоид и поверхности Каталана [1; 5; 19; 22; 27–29], которые можно увидеть практически повсюду. Согласно классической классификации в русскоязычной учебной литературе, линейчатые поверхности по количеству направляющих делятся на три типа: с тремя, двумя и одной направляющей. В свою очередь, линейчатые поверхности с тремя направляющими делятся на [5]: 1) поверхность общего вида — с тремя криволинейными направляющими; 2) дважды косой цилиндроид — с двумя криволинейными и одной прямолинейной направляющей; 3) дважды косой коноид — с двумя прямолинейными и одной криволинейной направляющей; 4) однополостной гиперболоид — с тремя прямолинейными направляющими. Линейчатая поверхность с тремя направляющими фактически представляет собой поверхность линейчатой конгруэнции иперболического типа, где две из трех направляющих являются директрисами, а третья — погружаемой в линейчатую конгруэнцию кривой. Таким образом, дважды косые коноиды являются поверхностями гиперболической конгруэнции прямых Кг (1,1), а однополостной гиперболоид является ее частным случаем при погружении прямой. Особое внимание конгруэнциям начали уделять в период развития проективной геометрии. В начале ХХ в. было построено множество натурных наглядных моделей по представлению пространственных кривых, являющихся линиями пересечения поверхностей, линейчатых поверхностей и линейчатых конгруэнций и их поверхностей (рис. 1). Практическое применение поверхностей конгруэнций прямых стало возможным с развитием синтетической и конструктивной геометрии [2; 6; 10–14]. В настоящее время изучение построения и визуализации таких поверхностей ведется как с точки зрения создания программно реализуемых алгоритмов проективной геометрии [7; 8; 19–21; 23; 25–30], так и с точки зрения получения параметрических уравнений конгруэнций и их поверхностей конструктивно-параметрическим методом [3; 4; 9; 15–18; 24]. В работах [18; 31] был предложен способ получения параметрических уравнений гиперболической конгруэнции прямых Кг (1,1) и ее поверхностей, а также рассмотрены некоторые частные случаи управления параметрами формы. Эти поверхности являются дважды косыми коноидами. Целью настоящей работы является получение параметрических уравнений линейчатых поверхностей, полученных погружением кривой в конгруэнцию гиперболического типа, в которой директрисами являются прямая и винтовая линии (цилиндрическая и коническая с постоянным шагом), а лучом — прямая. Данный тип поверхности относится к категории дважды косых цилиндроидов.
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