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Russian Federation
Probably, it is impossible to find such industry where the ruled surfaces would not be used. They are used in agriculture, in the heavy and light industries, in construction, in aircraft manufacturing, and in military art. Ruled surfaces are used in the design of wings, tail and partially fuselage of aircraft, car bodies, in the project engineering of slopes and embankments of auto-roads, abutments of bridge supports, transitions from a vertical quay to inclined walls of embankments, various hydraulic structures, towers, masts, cooling towers, vaults and arches, overlaps of pavilions, circuses, stadiums and other building structures, as well as in the calculation of solar exposure. This paper deals with the formation of ruled surfaces in a single method of their definition. A number of examples for definition of ruled surfaces have been presented. These examples show that in general for definition of ruled surfaces it is required to have three guides and three geometric conditions characterizing the position of a rectilinear generator with respect to each of the guides. Both surfaces and lines can act as guides. The plane is selected separately from other surfaces. The geometric conditions are the intersection with the guide line and the tangent or intersection at a certain sharp angle with the guide surface. The table of 19 variants for guides has been given. An attempt to classify surfaces does not even consider in this paper since it is impossible to classify ruled surfaces, even within its class, due to the lack of a criterion showing their belonging to one or another species. It can be concluded that the classification of surfaces may be used only for educational purposes and in cases where the surface name is obvious.
surfaces, ruled surfaces, geometry, descriptive geometry, surface formation.
Введение
Линейчатые поверхности имеют большое значение в практической деятельности человека [2; 6; 10; 12; 14–17]. Они широко применяются в различных областях науки, техники [3; 6; 20; 21; 30; 31; 33], строительства [4; 9; 14; 28; 32]. Линейчатые поверхности используются при конструировании крыльев, хвостового оперения и частично фюзеляжа самолетов, кузовов автомобилей, при проектировании откосов и насыпей автомобильных дорог [18; 19; 22–25; 28; 29], устоев мостовых опор, переходов от вертикальной причальной к наклонным стенкам набережных, различных гидротехнических сооружений, башен, мачт, градирен, сводов и арок, перекрытий павильонов, цирков, стадионов и других строительных сооружений, при расчете инсоляции [11]. Примерами практического применения линейчатых поверхностей в технике являются зубчатые передачи [5; 7; 13], в том числе с гиперболическим зацеплением, сцепные муфты, шнековые питатели, пружины с прямоугольным сечением прутка; прямоугольная, треугольная, трапецеидальная, упорная резьбы, нарезка в орудийных и винтовочных стволах, винты самолетов и кораблей, лопатки различного назначения турбин [8]. Этим перечнем далеко не исчерпываются возможности практического использования линейчатых поверхностей. В аналитической и дифференциальной геометриях линейчатые поверхности, как и другие, задаются уравнением: F(x, y, z) = 0, (1)
в котором устанавливается зависимость между координатами точек, принадлежащих этой поверхности. В зависимости от характера уравнения поверхность будет называться алгебраической n-го порядка, если уравнение алгебраическое n-ной степени, то трансцендентной, если уравнение трансцендентное, выражает тригонометрическую или логарифмическую зависимость. Произвольная секущая плоскость пересекает алгебраическую поверхность n-го порядка по кривой того же порядка. Произвольная прямая линия пересекает алгебраическую поверхность в точках, количество которых равно порядку этой поверхности. В начертательной геометрии чаще всего поверхность задается кинематическим способом. При этом поверхность получается в результате непрерывного перемещения в пространстве какой-либо линии, называемой образующей, по определенному закону. Поверхность, полученная в результате перемещения прямолинейной образующей, называется линейчатой.
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