In this paper the visibility concept in the context of modeling of multidimensional spaces’ objects is clarified. It is concluded that such model’s visibility should be defined as unambiguity and completeness of information presented in this model and consistent with the student’s experience in the area of modeling a space of higher dimension (3D) by elements of spaces of lower dimension (2D). Such possibilities are presented by the generalized complex drawing. Examples for objects 4D-modeling using two 3D or three 2D flat projections are presented, some properties of the 4D generalized drawing are listed. The solution of problems with 4D-objects is considered on the example of 4D-pyramid section construction, and deploying its lateral surface. It is shown that to simplify the solution of these problems is required a system allowing automatically perform repetitive sequences of constructions. A list of elementary constructions is presented, and a method for recording of composite constructions and based on them algorithms for problems solving is shown. It is demonstrated that a 3D-scan of 4D-pyramid’s lateral surface, constructed with 2D drawing, can be imported into CAD as a 3D-model. The deploying of the 4D-cone’s lateral surface is considered. The resulting scan’s surface 3D-model imported into CAD is shown. Cases are indicated when a multidimensional space’s object 3D-model may be more visual than a flat one. As an example, 2D-models for imaginary continuations of lines and circles of the complex plane (simulated by Euclidean 4D-space) are presented. Two 3D-projections for imaginary continuations of a circle with a real radius as 3D-space surfaces are shown. It is noted that in order to combine in an educational course the multidimensional space’s objects modeling and work in CAD the tasks on designing of complex technical surfaces by means of output in multidimensional space are suitable. A brief review of sources is given, in which theoretical foundations and the use of key geometrical methods for surfaces construction are considered; an example of a surface constructed by a progressive key method and imported into CAD is shown. The concept of a product’s electronic model (PEM) is described, in which the modeled object’s 3D-simulator as its visual representation is combined with numerous 2D-layers, which elements automatically perform geometrical and graphical calculations in spaces of any dimensions, and control 3D-model’s dimensions and shape through constructive and parametric links. Conclusions are drawn about the possibility of visual multidimensional modeling in the educational process, the advantages of using a complex drawing for solving of problems with multidimensional objects, the need to use special systems of constructive geometric modeling that automate repetitive sequences of constructions. It is also concluded that multidimensional objects’ 2D-models can and should be directly involved in the PEM formation.
multidimensional geometry, general complex drawing, Radishchev hyper epure, descriptive geometry of multidimensional space, 3D modeling, geometrical constructions, surface constructing, CAD.
Введение
В статье рассматриваются вопросы, связанные с построением моделей объектов многомерных евклидовых пространств в учебном процессе. Многочисленные публикации последних лет [10; 20–22; 29; 43] показывают, что моделирование объектов многомерных пространств имеет большое значение для решения практических задач, однако вопросы построений в многомерных пространствах, как правило, освещаются лишь в научных статьях и монографиях [8; 10; 24; 37; 38; 41]. Систематическое решение задач, связанных с построениями в многомерных пространствах, в учебном процессе не рассматривается. Попытка включения в курс «Наглядной инженерной геометрии» разделов, связанных с многомерным моделированием, в [35; 36] вызывает много вопросов. В частности, предлагаемый в [35] подход основан на рассмотрении следующих тем: пересечение геометрических фигур плоскостью, прямой, пересечение геометрических фигур между собой, структура и развертки n-мерных многогранников, решение метрических задач, которые излагаются, за исключением разверток, исключительно на примерах
трехмерного моделирования. В работе [36] производится попытка пояснить значение термина «наглядность» в контексте нового курса, но связь многомерных моделей с фазовыми пространствами системы в физической химии и материаловедении, с многокоординатной обработкой на станках с ЧПУ только декларируется без каких-либо наглядных примеров, а все практическое моделирование осуществляется при помощи «современных информационных технологий» — CAD-систем, по сути своей, не приспособленных для работы с пространствами более трех измерений. Сказанное свидетельствует о противоречиях, которые становятся очевидными при попытках увязать элементы многомерного моделирования с геометро-графическими курсами, базирующимися на использовании CAD-систем. Для преодоления этих противоречий требуется уточнить понятие наглядности в контексте многомерного моделирования и определить требования к компьютерным системам геометрического моделирования для решения задач, связанных с объектами многомерных пространств.
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