DOI: 10.14489/vkit.2014.07.pp.014-020

Короткий В. А.
КОМПЬЮТЕРНОЕ МОДЕЛИРОВАНИЕ ФИГУР ЧЕТЫРЕХМЕРНОГО ПРОСТРАНСТВА
(с. 14-20)

Аннотация. Предложен способ изображения фигуры четырехмерного пространства на экране компьютера с помощью чертежа, со-стоящего из двух или более трехмерных проекций данной фигуры. Рассмотрены инварианты ортогонального проецирования точек четырехмерного пространства на картинную гиперплоскость. Приведено доказательство теоремы об изображении вза-имно перпендикулярных прямых, плоскостей и гиперплоскостей на трехмерном чертеже. Приведены примеры моделирования фигур четырехмерного пространства.

Ключевые слова: гиперэпюр Наумович; трехмерная проекция; размерность пространства; поляритет; абсолют; гиперплоскость; гиперсфера.

Korotky V. A.
COMPUTER MODELING OF FOUR-DIMENSIONAL FIGURES
(pp. 14-20)

Abstract. Provided a method for geometric modeling figures four-dimensional space (the "giperdrawing Naumovich") with the aid of computer graphics. Three-dimensional giperdrawing has the minimum difference between the dimensions of the original (four) and paintings (three-dimensional) spaces. This greatly simplifies the design solution of geometric problems on four-dimensional figures compared to solving the same problems on a flat projection model. Three-dimensional giperdrawing study of four space consists of two or more three-dimensional projections of the figure on the mutually perpendicular three-dimensional subspace, combined rotation around the plane of intersection. We consider some of the orthogonal projection invariants points on four-dimensional space coordinate hyperplanes. In particular, it is shown that the n-plane perpendicular to the picture hyperplane orthogonally projected (n – 1)-plane, where n ≤ 3. To prove two theorems on the image mutually perpendicular lines, planes and hyperplanes in three-dimensional giperepdrawing. The first theorem is a "four-dimensional analogue" of the theorem on projections of a right angle in three-dimensional space. The second theorem – "four-dimensional analogue" of the theorem about the image of mutually perpendicular lines and planes of three-dimensional space. We consider five examples of computer simulation of four-dimensional figures. In the first example, we solve the problem of constructing a perpendicular from this point of four-dimensional space onto a plane in general position. In the second example, the geometric model is composed of two-dimensional sphere embedded in a hyperplane in general position. In the third example given model of a sphere embedded in projecting the hyperplane. The fourth example is given hypersphere model and definition of ethylene point on its surface. In the fifth example, found the intersection of the hypersphere and direct the overall situation. All examples are considered as three-dimensional giperepdrawing Naumovich, and on the plane of the projection model (orthographic drawing on the generalized Monge). Possibilities of modern three-dimensional computer graphics allow automate the transition from a three-dimensional giperdrawing to a flat drawing at any stage of the modeling process.

Keywords: Giperdrawing Naumovich; Three-dimensional projection; Dimension of the space; Polâritet; Absolute; Hyperplane; Hypersphere.