CYCLIDES AS A SPECIAL TYPE OF CYCLICAL ELLIPTICAL PEDAL SURFACES

Abstract

Intensive use of geometrical modelling started with the development of computer graphics and CAD/CAM technologies, comprising a wide area of mathematical disciplines. Geometrical modelling is a synthesis of the geometry and computer graphics, which enable us to develop complex mathematical models that would be rather difficult to display without using a computer. We recognize two basic research methods of geometrical models, the synthetic and the analytic one. The former method makes use of geometrical constructions whereas the latter characterizes geometrical objects by numerical data. Both methods have been applied in this contribution. The aim of this research is to show central cyclides (Dupin´s cyclides) as a special kind of the cyclical elliptical pedal surfaces. Firstly, using the method of synthetic geometry, a new class of such surfaces in the 3-dimensional Euclidean space (model of the projective space) is defined. Geometrical construction of these surfaces is dependent on the given ellipse and the position of the pole P. It is the point at which the pencil of perpendicular planes passes to the plane of the ellipse, wherein the generating circles of the surface are lying. The parametric equations of the cyclical elliptical surface are derived applying the method of analytic geometry. We classify the surfaces according to the number of generating circles with a zero radius. The evolute of the ellipse divides the plane of the ellipse into two areas, Ω1 and Ω 2. The shape of the surface depends on the position of the pole P in the areas of Ω1 and Ω2. A surface can have 4, 3 or 2 circles with a zero radius. We then describe the relationship between these surfaces and the Dupin´s cyclides, obtained in the case when the pole P is the point on the major axis of the ellipse. Finally, the transformation of these surfaces is shown by changing of the orthonormal base in the parameterization of the generating circles of the surface. The resulting surfaces obtained by the parametric approach are visualized in the MAPLE program environment.