Graphical Model of the Biquadratic Transformation

Abstract

For the first time Cremona collected initial information and are the foundations of the classical theory of non-linear birational (Cremona) transformations of the plane and three-dimensional space [1]. Hudson in his monograph gave a more complete overview of the main issues of the classical theory of non-linear plane and three-dimensional space of birational transformations [2]. The main result of this paper is to develop the theoretical foundations of the construction of graphical models biquadratic plane transformations. The spatial mapping scheme of the two surfaces of the second order, established new patterns of producing four-four-correspondences between the two planes misalignment and theoretical principles modeling biquadratic canonical transformations of the plane. For graphical model and biquadratic transformation as well as for the study of their properties, biquadratic convert binary plane were mapped onto the plane of the two surfaces of the second order. At the same time dealt with three cases: (a) a combination of no ruled surfaces of the second order; (b) a combination of cylindrical and conical surfaces of the second order; (c) a combination of hyperboloid of second order [3]. Therefore, this article is dedicated to the development of the theory of building a graphical model transformation biquadratic plane misalignment between the planes and the development of the theory of biquadratic graphical model transformation plane and study their properties. Considered three cases: a combination of no ruled surfaces of the second order; combinations of conical and cylindrical surfaces of the second order; combination of hyperboloid of second order. Developed graphical models biquadratic twelve canonical transformations of the plane.