Реконструкция квадратичного кремонова преобразования

Abstract

The geometric correspondence between the points of two planes can be considered well defined only when base data for its establishing is available, and a construction method by which its possible on the basis of these data for each point in one plane to find the corresponding points in the other one. Quadratic Cremona transformation can be specified by pointing out in the combined plane seven pairs of corresponding points. Naturally there is a need to establish a method for constructing any number of corresponding points. An outstanding Russian geometer K.A. Andreev indicated the linear construction based on the consideration of two correlations by which for each eighth point in the one plane is found the corresponding point of the other one. But in his work was not set up a problem to construct excluded (fundamental) points of quadratic Cremona transformation specified by seven pairs of points. There are many constructive ways to obtain the quadratic transformation in the plane. For example, it can be obtained by using two pairs of projective pencils of straight lines with vertices at the fundamental points (F-points). K.A. Andreev noted that this method for establishing of quadratic correspondence spread only to those cases when all F-points are the real ones. This statement is true for the 19th century’s level of geometric science, but today it’s too categorical. The theory of imaginary elements in geometry allows to develop a universal algorithm for construction of corresponding points in a quadratic transformation, given both by real and imaginary F-points. Summarizing the K.A. Andreev task, we come to the problem of finding the fundamental points (F-points) for a quadratic transformation specified by seven pairs of corresponding points. Almost one and half century the K.A. Andreev generalized task remained unsolved. The formation of this task’s constructive solution algorithm and its practical implementation has become possible by means of modern computer geometric modeling. According to proposed algorithm, the construction of F-points is reduced to the construction of second order auxiliary curves, on which intersection are marked the required F-points. The result received in this paper is used for development of the Cremona transformations’ theory, and for further application of this theory in the practice of geometric modeling.