Computer geometric modeling of solutions to the systems of quadratic equations of the same kind

Abstract

Abstract In the present paper the opportunity of computer geometric modeling of solutions to systems of algebraic equations of the same kind is researched. The geometric interpretation of such systems and their solutions are known in classical geometry as the Apollonius problem. It is demonstrated that in space of dimension equal to the number of equations of the system, the latter can be given a cyclographic interpretation. This significantly simplifies solution to the initial system as well as the Apollonius problem itself. As a result, the system of equations receives an analytical, i.e. exact solution, while constructive solution of the Apollonius problem is reduced to solution of a positional task of finding common points between a straight line and a cone of revolution. Algorithms for analytical and constructive solutions can be generalized to a space of n dimensions with preserving the analyticity of solutions. Algorithms are realized by means of the tools of CAD systems and computer algebra.