Cyclographic Interpretation and Computer Solution of One System of Algebraic Equations

The subject of this study is an algebraic equation of one form and a system of such equations. The peculiarity of the subject of research is that both the equation and the system of equations admit a cyclographic interpretation in the operational Euclidean space, the dimension of which is one more than the dimension of the subspace of geometric images described by the original equations or system of equations. The examples illustrate the advantages of cyclographic interpretation as the basis of the proposed solutions, namely: it allows you to get analytical, i.e. exact solutions of the complete system of equations of the considered type, regardless of the dimension of the subspace of geometric objects described by the equations of the system; in the geometric version of the solution of the system (the Apollonius and Fermat problems), no application of any transformations (inversions, circular transforms, etc.) is required, unlike many existing methods and approaches; constructive and analytical solutions of the system of equations, mutually complementary, are implemented by available means of graphic CAD and computer algebra. The efficiency of cyclographic interpretation is shown in obtaining an analytical solution to the Fermat problem using a computer algebra system. The solution comes down to determining in the operational space the points of intersection of the straight line and the 3-α-rotation cone with the semi-angle α = 45° at its vertex. The cyclographic images of two intersection points in the operational space are the two desired spheres in the subspace of given spheres. A generalization of the proposed algorithm for the analytical solution of the Fermat problem for n given (n – 2)-spheres in (n – 1)-dimensional subspace. It is shown that in this case the analytical solution of the Fermat problem is reduced to determining the intersection points of the straight line and the (n – 1)-α-cone of rotation in the operational n-dimensional Euclidean space.

The Weighted Fermat Triangle Problem

We completely solve thegeneralized Fermat problem: given a triangle , , and three positive numbers , , , find a point for which the sum is minimal. We show that the point always exists and is unique, and indicate necessary and sufficient conditions for the point to lie inside the triangle. We provide geometric interpretations of the conditions and briefly indicate a connection with dynamical systems.