A Constructive Algorithm for Transforming Conics in a Circle on Imaginary Laguerre Points

The article is devoted to the consideration of the properties of Laguerre points in their constructive relationship with the solution of the problem of transforming a pair of conical curves into two circles. Geometric schemes are presented that reveal the principle of specifying a collinear transformation that establishes a correspondence between conics and their images in the form of circles, including if the Laguerre points defining this transformation are imaginary. In the solution of the problem, a geometric scheme for the formation of the main conjugations of conics is given, one of which is a circle. The conclusion is substantiated that the radical axis of the bundle of circles is a circle that has split into a pair of straight lines, one of which is infinitely distant. An algorithm for the formation of a series of elliptic curves induced by Laguerre points is presented.