engleski

Circular Curves of Order Four in Isotropic Plane Produced by Inversion

The problem will be studied on the model of an isotropic plane with the absolute figure (f, F), F incident with f, in finiteness. A curve in the isotropic plane is circular if it passes through the absolute point F. If it does not share any common point with the absolute line f except the absolute point, it is entirely (completely) circular. An involutive mapping of the isotropic plane, where any point and its image are conjugate with respect to a fixed circular conic (a special hyperbola or a circle) q, and simultaneously lie on the lines of a fixed pencil with the vertex P on the absolute line, is called the automorphic quadratic inversion. The conic q is called the fundamental conic, and the vertex P is called the pole of the inversion. The image of the generating conic k is the quartic K. There are five types of the automorphic inversion. For each of them the conditions that the generating conic has to fulfill in order to obtain a circular quartic of certain type will be determined by using the synthetic (constructive) method. It will be demonstrated that an entirely circular quartic can be constructed only when the fundamental conic is a circle, the pole is the absolute point and the generating conic intersects the absolute line in two points different from the absolute point. The same conclusions can be carried out using analytical method on the Euclidean model of the isotropic plane.

isotropic plane; automorphic quadratic inversion; circular quartic

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