engleski

Circular curves of the 3rd class in the quasi-hyperbolic plane

The quasi-hyperbolic plane is one of the nine Cayley-Klein projective metrics where the metric is induced by an absolute figure F_{; ; QH}; ; ={; ; F, f_1, f_2}; ; , consisting of two real lines f_1 and f_2 incidental with the real point F. A curve of class n is circular in the quasi-hyperbolic plane if it contains at least one absolute line. In this presentation we will study the curves of the 3rd class obtained by projective mapping i.e. obtained by projectively linked pencil of curves of the 2nd class and range of points. The conditions that pencils, range and projectivity have to fullfil in order to obtain a circular curve of the 3rd class of a certain type of circularity will be determined analytically. It will be shown that the curves of 3rd class of all types (depending on their position with respect to the absolute figure) can be constructed by using these results.

quasi-hyperbolic plane; circular curve of 3rd class; projective mapping

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