Abstract
The development of the theory of complex numbers is very closely connected with the geometrical interpretation of ordinary complex numbers as points of a plane. The circle that is the locus of points for which the ratio of the distances from two given points S1 and S2 has a fixed value α is often called the circle of Apollonius of these two points. The chapter discusses dual numbers as oriented lines of a plane. The points of the Lobachevskii plane can be represented as points in the interior of the unit circle Σ; the lines of the Lobachevskii plane are then represented by the diameters of the circle Σ and arcs of circles that are orthogonal to the circle Σ. The Poincaré model may be considered as a representation of the Lobachevskii plane on the plane of a complex variable; it enables one to set up a correspondence between ordinary complex numbers and points of the Lobachevskii plane.
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