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The inverse Gudermannian in the hyperbolic geometry

ABSTRACT We present the first applications of the inverse Gudermannian in the geometry of the hyperbolic spaces of positive curvature. A hyperbolic space of positive curvature can be realised on the ideal domain of the Lobachevskii space . In this paper, the metric dependences for various figures of the plane are written down by the means of the inverse Gudermannian. The volume formula for a finite light cone of the space is obtained.