Trigonometry in extended hyperbolic space and extended de Sitter space - Mathematics > Metric GeometryReportar como inadecuado




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Abstract: We study the hyperbolic cosine and sine laws in the extended hyperbolic spacewhich contains hyperbolic space as a subset and is an analytic continuation ofthe hyperbolic space. And we also study the spherical cosine and sine laws inthe extended de Sitter space which contains de Sitter Space $S^n 1$ as a subsetand is also an analytic continuation of de Sitter space. In fact, the extendedhyperbolic space and extended de Sitter space are the same space only differ by-1 multiple in the metric. Hence these two extended spaces clearly show andapparently explain that why many corresponding formulas in hyperbolic andspherical space are very similar each other. From these extended trigonometrylaws, we can give a coherent and geometrically simple explanation for thevarious relations between the lengths and angles of hyperbolic polygons andrelations on de Sitter polygons which lie on $S^2 1$.



Autor: Yunhi Cho

Fuente: https://arxiv.org/







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