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We study the hyperbolic cosine and sine laws in the extended
hyperbolic space which contains hyperbolic space as a subset and is
an analytic continuation of the hyperbolic space. And we also study
the spherical cosine and sine laws in the extended de Sitter space
which contains de Sitter space S^n_1 as a subset and is also an
analytic continuation of de Sitter space. In fact, the extended
hyperbolic 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 and apparently explain that why many
corresponding formulas in hyperbolic and spherical space are very
similar each other. From these extended trigonometry laws, we can
give a coherent and geometrically simple explanation for the various
relations between the lengths and angles of hyperbolic polygons, and
relations on de Sitter polygons which lie on S^2_1, and tangent
laws for various polyhedra.
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참고문헌 (14)
참고문헌 신청
B. O’Neill / 1983 / Semi-Riemannian Geometry / Academic Press
D. Derevnin / / On the volume of spherical Lambert cube / arXiv:math. MG/0212301
D. V. Alekseevskij / 1993 / Geometry of spaces of constant curvature, Geometry, II / Encyclopaedia Math. Sci 29 : 1 ~ 138
E. B. Vinberg / 1993 / Volumes of non-Euclidean Polyhedra / Russian Math. Surveys 48 (2) : 15 ~ 45
J. G. Ratcliffe / 1994 / Foundations of Hyperbolic Manifolds, Graduate Texts in Mathematics / Springer-Verlag : 149 ~
D. Derevnin / / On the volume of spherical Lambert cube / arXiv:math. MG/0212301
D. V. Alekseevskij / 1993 / Geometry of spaces of constant curvature, Geometry, II / Encyclopaedia Math. Sci 29 : 1 ~ 138
E. B. Vinberg / 1993 / Volumes of non-Euclidean Polyhedra / Russian Math. Surveys 48 (2) : 15 ~ 45
J. G. Ratcliffe / 1994 / Foundations of Hyperbolic Manifolds, Graduate Texts in Mathematics / Springer-Verlag : 149 ~
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