A necessary and sufficient condition for a ``sphere'' to separate points in euclidean, hyperbolic, or spherical space
Authors:
J. E. Valentine and S. G. Wayment
Journal:
Trans. Amer. Math. Soc. 176 (1973), 285295
MSC:
Primary 50C05; Secondary 50B10
MathSciNet review:
0313927
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Abstract: The purpose of this paper is to give conditions wholly and explicitly in terms of the mutual distances of points in nspace which are necessary and sufficient for two of the points to lie in the same or different components of the space determined by the sphere which is determined by of the points. Thus in euclidean space we prove that if the cofactor of the element in the determinant is nonzero then lie in the same or different components of (where denotes the sphere or hyperplane containing the remaining points) if and only if or , respectively. In hyperbolic space the result is: if the cofactor of the element in the determinant is nonzero then lie in the same or different components of (where denotes the hyperplane, sphere, horosphere, or one branch of an equidistant surface containing the remaining points) if and only if or , respectively. For spherical space we obtain: if the cofactor of the element in the determinant is nonzero then lie in the same or different components of (where denotes the sphere containing the remaining points which may be an dimensional subspace) if and only if or respectively.
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 A. Cayley, On a theorem in the geometry of position, Cambridge Math. J. 2 (1841), 267271.
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 G. Darboux, Sur les rélations entre les groupes de points, de cercles et de sphères dans le plan et dans l'espace, Ann. Sci. de l'E.M.S. (2) 1 (1872), 323392. MR 1508589
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 D. M. Y. Sommerville, The elements of nonEuclidean geometry, G. Bell, London, 1914.
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 J. E. Valentine, An analogue of Ptolemy's theorem in spherical geometry, Amer. Math. Monthly 77 (1970), 4751. MR 40 #7932. MR 0254725 (40:7932)
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 , An analogue of Ptolemy's theorem and its converse in hyperbolic geometry, Pacific J. Math. 34 (1970), 817825. MR 42 #5151. MR 0270261 (42:5151)
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 J. E. Valentine and E. Z. Andalafte, A metric characterization of ``spherical'' surfaces in ndimensional hyperbolic space, J. Reine Angew. Math. 251 (1971), 142152. MR 0293481 (45:2558)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197303139274
PII:
S 00029947(1973)03139274
Keywords:
Components,
determinant,
euclidean,
hyperbolic,
sphere,
spherical
Article copyright:
© Copyright 1973 American Mathematical Society
