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Date of Publication:2009-01-01

Journal:大连理工大学学报

Affiliation of Author(s):数学科学学院

Issue:1

Page Number:152-156

ISSN No.:1000-8608

Abstract:The Crofton formula in the Euclidean plane reveals the relationship between the length of a plane curve segment and the measure of the set of lines intersecting with it. It can be used to compute the approximated value of the length of a complicated plane curve segment. The Crofton formula for arbitrary curve segment in the n dimensional real hyperbolic space is concerned. At first, the n dimensional real hyperbolic space H+n(-1) is considered as the set of all h-unit time-like vectors in the n+1 dimensional Minkowski space R1n+1. Then, the set of the n-1 dimensional complete totally geodesic hypersurfaces of H+n(-1) is transfered to the set of the h-unit space-like vectors in R1n+1via the one to one correspondence between the n dimensional oriented linear subspace and its h-unit normal vector. Finally, the Crofton formula for an arbitrary curve segment in the n dimensional real hyperbolic space is obtained via computing the invariant measure of the set of all h-unit normal vectors of the n dimensional linear spaces in n+1 dimensional Minkowski space intersecting with the given curve segment.

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