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Given two points A and B on the surface of the hyperboloid x^2+y^2-z^2=1. How to find the shortest distance between them along the surface?

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You could use the well known results on geodesics of surfaces of revolution (due to Clairaut), explained in standard textbooks on differential geometry of surfaces. Try do Carmo's book, Differential Geometry of Curves and Surfaces, for example. This question is not appropriate for this web site, which is for current research problems in mathematics. Try math.stackexchange.com. –  Ben McKay Jan 18 '12 at 10:10
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closed as too localized by S. Carnahan Jan 18 '12 at 23:50

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1 Answer

Using the following formula $$ \cosh(d(A,B))=\frac{|q(A,B)|}{|q(A)|^{-1/2}|q(B)|^{-1/2}}, $$ where $$ q(A,B)=A^t I_{2,1} B = a_1b_1+a_2b_2-a_3b_3 $$ with $I_{2,1}=diag(1,1,-1)$.

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But given A = (1, 0, 0), B = (-1, 0, 0), d(A, B) = 0 computed from the formula above but it's not true. As far as I understand q(A) = q(A, A), right? –  Igor Demidov Jan 20 '12 at 9:53
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