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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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closed as too localized by S. Carnahan Jan 18 '12 at 23:50

This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

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 – Ben McKay Jan 18 '12 at 10:10

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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