I wish to find the minimum path length between two points $P_1(\sqrt2,0,-1)$ and $P_2(0,\sqrt2,1)$ on a hyperbolic surface $S =\{(x,y,z)\in R^3\ |\ x^2+y^2-z^2=1\}$

I faintly recall studying something similar when I was into analytical mechanics, calculus of variation. Nevertheless, I have no idea how to solve this... I'd prefer elementary solutions if possible

Thanks in advance

  • $\begingroup$ Do you have any differential geometry or surfaces background? $\endgroup$ – Semsem Mar 30 '14 at 15:00
  • 1
    $\begingroup$ This is a ruled surface, the solution may be very short! $\endgroup$ – Alan Mar 30 '14 at 15:26
  • $\begingroup$ @Semsem none, i guess $\endgroup$ – Chanhee Jeong Apr 7 '14 at 13:41

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The hyperboloid has parametric equation: $$ x(t,u) = \cos(u) - t \sin(u)$$ $$ y(t,u) = \sin(u) + t \cos(u)$$ $$ z(t,u) = t $$ with $ u \in [0,2\pi) , t \in R $.

Put $ u= \pi/4 $ in the above equation to recover the line through the points $P_1(\sqrt2,0,-1)$ and $P_2(0,\sqrt2,1)$ .

( I'd like to see a general discussion of geodesics on quadric surfaces , evidently the ellipsoid is the most interesting case. )


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