Finding the shortest path length on a curved surface(hyperboloid)

Question

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

Answer 1

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. )