I have a curved surface $z=x^2-y^2$. Given a point (x,y) on the surface, how do I calculate the (3D) distance from it to the origin (0,0,0)? By distance I am referring to the distance on top of the surface, not the straight line that can easily be calculated.
What I have found is that on the line (x,y), the surface behaves like the parabola $(1-y/x) (\sqrt{x^2+y^2})^2=(1-y/x)(x^2+y^2)$. Therefore, I can find the length using the formula $d(x,y)=\frac{1}{2(1-y/x)}\int_{0}^{\sqrt{x^2+y^2}}{\sqrt{1+t^2}dt}$.
However this result cannot be correct:
upon using Desmos to plot the equation $d(x,y)=1$ to see how a circle on the hyperbolic surface look like, the result was not a closed shape:
It also appears like the distance close to $x=0$ but far away from $(0,0,0)$ is 1 unit distance from the origin, when in fact it's much further away. Also, in the origin it appears as if the distance is 1 unit, when obviously it should be 0.
Have I done some of my calculations wrong, am I misinterpreting the results, or am I missing something?
Thanks in advance.