@vv 
Any point on the surface can be obtained as a single-valued function of (l1, l2), where l1 and l2 are the lengths of curves. (Within the framework of the numerical solution, of course.) And it has no relation to parametrisation of curves.
I would advise you to first look carefully algorithm, and then draw conclusions.

@vv 
 You have shown the exact solution. I showed the approach to the approximate solution. x, y, z depend on the lengths of the curves on the surface. Curves cover the surface. This is shown in the form of animation, and all that is required in the program.

@vv 
Thank you very much.
But the real problems are almost never associated with the exact solutions, so I do not even think about particular cases.
 I'll show more examples with "torus" surfaces.

@hitstudent 

Example numerical parameterization with a similar surface. If you ever wanted like to parse algorithm, I think that the example will be waiting for you here.
(x1^4+x2^4-2.)^2+x3^4 -1.=0;

EXAM_КВ_БУБЛИК.mw

        The spiral on the surface of "Himmelblau":  z= - (x^2+y - 0.3)^2 - (x+y^2 - 0.7)^2+5;
        Rotates the plane, it intersects the surface. Along the curve of intersection of the plane and the surface is laid distance from the center of rotation depending on the rotation angle. Geodesic is not used.

      Cone: "true" spiral and "untrue" spiral
(program is the same as that for the cylinder).

@vv      However, you have a very well-received "untrue" spiral on the cylinder.
         But what do we do with “bad”' surfaces? For example, as this surface: z = 0.01exp(x)/(0.01+x^4+y^4);

@vv The distance from the center of the curve is a continuous monotonic function of the rotation angle. Distance is measured along the surface.
And example without animation:

@vv I wrote the "curve type of spiral". As for the definition; then, say that there is no unambiguous definition.

https://en.wikipedia.org/wiki/Spiral

     Simply as variant location of the curve type of spiral, and as an entertainment. For example, on the cylinder of radius 1. On this curve may be an infinite number of points of self-intersection.

SPIRAL.mw     

If I needed an angle between the vectors, I'd programmed the following algorithm: put vectors at one point, then set aside for each of the vectors equidistant point, construct a circle, and define the angle by circumferential length.

    The crank rotates uniformly around the axis oX2. The red line is defined by two equations:
(x1+.5*sin(5*x3))^2+(x2-2)^2+(x3+.5*sin(2*x1))^2-9=0;
x1^6+x2^6+x3^6-12=0;
Today make kinematic analysis of this mechanism is only possible using the proposed method.
   
     Kinematic analysis.  
General view:



    and the plane X1oX3:

@tomleslie  Geodesic itself is not the shortest distance between two points on the surface. To do this, choose the direction. (It is about surfaces at all.)

Backup plan (if you like).

POINTS_ON_sphere.mw 

@necron  For other solutions you can try to organize a loop, when you see the graph