In order to make some illustrative figures, I would like to be able to find the equation of a vector function $$ r(t)=\left< x(t),y(t),z(t)\right> $$ such that it passes through 3 specified points $A=(A_x,A_y,A_z)$, $B=(B_x,B_y,B_z)$ and $C=(C_x,C_y,C_z)$ all of which are on a specified surface $f(x,y)$; and such that $r(t)$ travels on the surface $f(x,y)$.
What I am doing right now is find the intersection between two surfaces, so that I can plot the resulting vector function.
For example, for $z_1(x,y) = - \frac15x^2 - \frac15y^2$ and $z_2(x,y)=x$, their intersection gives the vector function $r(t)=\left< t,\pm \sqrt{-5t-t^2},t\right>$. I can then plot it within the desired $t$ domain so that the figure shows $r(t)$ smoothly travelling on $z_1(x,y)$.
However, this process becomes tedious when I need that two different vector functions that have the same endpoints on the surface $z_1(t)$ (I have to establish another intersection), because then I have to guess the appropriate intersecting surfaces that give me the desired result.
What would be a better procedure?