How to project a parametric curve to a free-form parametric surface

A numerical approach using Newton-Raphson method:

For a fixed point of the curve corresponding to the value $t=t_0$, let us define:

$$\begin{cases}f(u,v)&:=&S_x(u,v)-r_x(t_0)\\g(u,v)&:=&S_y(u,v)-r_y(t_0)\end{cases}$$

The recurrent vector sequence

$$\binom{u_{n+1}}{v_{n+1}}=\binom{u_{n}}{v_{n}}-\begin{pmatrix}\partial S_x/\partial u&\partial S_x/\partial v \\ \partial S_y/\partial u&\partial S_y/\partial v\end{pmatrix}^{\color{red}{-1}}\binom{f(u_n,v_n)}{g(u_n,v_n)}$$

will converge (under the condition to be in a suitable vicinity of the solution) to a solution $\binom{u_{*}}{v_{*}}$, i.e., values of parameters $u$ and $v$ such that

$$\begin{cases}f(u_{*},v_{*})&=&0\\g(u_{*},v_{*})&=&0\end{cases} \iff \begin{cases}S_x(u_{*},v_{*})&=&r_x(t_0)\\S_y(u_{*},v_{*})&=&r_y(t_0).\end{cases}$$

Remark: the value of the $z$ coordinate is of course $z(u_{*},v_{*})$.