The problem is simple: we have a curve defined by some equation:
Where $g$ may be piecewise or feature a number of conditions, in other words, be very complicated, but the curve is supposed to be at least continuous.
What would be the fastest and the most stable numerical method which outputs any point $(x_0,y_0)$ such that:
I don't really care which point, as I can then use it to find other points all along the curve (due to its continuity). But finding the initial point is not that simple, because the curve may be finite and situated somewhere far from the origin.
One idea I have is to sweep the plane with a strtaight line coming from the origin:
$$y= x \tan \theta_n$$
$$\theta_n=n \theta_0, \qquad n=0,1,2,\dots$$
Then we search for a numerical solution to:
$$g(x,x \tan \theta_n)=0$$
If needed, we can try a few initial guesses until Newton's (or another) method converges or we are reasonably certain there's no solution. Then we increase the angle and try again.
But I don't think this method is the best, so I hope there are some other options.