Suppose $f(x,y)$ is a complex function of two real arguments with roots* that are not discrete points but lie in curves. (Is there are term for this characteristic?) An example is shown below: the black curves show all the points in the $(x, y)$ plane where $f(x, y) = 0$. What is the best way to find these curves numerically, within a rectangular region?
The obvious solution is to consider 1-D slices along the $x$- or $y$-axis, and use standard 1-D root-finding algorithms to find the discrete roots along these slices. These points can then be joined up appropriately to form the curves. However I wonder if there is a more efficient strategy, taking into account 2-D information.
Answers can assume some of the properties shown in the example plot. The curves do not terminate within the rectangular region, they do not intersect, and they always have a negative gradient.
*Definition of root: a point $(x_r, y_r)$ such that $f(x_r, y_r) = 0$.