In commercial CAD software, one can find ALL the intersections of two plane curves very easily.

As it is shown in this case, one of the curve is a B-spline and the other is a polynomial. Both of the curves are planar curves (meaning located on the same plane). The equations for the curves are known to the CAD software.

I understand there exist some analytical ways to find the intersections if the curves are conics, lines or even some polynomials. But usually, to find curve-curve intersections, we will use numerical methods such as Newton's method by providing an initial guess.

But how do we find ALL the intersections with efficiency? Do we just apply different initial guesses?

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