I've read that one can quickly test whether two n-degree curves f(t) and g(t) intersect (without finding the roots of an implicitized n*n polynomial) by checking if the determinant of their Bezout matrix is zero. But I am confused about how to actually apply this to two parametric curves, since I have 4 equations instead of two: x1(t), y1(t), x2(t), y2(t) (this is assuming that the two equations can't be represented in explicit form, in my case I'm using two quadratic bezier curves for example). I was trying to see if I can test x1 against x2 and then y1 against y2 independently, but did not get a zero resultant for two intersecting curves (and thinking about it afterwards, it makes sense since the t parameter will not necessarily match between the two curves even if x and y intersect). Another idea I had was computing the Bezout resultant of x1(t)-x2(t) and y1(t)-y2(t) since that seems like it would correspond to the difference between the two curves, but I realized yet again that that wouldn't work since the t is independent between the two curves and these equations do not treat it as such. Can someone point me in the right direction?

Thanks