I'm currently studying the implicitization of bezier curves (that is, finding a function that f(x, y) = 0 for any x and y pairs of a curve p(t)) as part of an algorithm for curve intersection. The approach I'm taking involves taking the resultant of two functions known as "moving lines." For a cubic curve, the two functions required are one quadratic and one linear:
a(t) = (a*x + b*y + c)*(1 - t)2 + (d*x + e*y + f)*(1 - t)*t + (g*x + h*y + i)*t2
b(t) = (k*x + l*y + m)*(1 - t) + (n*x + o*y + p)*t
In the paper I'm reading, a process for taking the resultant of two functions of similar degree, which I assume to be Bezout resultant, is presented, but not for functions of differing degree. From what I can gather, I need a resultant process that eliminates two variables, t and (1 - t), in functions of differing degree. As previously stated, the Bezout resultant is already out due to the differing degrees of my functions. Additionally, the Dixon approach seems to be inapplicable as it only works with a system of n+1 polynomials of degree n. I read that the Sylvester resultant could be used iteratively to eliminate two variables, but I couldn't find any details on that process.
What I'd like to ask is if anyone can provide details or a source on using Sylvester's resultants to iteratively remove more than one variable. However, if you know of another method to obtain the resultant of two polynomials of degree n and n - 1 in two variables, I'd be just as grateful. I apologize if this is not a suitable question for this site.