Curve implicitization using moving lines

Abstract It is a classical result that two corresponding pencils of lines intersect in a conic section, and likewise any conic section can be expressed as the intersection of two pencils of lines. We here extend the idea of pencils to higher degree families lines, and show that any planar rational curve can be expressed as the intersection of two families of lines. This extension leads to a more efficient implicitization algorithm for curves, in which, for example, the implicit equation of a degree four rational curve can generally be expressed as the determinant of a 2 × 2 matrix (Bezout's resultant produces a 4 × 4 matrix and Sylvester's resultant an 8 × 8 matrix).