Rational-Ruled Surfaces: Implicitization and Section Curves

Abstract This paper shows how to express the implicit equation of a degree 1 × n tensor product rational surface as a determinant of dimension at most n × n . If the rational surface has base points whose multiplicities sum to p , the implicit equation can still be expressed in a single determinant with no extraneous factors. As part of the derivation, it is shown that the given ruled surface can be expressed using two directrices whose degrees sum to 2 n - p . It is also noted that any plane section curve of such a surface patch is rational, and formulas for the control points of the rational Bezier representation of that curve are given.