Algebraic properties of plane offset curves

Abstract

We consider the offsets to a polynomial or rational parametric generator r(t) as algebraic curves, specified by implicit equations f{hook}o(x, y)=0. Efficient resultant formulations for computing these irreducible equations-which represent simultaneously the 'exterior' and 'interior' offsets-are presented, and certain special conditions, which cause the degree of f{hook}o(x, y) to be diminished, are identified. These include the presence of cusps, tangencies to the line at infinity, and passages through the circular points at infinity. (In the case of a rational generator with cusps or tangencies to the line at infinity coincident with the circular points, we remedy an apparent defect in the classical degree formula due to Cayley.) For polynomial or rational generators, the theory of singular points of algebraic curves offers further insight into the nature of the 'cusps' and 'extraordinary points' of offset curves, which we have previously described from a purely analytic perspective. Algebraic methods based on polynomial resultant and real-root isolation procedures also furnish an algorithmic basis for identifying the self-intersections of offset curves, and hence computing the trimmed offsets which are desired in most practical applications. © 1990.