Using μ-bases to implicitize rational surfaces with a pair of orthogonal directrices

A rational surfaceS(s,t)=(a(t)a⁎(s),a(t)b⁎(s),b(t)c⁎(s),c(t)c⁎(s))S(s,t)=(a(t)a⁎(s),a(t)b⁎(s),b(t)c⁎(s),c(t)c⁎(s)) can be generated from two orthogonal rational planar directrices: P(t)=(a(t),b(t),c(t))P(t)=(a(t),b(t),c(t)) in the xz  -plane and P⁎(s)=(a⁎(s),b⁎(s),c⁎(s))P⁎(s)=(a⁎(s),b⁎(s),c⁎(s)) in the xy  -plane. Moving a scaled copy of the curve P⁎(s)P⁎(s) up and down along the z  -axis with the size controlled by the curve P(t)P(t), we get the surface S(s,t)S(s,t). For example, when P⁎(s)P⁎(s) is a circle with center at the origin, the surface S(s,t)S(s,t) is a surface of revolution. Many other useful and interesting surfaces whose cross sections are not circles can also be generated in this manner. We provide a new technique to implicitize this kind of rational surface using μ  -bases. Let P(t)P(t) be a rational planar curve of degree n with a μ-basis consisting of two moving lines of degree μ   and n−μn−μ, and let P⁎(s)P⁎(s) be a rational planar curve of degree m with a μ  -basis consisting of two moving lines of degree μ⁎μ⁎ and m−μ⁎m−μ⁎. From the μ  -bases for these two directrix curves P(t)P(t), P⁎(s)P⁎(s), we can easily generate a μ  -basis for the surface S(s,t)S(s,t) consisting of three moving planes that follow the surface with generic bidegrees (m−1,μ)(m−1,μ), (m−1,n−μ)(m−1,n−μ), (m,0)(m,0). To implicitize the surface S(s,t)S(s,t), we construct a (2m−1)n×(2m−1)n(2m−1)n×(2m−1)n sparse resultant matrix Rs,tRs,t for these three polynomials. We show that det(Rs,t)=0det(Rs,t)=0 is the implicit equation of the surface S(s,t)S(s,t) with a known extraneous factor of degree (m−1)n(m−1)n. To decrease the size of this matrix and to eliminate entirely the extraneous factor, we construct a new mn×mnmn×mn Sylvester style sparse matrix Ss,tSs,t from four moving planes that follow the surface S(s,t)S(s,t). We prove that det(Ss,t)=0det(Ss,t)=0 is the exact implicit equation of the surface S(s,t)S(s,t) without any extraneous factors. Examples are presented to illustrate our methods.

► We initiate the study of rational surfaces with orthogonal directrices.
► We compute a μ-basis for these rational surfaces from μ  -bases for their directrices.
► We construct a sparse Sylvester style resultant matrix for three bivariate polynomials of bidegrees (m1,n1)(m1,n1), (m1,n2)(m1,n2), (m2,0)(m2,0).
► We use this sparse resultant in conjunction with our μ-bases to implicitize rational surfaces with orthogonal directrices.