Using moving planes to implicitize rational surfaces generated from a planar curve and a space curve

A rational surfaceS(s,t)=(A(s)a(t),B(s)b(t),C(s)c(t),C(s)d(t)) can be generated from a rational planer curve P∗(s)=(A(s),B(s),C(s)) and a rational space curve P(t)=(a(t),b(t),c(t),d(t)). Let P∗(s) pass through the point (1,1,1). Then the surface S(s,t) goes through the space curve P(t). Moreover on each z=z∗-plane, the cross section of the surface S(s,t) is a stretching or shrinking of the planar curve P∗(s), such that the point (1,1, z∗,1) travels to the point P(t)∩{z=z∗}. Using moving planes, we provide a new technique to implicitize this kind of rational surface. We find four moving planes that follow the surface from which we construct a sparse matrix whose size is just the degree of the surface with entries linear in x, y, z, w. We prove that the determinant of this matrix is the exact implicit equation of the surface S(s,t) without any extraneous factors. Examples are presented to illustrate our methods.