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

• Initiate study of rational surfaces generated from a planar curve and a space curve.
• We construct moving planes that follow the rational surfaces.
• We construct a sparse Sylvester style matrix for four special bidegrees polynomials.
• Use this resultant in conjunction with moving planes to implicitize the surfaces.

A rational surfaceequation(1)S(s,t)=(A(s)a(t),B(s)b(t),C(s)c(t),C(s)d(t))S(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))P⁎(s)=(A(s),B(s),C(s)) and a rational space curve P(t)=(a(t),b(t),c(t),d(t))P(t)=(a(t),b(t),c(t),d(t)). Let P⁎(s)P⁎(s) pass through the point (1,1,1)(1,1,1). Then the surface S(s,t)S(s,t) goes through the space curve P(t)P(t). Moreover on each z=z⁎z=z⁎-plane, the cross section of the surface S(s,t)S(s,t) is a stretching or shrinking of the planar curve P⁎(s)P⁎(s), such that the point (1,1,z⁎,1)(1,1,z⁎,1) travels to the point P(t)∩{z=z⁎}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)S(s,t) without any extraneous factors. Examples are presented to illustrate our methods.