Hermite interpolation of rational space curves using real algebraic surfaces

We present a simple characterization of the lowest degree, implicitly defined, real algebraic surfaces, which smoothly contain any given number of points and algebraic space curves, of arbitrary degree. The characterization is constructive, yielding efficient algorithms for generating families of such algebraic surfaces. Smooth containment of space curves yields C1-continuous surface fitting, and is a generalization of standard Hermite interpolation applied to fitting curves through point data, equating derivatives at those points. We deal with the containment and matching of “normals” (vectors orthogonal to tangents), possibly varying along the entire span of the space curves. Such Hermite interpolated surfaces prove useful as “blending” or “joining” surfaces for solid models as well as “fleshing” surfaces for curved wireframe models.