Any practical curve-blending algorithm should produce blending curves without singularities or self-intersections. In addition, in a parametric modeling environment, the blending curve (and its tangent directions) should depend continuously on the boundary conditions. We prove that, in any dimension, no algorithm having all these properties exists. Any curve-blending algorithm will either exhibit discontinuous dependence on the boundary conditions or produce curves with singularities and self-intersections for a wide range of boundary conditions. Furthermore, we show that these unavoidable singularities must be severe (as measured against any practical criteria).
We also investigate analogous questions for surface-blending algorithms and prove that any algorithm will either exhibit discontinuous dependence on the boundary conditions or produce selfintersecting surfaces for some simple boundary conditions.
We discuss the practical significance of these results for CAD/CAM designers, and we present the results of an experimental comparison of several "industrial" planar curve-blending algorithms. The special case of Hermite blending curves is investigated in an appendix.