On Knot Modifications of B-Spline or NURBS Surface

Modifying some knots of a B-spline or NURBS surface of order k×h to generate a family of B-spline or NURBS surface is presented. We show that the envelope of the family is a B-spline or NURBS surface defined by the same control points, and its order is (k-a)×(h-b), where a,b are the multiplicity of the modified knots. Moreover, any order partial derivatives of the B-spline surface and the family of B-spline surfaces differ only in a multiplier. The presented results can be served as a theoretical guidance for surface modeling and shape modifications in computer aided design today.