Adjusting Curvatures of B-spline Surfaces by Operations on Knot Vectors

The knot vectors of a B-spline surface determine the basis functions hereby, together with the control points, the shape of the surface. Knot manipulations and their influence on the shape of curves have been investigated in several papers (see e.g. [4] and [5]). The computations can be made very efficiently, if the basis functions and the vector function of the B-spline surface are represented in matrix form (see [1] and [6]). In our latest work [2] we summarized the knot manipulation techniques and the corresponding computations in matrix form. We also developed an algorithm for a direct knot sliding, how a knot can be repositioned in one step instead of inserting a new knot value, then removing an old one from the knot vector. In this paper we analyse the effect of varying knot intervals on the Gaussian curvature of a B-spline surface at a given point. We present an algorithm for the deformation of a B-spline surface, so that it should go through a given point with a given Gaussian curvature. The result of this deformation is, that a sphere with a given radius will fit tangential the reshaped surface at the given point with equal Gaussian curvatures. In applications the same situation arises, when a ball-end tool is pushed into a surface during processing. In our algorithm we use only linear interpolation equations besides the repositioning of knot values, in order to get numerically stable and effective solutions.