ABSTRACT
In this paper, continuing work in [1], procedures are considered for computing discrete plane curves subject to constraints on the numerical curvatures, to be piecewise of one sign, monotone, and convex, in a strong sense. Both, linearized and nonlinear constraints for numerical curvatures are analyzed. Subject to such constraints, existence of interpolating Gl and piecewise G2curves satisfying sufficiently strong curvature constraints is shown. The dis crete curves themselves are obtained by a nonlinear optimization algorithm which simultaneously enforces the constraints and ensures the curve deviates from data points by the least possible amount. Due to the constraints, a very small number of data points is required, so this procedure lends itself naturally to CAGD applications. Computing discrete curves (and surfaces) can be a convenient tool in CAGD for such operations as offsets, trimming, and quality control. If needed, many procedures are available for interpo lating or approximating a discrete curve by a smooth parametric curve of smoothness class Gk for sufficiently large k [3]. As observed by several au thors, constraints on the curvature are essential to guarantee quality of curves [1,3,5]. Given a discrete curve, existence of an interpolating or approximating G1-curve satisfying sufficiently strong curvature constraints poses interesting and challenging mathematical problems, which seem to have been little ex plored. In the present report, the results of [1] are improved, strengthened, and generalized.
