Smooth curve interpolation with generalized conics

Abstract

Efficient algorithms for shape preserving approximation to curves and surfaces are very important in shape design and modelling in CAD/CAM systems. In this paper, a local algorithm using piecewise generalized conic segments is proposed for shape preserving curve interpolation. It is proved that there exists a smooth piecewise generalized conic curve which not only interpolates the data points, but also preserves the convexity of the data. Furthermore, if the data is strictly convex, then the interpolant could be a locally adjustable GC2 curve provided the curvatures at the data points are well determined. It is also shown that the best approximation order is script O sign(h6). An efficient algorithm for the simultaneous computation of points on the curve is derived so that the curve can be easily computed and displayed. The numerical complexity of the algorithm for computing N points on the curve is about 2N multiplications and N additions. Finally, some numerical examples with graphs are provided and comparisons with both quadratic and cubic spline interpolants are also given.