Convex Preserving Scattered Data

We use bivariate C 1 cubic splines to deal with convexity preserving scattered data interpolation problem. Using a necessary and suucient condition on Bernstein-B ezier polynomials, we set the convexity preserving interpolation problem into a quadratically constraint quadratic programming problem. We show the existence of convexity preserving interpolatory surfaces under certain conditions on the data. That is, under certain conditions on the data, there always exists a convexity preservation C 1 cubic spline interpolation if the tri-angulation is reened suuciently many times. We then replace the quadratical constrains by three linear constrains and formulate the problem into linearly constraint quadratic programming problems in order to be able to solve it easily. Certainly, the existence of convexity preserving interpolatory surfaces is equivalent to the feasibility of the linear constrains. We present a numerical experiment to test which of these three linear constraints performs the best. x1. Introduction In computer aided geometric design, we often encounter to design an interpolation surface with convexity or concavity property. Typically, given a set of scattered data 's may be obtained from a convex function), we need to nd a smooth surface S (e. and s is convex. Many researchers have derived necessary and suucient conditions to ensure the convexity of Bernstein-B ezier polynomials over triangular domain and have studied Supported by the National Science Foundation under grant DMS-9870187.