Shape Preserving Interpolation to Convex Data by Rational Functions with Quadratic Numerator and Linear Denominator

By using rational functions of the type quadratic/linear (with quadratic numerator and linear denominator), we show in this paper how to construct, in a straightforward way, a convex C1 interpolant to convex data. In Delbourgo & Gregory (1985b) a cubic/quadratic rational form is discussed for this problem. A special case arises there which allows a reduction to the quadratic/linear form of the present paper. This simpler form was not evident at the time and we give here an independent account of the relevant theory which we support by numerical examples. Finally we examine the consistency conditions (a set of nonlinear equations) for second-derivative continuity. We prove that a unique solution exists which satisfies the convexity requirements.