In a 1980 paper [SIAM J. Numer. Anal., 17 (1980), pp. 238–246] the authors developed a univariate piecewise cubic interpolation algorithm which produces a monotone interpolant to monotone data. This paper is an extension of those results to monotone $\mathcal{C}^1 $ piecewise bicubic interpolation to data on a rectangular mesh. Such an interpolant is determined by the first partial derivatives and first mixed partial (twist) at the mesh points. Necessary and sufficient conditions on these derivatives are derived such that the resulting bicubic polynomial is monotone on a single rectangular element. These conditions are then simplified to a set of sufficient conditions for monotonicity. The latter are translated to a system of linear inequalities, which form the basis for a monotone piecewise bicubic interpolation algorithm.

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Monotone Piecewise Cubic Interpolation

In this paper we discuss the design of algorithms for interpolating discrete data using $C^1 $-quadratic splines in such a way that the monotonicity and/or convexity of the data is preserved. The analysis culminates in an interactive algorithm which takes full advantage of the flexibility which quadratic splines permit.

/pdf/on-shape-preserving-quadratic-spline-interpolation-17m5yddmt0.pdf

On Shape Preserving Quadratic Spline Interpolation

Algorithms are presented for computing a quadratic spline interpolant with variable knots which preserves the monotonicity and convexity of the data. It is also shown that such a spline may not exist for fixed knots.

https://www.ams.org/mcom/1978-32-144/S0025-5718-1978-0481734-6/S0025-5718-1978-0481734-6.pdf

Interpolation by convex quadratic splines

A general setting is given for smooth interpolating splines depending on a parameter such that as this parameter approaches infinity the spline converges to the piecewise linear interpolant. The theory includes the standard exponential spline in tension, a rational spline, and several cubic splines. An algorithm is given for one of the cubics; the parameter for this example controls the spacing of new knots which are introduced.

/pdf/alternatives-to-the-exponential-spline-in-tension-2l3y9jw2h5.pdf

Alternatives to the exponential spline in tension

Publisher Summary This chapter presents a bibliography of multivariate approximation. Multivariate interpolation and approximation have always been an important part of general Approximation Theory, however, in the past ten years or so, there has been a significant increased interest in multivariate problems. A very substantial number of both theoretical and practical results have been obtained on a variety of topics, ranging from the classical tensor-product methods to the newfangled multivariate splines. In addition to their applicability to such classical problems as data fitting and numerical solution of operator equations, modern multivariate methods have become important in several new applied areas, such as computer-aided design, robotics, and computer-aided tomography.

R. Peter Dube

Papers

Univariate Blending Functions and Alternatives

Tension in a bicubic surface patch