This paper describes an algorithm for monotone interpolation to monotone data on a rectangular mesh by piecewise bicubic functions. In [SIAM J. Numer. Anal. 22 (1985), pp. 386–400] the authors developed conditions on the Hermite derivatives that are sufficient for such a function to be monotonic. The present paper rewrites some of these conditions and presents a much simpler five-step algorithm for satisfying them that produces a visually pleasing monotone interpolant. The result of the algorithm does not depend on the order of the independent variables nor on whether the inequalities are swept left-to-right or right-to-left.

An algorithm for monotone piecewise bicubic interpolation

A method for constructing shape-preserving surfaces interpolating arbitrary sets of data on rectangular grids is presented. The surfaces are tensor product splines of arbitrary continuity class which extend some properties of univariate shape-preserving splines given in earlier papers.

Shape-preserving bivariate interpolation

Computer graphics and CAD literature—A keyword-indexed bibliography

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.

A bibliography of multivariate approximation

An algorithm plotting contour lines for discrete values zij, given at the nodes of a rectangular mesh is described. A bicubic Hermite polynomial f(x, y) is determined for every rectangle of the mesh, interpolating the zij and the derivatives zx, zy, and zxy. The derivatives are optionally computed by the algorithm. The contours found are normally smooth curves. They consist of polygons approximating intersections with the bicubics. It is possible to fill the areas between them with certain colors or patterns. This is done with a piecewise technique rectangle by rectangle. The method for finding the points of the polygons is shortly reviewed, and some numerical problems are pointed out. The algorithm has a flexible, easy-to-use interface and is easily installed with all plotting systems, provided that a fill-area command is available. A GKS interface may be used.

R. E. Carlson

Papers

Monotonicity preserving bicubic interpolation: A progress report