C1 Convexity-Preserving Interpolation of Scattered Data

We describe a detailed computational procedure which, given data values at arbitrarily distributed points in the plane, determines if the data are convex and, if so, constructs a smooth convex surface that interpolates the data. The method consists of constructing a triangulation of the nodes (data abscissae) for which the triangle-based piecewise linear interpolant is convex, computing a set of nodal gradients for which there exists a convex Hermite interpolant, and constructing a smooth convex surface that interpolates the nodal values and gradients. The method involves two data-dependent triangulations along with a straight-line dual of each, and we describe some interesting relationships among them.

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