Convexity preserving splines over triangulations

Abstract

A general method is given for constructing sets of sufficient linear conditions that ensure convexity of a polynomial in Bernstein-Bézier form on a triangle. Using the linear conditions, computational methods based on macro-element spline spaces are developed to construct convexity preserving splines over triangulations that interpolate or approximate given scattered data.

On the convexity of piecewise-defined functions

“…Definition 3.1 (compatible systems of sets) Assume (11). We say that A is a compatible system of sets if (12) i ∈ I j…”

Integer Generalized Splines on Cycles

“…Mathematicians chose the term spline to refer to the piecewise polynomial functions used to create smooth curves. This method of constructing curves is used in data interpolation [12,20], in computer design software to create smooth surfaces [13,18], and partial differential equations to find numerical solutions [9,14].…”