Smooth bivariate shape-preserving cubic spline approximation

• Shape-preserving splines are defined on triangulations of a new type.
• The key element is a combination of two-sided and classic Clough–Tocher macros.
• Formulas for spline coefficients are simple, explicit with respect to local data.
• Proofs of convexity, monotonicity and positivity preservation are included.

Given a piece-wise linear function defined on a type I uniform triangulation we construct a new partition and define a smooth cubic spline that approximates the linear surface and preserves its shape. The key piece is a new macro-element that has the ability to combine six independent gradients coming together at an interior vertex in a smooth yet shape-preserving fashion. The shape of the resulting spline surface follows local changes in the shape of the piece-wise linear interpolant without overshooting. We prove that convexity, positivity and monotonicity of the spline depend on the local data only. Computational scheme for Bernstein–Bezier spline coefficients is local and fast. Numerical examples highlight unique shape-preserving properties of the spline.

Figure optionsDownload high-quality image (64 K)Download as PowerPoint slide