Progressive iterative approximation for triangular Bézier surfaces

Recently, for the sake of fitting scattered data points, an important method based on the PIA (progressive iterative approximation) property of the univariate NTP (normalized totally positive) bases has been effectively adopted. We extend this property to the bivariate Bernstein basis over a triangle domain for constructing triangular Bézier surfaces, and prove that this good property is satisfied with the triangular Bernstein basis in the case of uniform parameters. Due to the particular advantages of triangular Bézier surfaces or rational triangular Bézier surfaces in CAD (computer aided design), it has wide application prospects in reverse engineering.

► The PIA (progressive iterative approximation property) of the univariate NTP basis is extended to the bivariate Bernstein basis over a triangle domain.
► This good property is satisfied with the triangular Bernstein basis in the case of uniform parameters.
► PIA algorithms are also useful for rational triangular Bézier surfaces of degrees 2 and 3.