Some smoothness conditions and conformality conditions for bivariate quartic and quintic splines

This paper is concerned with a study of some new formulations of smoothness conditions and conformality conditions for multivariate splines in terms of B-net representation. In the bivariate setting, a group of new parameters of bivariate quartic and quintic polynomials over a planar simplex is introduced, new formulations of smoothness conditions of bivariate quartic C-1 splines and quintic C-2 splines are given, and the conformality conditions of bivariate quartic C-1 splines are simplified.