A note to “Dimensions of spline spaces over unconstricted triangulations” [J. Comput. Appl. Math. 192: 320–327, 2006]

Let Omega be a regular triangulation of a two dimensional domain and S_n ^r(Omega) be a vector space of functions in C^r whose restriction to each small triangle in Omega is a polynomial of total degree at most n. Dimensions of bivariate spline spaces S_n ^r(Omega) over a special kind of triangulation, called the unconstricted triangulation, were given by Farin in the paper [J. Comput. Appl. Math. 192(2006), 320-327]. In this paper, a counter example is given to show that the condition used in the main theorem in Farin´s paper is not correct, and then an improved necessary and sufficient condition is presented.