A note to “imensions of spline spaces over unconstricted triangulations”[J

Let Ω be a regular triangulation of a two dimensional domain and S n(Ω) be a vector space of functions in C r whose restriction to each small triangle in Ω is a polynomial of total degree at most n. Dimensions of bivariate spline spaces S n(Ω) 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. 1. A counter example In [2], by introducing two kinds of construction operations, called a flap and a pair of triangles respectively, the so-called unconstricted triangulation was first defined, which can be obtained by recursively adding a flap or a pair of triangles to a subtriangulation started from a single triangle. In Section 3 in [2], the dimension of S 3(Ω) over the unconstricted triangulation was given. Then in Section 4, the construction of a minimal determining set for the spline space S n(v ) over a star v was further considered. And finally in Section 5, the dimension of the spline space S n(Ω) over the unconstricted triangulation Ω was determined by recursively using the results over stars presented in Section 4. For a star v which is obtained from v −2 by adding a pair of triangles, δ n(b) was defined in [2] as δ n(b) = dimS r n(v )− dimS n(v −2), (1) where b is the valence of an interior vertex v. The key step in the proof of the theorem in Section 5 in [2] is based on the statement “if δ n(b) ≥ 0 then a minimal determining set for S n(v ) can be obtained by adding some other Bézier ordinates to the minimal determining set for S n(v −2)”. However, it is found in this section that this statement is not always true. Here is a counter example. Counter example. Let us take b = 5, n = 5 and r = 2, and let v −2 = Δv1vv2 ∪ Δv2vv3 ∪ Δv3vv4 with ∠v1vv3 ∈ (π2 , π). The star v is obtained by adding a pair of triangles Δv1vv5 ∪ Δv5vv4 to v −2, where ∠v3vv4 ∈ (0, π2 ) and ∠v1vv5 ∈ (0, π2 ), as shown in Fig. 1. We first consider the spline space S 5(v −2). It follows from [4], [5] that dimS 5(v −2) = 33. And a minimal determining set for S 5(v −2) can be easily chosen as the Bézier ordinates with respect to the domain points marked by “•” as shown in Fig. 1(a), which is denoted by P 2 5 (v −2).