*2.1. Review of the Cubic Triangular Bases of Zhu And Han*

Zhu and Han [44] proposed a new cubic triangular basis with three exponential parameters α, β, γ. Since we are dealing with triangulation, the barycentric coordinate (*u*, *v*, *w*) on the triangle *T*<sup>1</sup> with vertices *V*1, *V*<sup>2</sup> and *V*<sup>3</sup> is defined by *u* + *v* + *w* = 1, where *u*, *v*, *w* ≥ 0. Set the point inside the triangle as *<sup>V</sup>*(*x*, *<sup>y</sup>*) <sup>∈</sup> *<sup>R</sup>*<sup>2</sup> (as shown in Figure 1), which can be expressed as:

$$V = \mu V\_1 + \upsilon V\_2 + wV\_3 \tag{1}$$

**Figure 1.** Triangle.

**Definition 1.** *Let the parameters* α, β, γ ∈ [2, ∞] *and the triangular domain D* = (*u*, *v*, *w*) *<sup>u</sup>* <sup>+</sup> *<sup>v</sup>* <sup>+</sup> *<sup>w</sup>* <sup>=</sup> <sup>1</sup> *; the following are cubic Bernstein–Bézier basis functions ([44]):*

$$\begin{split} \left| B\_{3,0,0}^{3} (\mu, \upsilon, w; a, \beta, \gamma) \right| &= u^{\alpha}, \, B\_{0,3,0}^{3} (\mu, \upsilon, w; a, \beta, \gamma) = v^{\beta}, \, B\_{0,0,2}^{3} (\mu, \upsilon, w; a, \beta, \gamma) \\ &= w^{\gamma}, \, B\_{2,1,0}^{3} (\mu, \upsilon, w; a, \beta, \gamma) \\ &= u^{2}v \, \left[ \frac{3 - 2u - v^{2 - \mu - 2}}{1 - u} \right], \, B\_{2,0,1}^{3} (\mu, \upsilon, w; a, \beta, \gamma) \\ &= u^{2}w \, \left[ \frac{3 - 2u - v^{2 - \mu - 2}}{1 - v} \right], \, B\_{1,2,0}^{3} (\mu, \upsilon, w; a, \beta, \gamma) \\ &= uv^{2} \left[ \frac{3 - 2v - v^{2 - \mu - 2}}{1 - v} \right], \, B\_{0,2,1}^{3} (\mu, \upsilon, w; a, \beta, \gamma) \\ &= v^{2}w \left[ \frac{3 - 2v - v^{2 - \mu - 2}}{1 - w} \right], \, B\_{0,1,2}^{3} (\mu, \upsilon, w; a, \beta, \gamma) \\ &= uv^{2} \left[ \frac{3 - 2v - v^{2 - \mu - 2}}{1 - w} \right], \, B\_{1,1,1}^{3} (\mu, \upsilon, w; a, \beta, \gamma) = 6uvw. \end{split} \tag{2}$$

Zhu and Han's triangular basis functions satisfy the following properties: Non-negativity:

$$B^3\_{i,j,k}(\mathfrak{u}, \mathfrak{v}, w; \mathfrak{a}, \mathfrak{v}, \mathfrak{y}) \ge \mathbf{0}, \ i+j+k=\mathbf{3}.$$

Partition of unity:

$$\sum\_{i+j+k=3} B^3\_{i,j,k}(u,v,w;\alpha,\beta,\gamma) = 1.$$

Symmetry:

$$\mathcal{B}^{3}\_{i,j,k}(\mathfrak{u},\mathfrak{v},\mathfrak{w};\mathfrak{a},\mathfrak{b},\mathfrak{v}) = \mathcal{B}^{3}\_{ijk}(\mathfrak{w},\mathfrak{v},\mathfrak{u};\mathfrak{y},\mathfrak{f},\mathfrak{a})\_{i,j}$$

For more details on the other properties, please refer to Zhu and Han [44].

Zhu and Han's triangular patches with three parameters α, β, and γ, and control points *bijk*, *i* + *j* + *k* = 3 are defined as

$$P(u,v,w) = \sum\_{i+j+k=3} b\_{ijk} B^3\_{i,j,k}(u,v,w; \alpha\_\prime \beta\_\prime \gamma)\_\prime \quad u+v+w=1\tag{3}$$

Figure 2 shows the Zhu and Han ordinates, and Figure 3 shows one patch of the Zhu and Han triangular with α = β = γ = 3 (i.e., cubic Bézier triangular).

**Figure 2.** The 10 quartic triangular ordinates (control points).

**Figure 3.** One patch (Zhu and Han [44]).
