*2.2. Quartic Zhu and Han Triangular Patches*

From Equation (2), let α = β = γ = 4, then we obtain the following ten quartic basis functions defined on the triangular domain:

$$\begin{aligned} \left| B\_{3,0,0}^{3}(u,v,w;a,\theta,\gamma) \right\rangle &= u^4, B\_{3,0,0}^{3}(u,v,w;a,\theta,\gamma) = v^4, B\_{0,0,3}^{3}(u,v,w;a,\theta,\gamma) \\ &= w^4, B\_{2,1,0}^{3}(u,v,w;a,\theta,\gamma) = u^2v \ (3+u), B\_{2,0,1}^{3}(u,v,w;a,\theta,\gamma) \\ &= u^2w \ (3+u), B\_{1,2,0}^{3}(u,v,w;a,\theta,\gamma) \\ &= uv^2 \ (3+v), B\_{0,2,1}^{3}(u,v,w;a,\theta,\gamma) \\ &= v^2w \ (3+v), B\_{1,0,2}^{3}(u,v,w;a,\theta,\gamma) \\ &= uv^2 \ (3+w), B\_{0,1,2}^{3}(u,v,w;a,\theta,\gamma) \\ &= vw^2 \ (3+w), B\_{1,1,1}^{3}(u,v,w;a,\theta,\gamma) = 6uvw. \end{aligned} \tag{4}$$

Figure 4 shows the quartic triangular basis on the triangular domain.

**Figure 4.** Quartic triangular basis functions.

Thus, the quartic Zhu and Han triangular patch can be defined by

$$\begin{aligned} P(\boldsymbol{\mu}, \boldsymbol{\upsilon}, \boldsymbol{w}) &= \boldsymbol{\mu}^4 b \boldsymbol{y}\_{30} + \boldsymbol{\upsilon}^4 b \boldsymbol{y}\_{30} + \boldsymbol{w}^4 b \boldsymbol{b}\_{003} + \boldsymbol{\mu}^2 \boldsymbol{\upsilon} \left( \boldsymbol{3} + \boldsymbol{\mu} \right) b \boldsymbol{z}\_{10} + (\boldsymbol{3} + \boldsymbol{\mu}) \boldsymbol{\upsilon}^2 \boldsymbol{w} b \boldsymbol{z}\_{201} \\ &+ (\boldsymbol{3} + \boldsymbol{\upsilon}) \boldsymbol{\upsilon}^2 \boldsymbol{w} b \boldsymbol{b}\_{120} + (\boldsymbol{3} + \boldsymbol{\upsilon}) \boldsymbol{\upsilon}^2 \boldsymbol{w} b \boldsymbol{b}\_{021} + (\boldsymbol{3} + \boldsymbol{\upsilon}) \boldsymbol{w}^2 \boldsymbol{w} b \boldsymbol{b}\_{102} \\ &+ (\boldsymbol{3} + \boldsymbol{\upsilon}) \boldsymbol{w}^2 \boldsymbol{v} b\_{012} + 6 \boldsymbol{\upsilon} \boldsymbol{w} \boldsymbol{v} b\_{111} \end{aligned} \tag{5}$$

The main advantage of Zhu and Han's quartic is that it only requires ten control points to construct one triangular patch; meanwhile, the quartic Bézier triangular will require 15 control points to produce one patch. Furthermore, when the quartic Bézier triangular is used for scattered data interpolation, an optimization method is required to produce the interpolated surface, as discussed in Saaban et al. [35], Piah et al. [36] and Hussain et al. [37,38]. However, if we apply the proposed quartic triangular patches for scattered data interpolation, the optimization is not required since we can employ the cubic precision scheme of Foley and Opitz [30] to construct a *C*<sup>1</sup> interpolated surface everywhere. So far, this is the first study to apply a a quartic triangular basis but with ten control points for scattered data interpolation.

Figure 5a shows examples of quartic Zhu and Han, and Figure 5b shows the quartic Bézier triangular patch.

**Figure 5.** Quartic triangular patches. (**a**) Quartic Zhu and Han [44]; (**b**) Quartic Bézier triangular.
