# Cubic \(q\)-Bezier Triangular Patch for Scattered Data Interpolation and Its Algorithm
## 2 \(q\)-Bezier Triangular Patch  
**Definition 1**.: _Given a non-negative integer \(n\) and three indices \(i,j,k\geq 0\) such that \(i+j+k=n\), the triangular \(q\)-Bernstein from [34] is defined as_\[\begin{split} B^{n}_{i,j,k}(u,v)&=\left[\begin{array}{c} n\\ k\end{array}\right]\left(\begin{array}{c}i+j\\ i\end{array}\right)u^{i}v^{j}\prod_{s=0}^{k-1}(1-q^{s}u-q^{s}v),\quad i,j,k \geq 0,\quad i+j+k=n\\ \text{where }q\in[0,1]\end{split} \tag{1}\]  
In order to construct a \(q\)-Bezier triangular patch for scattered data interpolation, the existing triangular Bernstein polynomials \(B^{n}_{i,j,k}(u,v)\) are extended to \(B^{n}_{i,j,k}(u,v,w)\), which is defined in Proposition 1.  
Proposition 1: _The triangular \(q\)-Bernstein polynomials from Equation (1) are extended to \(B^{n}_{i,j,k}(u,v,w)\), which are given by_  
\[\begin{split} B^{n}_{i,j,k}(u,v,w)&=\left[\begin{array}[] {c}n\\ k\end{array}\right]\left(\begin{array}{c}i+j\\ i\end{array}\right)u^{i}v^{j}\prod_{s=0}^{k-1}(1-q^{s}+q^{s}w),\quad i,j,k \geq 0,\quad i+j+k=n\\ \text{where }q\in[0,1]\end{split} \tag{2}\]  
Proof: Given a non-negative integer n and three indices \(i,j,k\geq 0\) such that \(i+j+k=n\),  
\[B^{n}_{i,j,k}(u,v) =\left[\begin{array}{c}n\\ k\end{array}\right]\left(\begin{array}{c}i+j\\ i\end{array}\right)u^{i}v^{j}\prod_{s=0}^{k-1}(1-q^{s}u-q^{s}v) \tag{3}\] \[=\left[\begin{array}{c}n\\ k\end{array}\right]\left(\begin{array}{c}i+j\\ i\end{array}\right)u^{i}v^{j}\prod_{s=0}^{k-1}(1-q^{s}u-q^{s}v)+q^{s}-q^{s} \tag{4}\]  
By factorizing Equation (4), we have  
\[B^{n}_{i,j,k}(u,v)=\left[\begin{array}{c}n\\ k\end{array}\right]\left(\begin{array}{c}i+j\\ i\end{array}\right)u^{i}v^{j}\prod_{s=0}^{k-1}(1-q^{s}+q^{s}(1-u-v)) \tag{5}\]  
Taking into account that \(w=1-u-v\), we can therefore rewrite \(B^{n}_{i,j,k}(u,v)\) as  
\[B^{n}_{i,j,k}(u,v,w)=\left[\begin{array}{c}n\\ k\end{array}\right]\left(\begin{array}{c}i+j\\ i\end{array}\right)u^{i}v^{j}\prod_{s=0}^{k-1}(1-q^{s}+q^{s}w) \tag{6}\]  
Definition 2: Let \(D:=\{(u,v,w)\mid u+v+w=1,u>  
**Theorem 1**.: _The basis function defined over a triangular domain has the following properties:_  
1. _Partition of unity:_ \(\sum_{i+j+k=3}B^{3}_{i,j,k}(u,v,w)=1\)_._
2. _Non-negativity:_ \(B^{3}_{i,j,k}(u,v,w)\geq 0\)_,_ \(i,j,k\in\mathbb{N}\)_,_ \(i+j+k=3\)_._
3. _Property at the corner points: for_ \(i,j,k\in\mathbb{N}\)_,_ \(i+j+k=3\)_, we have_ \[B^{3}_{i,j,k}(1,0,0) =\begin{cases}1,&i=3,\\ 0,&i\
eq 3,\end{cases}\] \[B^{3}_{i,j,k}(0,1,0) =\begin{cases}1,&j=3,\\ 0,&j\
eq 3,\end{cases}\] \[B^{3}_{i,j,k}(0,0,1) =\begin{cases}1,&k=3,\\ 0,&k\
eq 3,\end{cases}\] _(d) When_ \(q=1\)_, the cubic_ \(q\)_-Bezier triangular is reduced to the standard cubic Bezier._  
**Definition 3**.: _A cubic \(q\)-Bezier triangular patch is defined as_  
\[R(u,v,w)=\sum_{i+j+k=3}B^{3}_{i,j,k}(u,v,w)b_{i,j,k} \tag{7}\]  
_where \(B^{3}_{i,j,k}(u,v,w)\) is defined by Equation (6) for all \(q\in[0,1]\)._  
Figure 2 presents a comparison of the interpolation surfaces for one cubic \(q\)-Bezier patch, with varying values of \(q\) where the values are 0.25, 0.5, 0.75, and 1.0. As the value of \(q\) increases, the surface gradually shifts closer to the control polygon, reflecting the influence of \(q\) on the overall interpolation. When \(q=1.0\), the surface conforms more closely to the shape of the control polygon, while for smaller \(q\) values (e.g., 0.25), the surface tends to be smoother and less influenced by the control points. This demonstrates the flexibility of the \(q\)-Bezier interpolation scheme in controlling the tension and shape of the interpolating surface.  
Figure 1: Bases for \(q\)-Bézier triangular.