# Cubic \(q\)-Bezier Triangular Patch for Scattered Data Interpolation and Its Algorithm
## 1 Introduction  
Scattered data interpolation is crucial in computational science, providing a method to estimate values at arbitrary points within datasets characterized by irregularly distributed data points. Unlike structured datasets, which are arranged on grids or meshes, scattered data often arise from real-world observations or simulations where data points are irregularly positioned in multidimensional space. Interpolating these data involves constructing a continuous function that approximates the dataset's underlying behavior, allowing for the inference of values at unobserved locations within the domain [1]. There are two primary approaches to scattered data interpolation: triangulation-based methods and meshless techniques. Triangulation-based approaches involve creating a mesh from the scattered data points. However, not all interpolation methods require such a mesh. For example, some variants of the Shepard method do not require creating a mesh; instead, barycentric coordinates are used to obtain the explicit expression of the polynomial interpolant [2]. Common basis functions used in these methods include Bezier, B-spline, and radial basis functions (RBFs). To mitigate noise in the scattered data, least squares approximation is often employed. [PERSON] [3] developed a local multi-level scattered data interpolation method by introducing layered datasets and adjusting compactly supported radial basis functions. [PERSON] and [PERSON] [4; 5] explored meshless methods using specific RBFs for interpolation, employing domain decomposition methods to generate symmetric saddle point systems. [PERSON] et al. [5] presented an innovative variant of the normalized RBFs known as the Squared Normalized RBFs (SN-RBFs), specifically designed for spatio-temporal data. This new formulationinvolves the normalization of the rows of the RBF matrix, a process that significantly improves the conditioning of the RBF matrix. As a result, the SN-RBF method enhances the robustness and accuracy of RBF interpolation, making it more suitable for a wide range of applications. [PERSON] et al. [6] enhanced the moving least squares (MLS) technique by incorporating polynomial bases to improve scattered data interpolation. [PERSON] et al. [7] examined constrained surface interpolation using the Shepard interpolant, addressing the problem with optimization techniques. [PERSON] and [PERSON] [8] reviewed the use of non-negative bivariate triangular splines for maintaining the shape of scattered data. [PERSON] and [PERSON] [9] investigated preserving non-negativity using macro-element spline spaces with Clough-Tocher macro-elements. [PERSON] et al. [10; 11] explored spatial interpolation for rainfall data and terrain data using cubic Bezier and cubic Ball triangular patches. These studies demonstrated that the scattered data interpolation approach by [PERSON] and [PERSON] [12] does not exhibit \(C^{1}\) continuity, prompting the establishment of new criteria for \(C^{1}\) continuity. Their method yielded a differentiable surface with a seamless appearance. [PERSON] and [PERSON] [13] introduced segmented bivariate Hermite interpolation using triangulation for precise surface reconstruction from extensive scattered datasets. [PERSON] et al. [14] developed a bivariate rational interpolation on a triangular domain but noted increased computation time with rational splines. [PERSON] et al. [15] suggested using polyharmonic splines to estimate noisy scattered data. [PERSON] and [PERSON] [16] developed a \(C^{1}\) triangular interpolant using a convex combination scheme, differing from [PERSON] and [PERSON]'s \(C^{1}\) cubic triangular convex combination technique [17]. [PERSON] and [PERSON]'s method using cubic Ball triangular patches yielded results equivalent to cubic Bezier triangular patches with reduced computation [12]. [PERSON] and [PERSON]'s study confirmed the non-continuous differentiability of [PERSON] and [PERSON]'s function [18]. [PERSON] et al.