In this paper, a ternary 4-point rational interpolation subdivision scheme is presented, and the necessary and sufficient conditions of the continuity are analyzed. The generalization incorporates existing schemes as special cases: Hassan–Ivrissimtzis’s scheme, Siddiqi–Rehan’s scheme, and Siddiqi–Ahmad’s scheme. Furthermore, the fractal behavior of the scheme is investigated and analyzed, and the range of the parameter of the fractal curve is the neighborhood of the singular point of the rational scheme. When the fractal curve and surface are reconstructed, it is convenient for the selection of parameter values.

For non-uniform control polygons, a parameterized four-point interpolation curve ternary subdivision scheme is proposed, and its convergence and continuity are demonstrated. Following curve subdivision, a non-uniform interpolation surface ternary subdivision on regular quadrilateral meshes is proposed by applying the tensor product method. Analyses were conducted on the updating rules of parameters, proving that the limit surface is continuous. In this paper, we present a novel interpolation subdivision method to generate new virtual edge points and new face points of the extraordinary points of quadrilateral mesh. We also provide numerical examples to assess the validity of various interpolation methods.

In many application fields (closed curve noise data reconstruction, time series data fitting, image edge smoothing, skeleton extraction, etc.), curve reconstruction based on noise data has always been a popular but challenging problem. In a single domain, there are many methods for curve reconstruction of noise data, but a method suitable for multi-domain curve reconstruction has received much less attention in the literature. More importantly, the existing methods have shortcomings in time consumption when dealing with large data and high-density point cloud curve reconstruction. For this reason, we hope to propose a curve fitting algorithm suitable for many fields and low time consumption. In this paper, a curve reconstruction method based on clustering and point cloud principal component analysis is proposed. Firstly, the point cloud is clustered by the K++ means algorithm. Secondly, a denoising method based on point cloud principal component analysis is proposed to obtain the interpolation nodes of curve subdivision. Finally, the fitting curve is obtained by the parametric curve subdivision method. Comparative experiments show that our method is superior to the classical fitting method in terms of time consumption and effect. In addition, our method is not constrained by the shape of the point cloud, and can play a role in time series data, image thinning and edge smoothing.