| Abstract:To enhance the flexibility of Catmull-Rom spline curves, this paper constructs spline curves with two shape parameters and spline curves with three shape parameters. First, the spline and spline basis functions are constructed based on the Catmull-Rom spline, and the fundamental properties of these basis functions, such as normality, quasi-symmetry, and linear independence, are analyzed. Then, spline curves and spline curves are constructed, and their symmetry, geometric invariance, affine invariance, and endpoint characteristics are discussed. Numerical experiments verify the shape adjustability and continuity of the spline and spline curves, demonstrating their flexibility and advantages in modeling complex curves. Additionally, a comparison with the error of Lagrange interpolation curves shows that spline curves have superior approximation effects. Finally, an optimal parameter selection method based on minimal bending energy is proposed, providing theoretical support and practical guidance for spline curve design. Examples illustrate that spline and spline curves enhance the flexibility of Catmull-Rom spline curves and improve the smoothness of the curves by selecting the optimal shape parameters. |
