B-spline Curve and Surface Interpolation and Approximation Algorithms

This text explores B-spline curve and surface interpolation and approximation algorithms, along with visualization techniques using B-splines. We provide MATLAB-formatted code for reference. B-spline curves and surfaces are essential tools in computer graphics, capable of describing various shapes and contours such as automobile bodies or ship hulls. The interpolation algorithms typically involve solving linear systems to determine control points that pass through given data points, while approximation algorithms focus on minimizing error metrics between the target shape and the B-spline representation. Our MATLAB implementation includes functions for: - Knot vector generation using uniform or chord-length parameterization - Basis function calculation via Cox-de Boor recursion algorithm - Control point computation through least-squares approximation - Surface visualization using parametric evaluation and mesh plotting These interpolation and approximation techniques provide powerful methods for processing and rendering complex shapes and curves. This article delves into practical applications of B-spline curves and surfaces, accompanied by executable code examples that demonstrate: 1. Curve interpolation through given data points using degree-regulated B-splines 2. Surface approximation with adjustable tolerance levels 3. Real-time visualization with customizable resolution parameters The provided MATLAB code includes comprehensive comments explaining key algorithmic steps and parameter tuning guidelines for optimal results.