B-Spline Curves and Cubic B-Spline Interpolation Algorithms

B-spline curves are parametric curves extensively used in computer graphics and numerical analysis. Compared to Bézier curves, B-splines offer the advantage of locality—modifying a control point only affects the local segment of the curve rather than the entire curve shape. Cubic B-spline curves are the most commonly used type due to their optimal balance between computational efficiency and smoothness. Each segment of a cubic B-spline is determined by four control points and maintains C² continuity at knot points, ensuring visually smooth and natural curves. The core components of B-spline curves include: Control Points: Define the curve's shape (though the curve may not pass through them directly) Knot Vector: Determines parameterization and segmentation of the curve Basis Functions: Weight functions defining control point influence, typically computed recursively via the Cox-de Boor algorithm In interpolation applications, cubic B-splines construct smooth curves passing through given data points. The interpolation process typically involves: 1. Knot Vector Determination: Select uniform or chord-length-based non-uniform knot vectors according to data point distribution 2. Control Point Calculation: Solve linear equations to ensure exact passage through data points using matrix inversion methods 3. Curve Generation: Compute the final curve using B-spline recursive formulas with obtained control points and knot vector B-splines excel not only in locality but also in flexibility for various interpolation needs, including irregular data fitting and smooth curve adjustments. These characteristics make them ideal for CAD systems, animation modeling, and path planning applications. Code implementation typically involves algorithms for basis function computation, linear system solving, and parametric evaluation using de Boor's algorithm.