de Boor Evaluation Algorithm for Constructing Quadratic and Cubic B-Spline Curves with Non-Interpolating Properties

When constructing smooth quadratic or cubic B-spline curves, the de Boor algorithm provides an efficient computational method. Unlike interpolation methods, non-interpolating B-spline curves don't necessarily pass through all control points but are influenced by them, exhibiting more natural shape transitions. In code implementation, this involves calculating basis functions and weighted sums of control points rather than solving interpolation equations.

For given control points, we first need to determine the knot vector. A common approach is to use open uniform knot vectors, ensuring the curve coincides with the first and last vertices of the control polygon at the starting and ending points. The choice of knot vector directly affects the curve's parameterization method, where in programming we typically define this as an array of monotonically increasing values.

During computation, the de Boor algorithm employs a recursive approach to progressively subdivide curve segments. For quadratic B-splines (degree 2), each curve segment is determined by three control points, while cubic B-splines (degree 3) require four control points. The algorithm performs linear interpolation within knot intervals through iterative calculations, ultimately obtaining coordinates of corresponding parametric points on the curve. The core implementation involves nested loops for basis function evaluation and point calculation.

Notably, locality is an essential characteristic of B-spline curves, meaning each control point only influences a limited range of the curve shape. This makes B-spline curves highly flexible in geometric design, allowing local adjustments without affecting the overall structure. Programmatically, this is achieved through the limited support property of B-spline basis functions.

To obtain a complete curve, we need to apply the de Boor algorithm to each valid knot interval. After calculating a sufficient number of points, we connect these points with line segments to approximate and display the smooth B-spline curve. In practical implementation, this typically involves sampling parameter values at regular intervals and computing corresponding curve points.