NURBS

Non-Uniform Rational B-Splines (NURBS) are a mathematical model used to represent curves and surfaces. They provide a generalization of B-splines and Bezier curves and are widely used in computer graphics, computer-aided design (CAD), and computer-aided engineering because they can precisely represent common analytic shapes such as lines, circles, and conic sections, while also supporting free-form shapes.

A NURBS curve of degree p is defined by a set of control points P_i, corresponding weights w_i, and a non-decreasing knot vector U. The curve is given by C(t) = [sum_i N_{i,p}(t) w_i P_i] / [sum_i N_{i,p}(t) w_i], for t in the parameter domain. Here N_{i,p} are the B-spline basis functions determined by U. The weights impart rational weighting to convex combinations of control points, allowing exact representation of conic sections.

A NURBS surface is defined by a grid of control points P_{i,j}, weights w_{i,j}, a pair of knot vectors U and V, and degrees p and q; the surface is C(u,v) = [sum_i sum_j N_{i,p}(u) N_{j,q}(v) w_{i,j} P_{i,j}] / [sum_i sum_j N_{i,p}(u) N_{j,q}(v) w_{i,j}]. This tensor-product form extends NURBS curves to two dimensions, enabling rich, controllable surface modeling.

Special cases: when all weights are equal, NURBS reduce to non-uniform B-splines; with uniform knots and equal weights, they mirror standard B-splines or Bezier forms. The non-uniform knot vector provides local control over the shape, improving flexibility for complex models.

Applications span CAD/CAM, automotive and aerospace design, shipbuilding, product design, animation, and visual effects. NURBS remain a standard due to their precision, versatility, and compatibility with existing curve and surface representations.