NURBSsplines

NURBS splines are Non-Uniform Rational B-Splines, a versatile mathematical representation for curves and surfaces used in geometric modeling and computer graphics. They extend B-splines by incorporating rational weights, which enables exact representation of a wider range of shapes, notably conic sections. NURBS form a standard in CAD/CAM, 3D modeling, and animation.

A NURBS curve is defined by a degree p, a set of control points P_i in space, a corresponding set of weights w_i > 0, and a nondecreasing knot vector U = {u_0, ..., u_m}. The curve is C(u) = (sum_i N_i,p(u) w_i P_i) / (sum_i N_i,p(u) w_i), where N_i,p are B-spline basis functions generated from U. The non-uniform knot vector controls parameter spacing and local refinement, while weights adjust influence of each control point.

Key properties include local control (each basis function has compact support), affine invariance (transforming control points and weights linearly transforms the curve), and partition of unity (sum of basis functions equals 1 over the parameter domain). With appropriate weights, circles and ellipses can be represented exactly, and the geometry can be refined through knot insertion without changing the curve’s shape.

NURBS extend to surfaces via tensor products of B-spline basis functions in two parameters (u,v), with a grid of control points and weights. This yields NURBS surfaces that can model complex, smooth surface geometry with relatively few parameters, and support operations such as knot insertion, refinement, and trimming.

NURBS are widely used in CAD/CAM, architectural modeling, automotive and aerospace design, and 3D animation. They provide a robust, flexible framework for designing precise, editable geometries and integrate with many commercial and open-source modeling systems.