NURBSkurmien

NURBSkurmien is a term used in computational geometry to denote a class of NURBS-based curves and surfaces in which curvature behavior is an explicit design objective. It extends the standard NURBS framework by integrating curvature control into the modeling process to achieve smooth and predictable curvature distributions across the geometry.

In a NURBSkurmien surface S(u,v), the geometric definition follows the conventional NURBS formula, with a control net P_ij, associated weights w_ij, and knot vectors U and V. In addition to the usual evaluation, NURBSkurmien introduces a curvature-oriented objective that guides control-point adjustment or weighting during construction. Concretely, the surface is given by S(u,v) = (sum_i sum_j N_i,p(u) M_j,q(v) w_ij P_ij) / (sum_i sum_j N_i,p(u) M_j,q(v) w_ij), where N_i,p and M_j,q are B-spline basis functions. The mean curvature H(u,v) is computed from first and second partial derivatives of S, and a typical formulation seeks to minimize an energy such as ∫∫ (H(u,v) - H_desired(u,v))^2 dudv or to enforce prescribed curvature values at selected parameter points.

NURBSkurmien surfaces retain the core strengths of NURBS, including local control via the control net and compatibility with knot insertion and refinement. The curvature objective can be implemented through optimization steps during design-time or during interactive editing, sometimes requiring iterative refinement that respects geometric fidelity to the original control points.

Applications are mainly in computer-aided design and animation where smooth curvature distribution is important, such as automotive body panels, industrial components, and character surfaces. The approach offers smoother curvature transitions and better control of isoparametric curves, but adds computational overhead and can complicate the guarantee of exact interpolation of a target boundary when curvature constraints are active.

Originating in theoretical discussions of curvature-driven modeling, NURBSkurmien remains a niche concept and is not yet standardized in commercial NURBS toolkits, though several research prototypes explore its feasibility.