NURBSpintojen

NURBSpintojen is a theoretical framework in geometric modeling that extends non-uniform rational B-splines (NURBS) by attaching spinorial data to control points to encode local surface orientation. It is intended to provide a unified representation of geometry and orientation, enabling more precise deformation, curvature control, and directional properties in modeling and simulation.

The name fuses NURBS with spinors, drawing on ideas from spin geometry and directional frames. Proponents describe NURBSpintojen as an extension rather than a replacement for conventional NURBS, designed to maintain compatibility with existing evaluation pipelines while enriching the local frame information.

Construction involves augmenting each control point Pij with a weight Wij and a spinor sij. The standard NURBS basis functions Ni,p(u) and Mj,q(v) remain in use for position interpolation, but the spinors contribute to a spin-weighted orientation field that travels along the surface. The resulting surface S(u,v) comes equipped with an orientation frame derived from the spin data, enabling consistent tangent and normal fields under deformations.

Properties include local control through traditional knot vectors and control nets, smoothness determined by degrees p and q, and a built-in mechanism for preserving orientation continuity across patches. The approach adds computational overhead due to spin calculations and requires specialized tooling or libraries.

Applications are envisioned in computer-aided design, animation, and material simulation where surface orientation matters, such as anisotropic materials or fabric-like modeling. The concept remains exploratory, with ongoing discussion about numerical stability, interoperability with standard NURBS workflows, and practical implementations.