Explicit Inversion of Planar NURBS Curves
This provides a theoretical foundation and practical method for inverting NURBS curves, benefiting geometric modeling and CAD applications.
The paper proves that a general planar NURBS curve has an inverse map defined by rational splines, providing explicit formulas for its computation. Examples demonstrate the approach's effectiveness.
We prove that a general planar NURBS curve parametrization $ϕ: [u_0,u_m] \xrightarrow{} C \subset \mathbb{R}^2$ admits an inverse map $ϕ^{-1}: C \xrightarrow{} [u_0,u_m]$ defined by rational splines. More specifically, we construct a family of rational spline functions on the curve $C$, present explicit formulas for their computation, and prove that the inverse parametrization admits a representation as a linear combination of these functions. Several examples are provided to illustrate the effectiveness of the proposed approach.