Explicit Inversion of Planar NURBS Curves

We prove that a general planar NURBS curve parametrization $\phi :[u_0,u_m]\stackrel{}{\to }C\subset {\mathbb{R}}^2$ admits an inverse map ${\phi }^{-1}:C\stackrel{}{\to }[u_0,u_m]$ 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.