Explicit Inversion of Planar NURBS Curves

TL;DR

This paper demonstrates that planar NURBS curves can be explicitly inverted using rational spline functions, providing formulas and examples to illustrate the method's effectiveness.

Contribution

It introduces a method to explicitly invert planar NURBS curves via rational splines, with explicit formulas and practical examples.

Findings

Explicit formulas for inverse NURBS parametrization are derived.

The inverse map can be represented as a linear combination of rational spline functions.

Examples confirm the practical effectiveness of the proposed inversion method.

Abstract

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