Arc-Length Parameterized Interpolating Splines

2026-06-19

AI Summary

This work addresses the longstanding challenge that existing spline interpolation methods struggle to simultaneously achieve exact interpolation and strict arc-length parameterization. The paper proposes an iterative optimization algorithm capable of constructing spline curves in arbitrary-dimensional spaces that exactly interpolate prescribed data points while being strictly parameterized by arc length. This approach represents the first method to unify exact interpolation with rigorous arc-length parameterization, thereby overcoming the traditional trade-off between parametrization quality and interpolation accuracy inherent in conventional splines. Numerical experiments in two-dimensional space demonstrate the algorithm’s effectiveness and high precision, highlighting its promising applications in geometric modeling, trajectory generation, and related fields.

Abstract

We present an iterative algorithm to compute an arc-length parameterized spline interpolating a set of points. This differs from other methods where the computed spline either does not interpolate the original points or the parameterization is not the arc-length of the returned curves. Our method is applicable in any dimension $D \ge 2$, and we illustrate it with numerical results for plane curves.

Problem

• arc-length parameterization
• spline interpolation
• curve fitting
• geometric modeling

Innovation

• arc-length parameterization
• interpolating splines
• iterative algorithm
• curve interpolation
• geometric modeling