Certified Approximation of Parametric Space Curves with Cubic B-spline Curves

TL;DR

This paper introduces a certified algorithm for approximating complex parametric space curves with cubic B-spline curves, ensuring high precision while preserving geometric features like topology and singular points.

Contribution

The paper presents a novel method to approximate parametric space curves with cubic B-spline curves, including a new optimization technique for weight selection and error control via tetrahedron subdivision.

Findings

High-precision approximation of high-degree space curves.

Few cubic B-spline segments needed for accurate approximation.

Ability to compute approximate implicit equations of curves.

Abstract

Approximating complex curves with simple parametric curves is widely used in CAGD, CG, and CNC. This paper presents an algorithm to compute a certified approximation to a given parametric space curve with cubic B-spline curves. By certified, we mean that the approximation can approximate the given curve to any given precision and preserve the geometric features of the given curve such as the topology, singular points, etc. The approximated curve is divided into segments called quasi-cubic B\'{e}zier curve segments which have properties similar to a cubic rational B\'{e}zier curve. And the approximate curve is naturally constructed as the associated cubic rational B\'{e}zier curve of the control tetrahedron of a quasi-cubic curve. A novel optimization method is proposed to select proper weights in the cubic rational B\'{e}zier curve to approximate the given curve. The error of the approximation is controlled by the size of its tetrahedron, which converges to zero by subdividing the curve segments. As an application, approximate implicit equations of the approximated curves can be computed. Experiments show that the method can approximate space curves of high degrees with high precision and very few cubic B\'{e}zier curve segments.