Certified Approximation of Parametric Space Curves with Cubic B-spline Curves
Liyong Shen, Chunming Yuan, Xiao-Shan Gao

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…
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Taxonomy
TopicsAdvanced Numerical Analysis Techniques · 3D Shape Modeling and Analysis · Computer Graphics and Visualization Techniques
