A rational cubic trigonometric approximation scheme of the generalized Cornu spirals

Area of Science:

• Computer-Aided Geometric Design
• Approximation Theory
• Computational Geometry

Background:

• Generalized Cornu spirals (GCS) are essential in various fields, including computer graphics and mechanical engineering.
• Accurate approximation of GCS is crucial for design and simulation tasks.
• Existing GCS approximation schemes have limitations in terms of accuracy and efficiency.

Purpose of the Study:

• To introduce novel approximation schemes for generalized Cornu spirals.
• To develop parametric rational cubic trigonometric Bézier curve-based methods for GCS approximation.
• To optimize free parameters within these schemes to minimize approximation errors.

Main Methods:

• Development of two new approximation schemes: a 2-parameter scheme and a 4-parameter scheme.
• Utilizing parametric rational cubic trigonometric Bézier curves for curve approximation.
• Optimization of scheme parameters by minimizing the maximum relative curvature error.

Main Results:

• The proposed [Formula: see text] and [Formula: see text]-approximation schemes effectively approximate generalized Cornu spirals.
• Optimization of free parameters significantly reduced relative curvature errors.
• Comparative analysis demonstrated superior performance over existing GCS approximation schemes.

Conclusions:

• The developed [Formula: see text] and [Formula: see text]-approximation schemes provide a more accurate and efficient method for approximating generalized Cornu spirals.
• Parametric rational cubic trigonometric Bézier curves are suitable for high-fidelity GCS approximation.
• The optimization strategy effectively enhances the approximation quality.