Modified progressive iterative approximation techniques for interoperable conversion of QT-Bézier and rational Bézier curve

Abstract

This paper presents an in-depth evaluation of modified Progressive Iterative Approximation (PIA) methods for basis curve conversion and approximation, addressing critical challenges in Quadratic Trigonometric (QT) Bézier and rational Bézier curves. The novelty of this study lies in the introduction of two distinct modified PIA methods designed to efficiently transform these curves. PIA Type 1 provides a new approach for converting QT-Bézier curves into standard Bézier curves through degree elevation, solving interoperability issues across software platforms. PIA Type 2 introduces a novel method for optimizing the shape parameter m, enabling efficient conversion of rational Bézier curves into QT-Bézier curves. By dynamically adjusting m, PIA Type 2 improves curve approximation without adding extra control points, offering a more efficient alternative to traditional methods that typically increase control points to enhance accuracy. The study demonstrates that PIA Type 1 achieves highly accurate conversions with minimal errors, while PIA Type 2 highlights the significance of selecting the optimal shape parameter for effective transformations. Both methods are grounded in a robust mathematical framework, allowing precise local adjustments and enhancing both accuracy and computational efficiency. These contributions address a critical gap in curve representation by providing flexible and efficient solutions for curve conversion and approximation.