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Fair curve designing by Said-Ball curve.

Sana Zafar¹, Maria Hussain¹

¹Department of Mathematics, Lahore College for Women University, Lahore, Pakistan.

|July 7, 2025

Summary

A new method for designing fair curves using rational cubic Said-Ball curves offers control over length and curvature variation. This approach optimizes curve aesthetics through energy functionals, yielding visually appealing and efficient designs.

Area of Science:

Computer-Aided Design
Geometric Modeling
Computational Geometry

Background:

Fair curves are essential in design for aesthetic appeal and functional efficiency.
Existing methods may lack sufficient control over curve properties like curvature variation.
Rational cubic Said-Ball curves offer a flexible framework for curve design.

Purpose of the Study:

To introduce a novel method for designing visually fair curves.
To achieve control over curve length and curvature variation.
To leverage rational cubic Said-Ball curves for enhanced design capabilities.

Main Methods:

Utilizing tangential continuous rational cubic Said-Ball curves (RCSBC).
Employing continuity conditions to fix control points.

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Constructing optimization problems with stretch and curvature variation energy functionals to determine optimal free parameters (weights).

Main Results:

A method for generating families of fair curves by adjusting free parameters.
Demonstrated effectiveness through numerical examples.
Successful application of the technique in two distinct design scenarios.

Conclusions:

The proposed method effectively generates fair curves with controlled properties.
The technique offers a flexible approach to curve design with adjustable parameters.
The method has practical applications in design and modeling.