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Summary

This study introduces a robust framework for drawing and editing Bézier curves on 3D surfaces using manifold averages. The new subdivision-based approach ensures stable interactive design on complex meshes.

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Area of Science:

Computer Graphics
Geometric Modeling

Background:

Bézier curves are fundamental in computer graphics but extending them to 3D surfaces is challenging.
Existing methods for manifold Bézier curves often suffer from discontinuities and fragility.

Purpose of the Study:

To develop a practical and robust framework for interactive Bézier spline manipulation on manifold surfaces.
To enable direct porting of 2D Bézier editing tools to 3D meshes.

Main Methods:

Implemented manifold extensions of recursive de Casteljau bisection and Lane-Riesenfeld subdivision schemes.
Developed algorithms for curve tracing, point evaluation, and point insertion on meshes.
Utilized repeated manifold averages for robust curve representation.

Main Results:

The subdivision-based approach proved robust and efficient, avoiding discontinuities common in direct extensions.
Achieved superior performance and correct results compared to state-of-the-art methods in bulk experiments.
Successfully ported 2D interactive editing functionalities and SVG drawing mapping to 3D meshes.

Conclusions:

Subdivision-based methods offer a robust and efficient solution for manifold Bézier curves.
The framework facilitates intuitive interactive design and editing of complex curves on 3D surfaces.