Variational Mesh Offsetting by Smoothed Winding Number

Area of Science:

• Computer Graphics
• Computational Geometry
• Geometric Modeling

Background:

• Surface mesh offsetting is crucial for shape modeling but faces challenges with intersection defects (implicit methods) or self-intersections (explicit methods).
• Existing methods struggle to balance robustness against intersection issues with precise shape control, such as preserving sharp features.

Purpose of the Study:

• To develop a novel variational framework for surface mesh offsetting that integrates the strengths of both implicit and explicit approaches.
• To enable flexible shape control, including sharp feature preservation and adherence to specific surface types (e.g., quadrics), while mitigating intersection problems.

Main Methods:

• A variational framework is proposed, treating mesh vertex locations as variables and utilizing a smooth winding-number field.
• An objective function is defined, enforcing that the input mesh lies on the offset contour of the field induced by the resulting mesh.
• Shape regularizations, such as sharp feature preservation and intersection penalties, are incorporated into the optimization problem.

Main Results:

• The proposed method successfully offsets meshes while preserving sharp features of the original shape.
• It allows for restricting specific mesh parts to quadric surfaces.
• The framework effectively alleviates intersection issues inherent in traditional offsetting techniques.

Conclusions:

• The variational framework offers a robust and versatile solution for surface mesh offsetting.
• It combines the advantages of implicit and explicit methods, providing superior shape control and intersection handling.
• The numerical friendliness due to field differentiability facilitates practical implementation and further development.