Triangular bubble spline surfaces

Area of Science:

• Computer Graphics
• Geometric Modeling
• Surface Reconstruction

Background:

• Generating smooth surfaces from discrete data is a fundamental problem in computer graphics and geometric modeling.
• Existing methods often struggle with complex network structures or achieving specific continuity conditions.

Purpose of the Study:

• To introduce a novel method for generating C^k-surfaces from compatible triangular networks of polynomial curves.
• To present a new 'bubble patch' concept for representing surface patches and ensuring continuity.

Main Methods:

• Construction of C^k-surfaces using a network of compatible surface strips defined by polynomial curves.
• Utilizing 'bubble patches' to represent individual surface patches and enforce C^k continuity conditions between them.
• Employing Gordon-Coons interpolation for triangle-based surface patch construction.

Main Results:

• A local, uniform construction scheme applicable to vertices of arbitrary valency.
• Generation of piecewise rational surfaces that interpolate the input network of polynomial curves.
• Successful generation and visualization of C^0, C^1, and C^2-surfaces using the proposed method.

Conclusions:

• The bubble patch method provides a robust and versatile approach for generating C^k-surfaces.
• The method ensures a simple C^k-condition between adjacent patches, generalizing to arbitrary k.
• The resulting surfaces exhibit high visual smoothness, demonstrated through reflection line analysis.