Least Degree G ¹-refinable Multi-sided Surfaces Suitable for Inclusion into C ¹ bi-2 Splines.

Area of Science:

• Computer-Aided Design (CAD)
• Geometric Modeling
• Spline Surfaces

Background:

• Smooth spline surfaces with n-sided facets (n≠4) often lack degrees of freedom upon refinement.
• This limitation hinders their use in detailed modeling and engineering analysis.

Purpose of the Study:

• To establish the minimum polynomial degree for multi-sided spline surfaces that guarantees flexibility under refinement.
• To ensure enhanced degrees of freedom for modeling and engineering applications.

Main Methods:

• Investigated geometrically smooth spline surfaces with generalized n-sided facets.
• Established a minimal polynomial degree for flexibility-increasing G¹-refinability.
• Proved sufficiency using two alternative G¹-refinable constructions.

Main Results:

• Identified degree bi-4 as the minimal requirement for smooth, multi-sided spline surfaces.
• Demonstrated that degree bi-4 is both necessary and sufficient for G¹-refinability.
• Developed constructions with good highlight line distributions.

Conclusions:

• Degree bi-4 ensures flexibility in multi-sided spline surfaces during refinement.
• This finding enhances the utility of spline surfaces in CAD and engineering analysis.
• The proposed methods offer improved geometric control and analysis capabilities.