Improved shape for refinable surfaces with singularly parameterized irregularities.

Area of Science:

• Computer-Aided Design
• Geometric Modeling
• Computer Graphics

Background:

• Singularly-parameterized surfaces traditionally exhibit poor highlight line distribution, limiting their use in primary design.
• Existing methods struggle to achieve high-quality surface representations for critical design applications.

Purpose of the Study:

• To explore graded, many-piece, everywhere C1 (continuously differentiable) singularly-parameterized surface caps.
• To mimic the shape of high-quality guide surfaces using these novel constructions.
• To analyze the trade-off between polynomial degree and surface quality in these designs.

Main Methods:

• Development of multi-piece, C1-continuous surface caps using singular parameterization.
• Investigation of varying polynomial degrees (bi-degree 5, 4, 3) for surface construction.
• Evaluation of highlight line distribution at both macroscopic and microscopic levels.

Main Results:

• Bi-degree 5 constructions show good macroscopic highlight line distribution, with minor flaws near the singularity.
• Lower-degree constructions (bi-4, bi-3) may necessitate guided subdivision for comparable macroscopic quality.
• Singularly-parameterized functions on these surfaces are easily refined, similar to subdivision surfaces.

Conclusions:

• Graded, C1 singularly-parameterized surface caps offer a viable approach for high-quality design surfaces.
• Polynomial degree significantly impacts surface quality and highlight distribution.
• The proposed method provides a flexible and refineable alternative for geometric modeling.