Streamline modeling with subdivision surfaces on the Gaussian sphere

Curvature and variation of curvature are the essential factors in determining the fairness of a surface. Unfortunately, most of the traditional surface representation schemes do not provide users with direct manipulation techniques of these quantities. Streamline modeling, a recently proposed free-form surface design methodology, is aimed at overcoming this shortcoming by allowing a user to control tangent vectors (and, consequently, curvature and variation of curvature) of the surface to be designed directly. A free-form surface is regarded as a set of streamlines: iso-parametric lines defined by blending directions of tangent vectors instead of blending positions of control points. This new surface design methodology can generate high quality smooth surfaces but requires much processing power for tangent vector blending. In this paper, we present subdivision based blending techniques of tangent vectors. These techniques can be used to develop subdivision techniques for curves and surfaces on the Gaussian sphere, such as Doo-Sabin, Catmull-Clark, and Kobbelt subdivisions. We also present new streamline modeling techniques based on the new tangent vector blending techniques. The new techniques reduce the processing time for the integration process required in streamline modeling. A prototype system based on the new techniques shows that free-form surface design using the streamline modeling methodology can achieve real-time performance.