Fair curve and surface design system using tangent control

Abstract

The paper proposes a generation system of unit quarternion curves and surfaces to design fair curves and surfaces. Curvature and variation of curvature are the primary factors that determine the quality of fair curves and surfaces; however not so many methods have been explored to directly manipulate them, especially in three dimensional space. We use unit quaternion curves in integral forms to specify the tangent of a curve and the first derivative of a surface in one parameter direction. This enables us to control their curvature and variation of curvature more than is possible with traditional parametric representations like Bezier and NURBS curves and surfaces. Using unit quaternion curves balances the efficiency of the design process and the fairness of the designed curves and surfaces. Our system can generate fair curves and surfaces within a short time and their quality is acceptable for industrial applications as well as CG animation.