Blending Basic Shapes By C-Type Splines and Subdivision Scheme

Abstract

In this article, we adopt the C-type spline of degree 2 to model and blend basic shapes including conics and circle arcs. The C-type spline belongs to the ωB-spline category of splines that are capable of blending polynomial, trigonometric and hyperbolic functions. Commonly used basic shapes can be exactly represented by these types of splines. We derive explicit formulas for the convenience of modeling the basic curves. The entire blending curve is C¹-continuous. In comparison with the existing best blending method by rational G² splines, which are rational splines of degree 3, the proposed method allows simpler representation and blending of the basic curves, and it can represent numerous basic shapes including the hyperbolic types. We also design a subdivision method to generate blending curves; this method is precise for the basic curves and approximate for the blending sections. The subdivision process is efficient for modeling and rendering. It has also proven to be C¹-continuous by the asymptotically equivalent theory and the continuity of stationary subdivision method. In addition, we extend the proposed methods to cases involving the modeling and blending of basic surfaces. We provide many examples that illustrate the merits of our methods.