A discrete spring model for generating fair curves and surfaces

The ability to generate fair curves and surfaces is important in computer graphics (CG), computer-aided design (CAD), and other geometric modeling applications. In this paper, we present an iteration-based algorithm for generating fair polygonal curves and surfaces that is based on a new discrete spring model. In the spring model, a linear spring, whose length approximately represents a curvature radius, is attached along the normal line of each polygon node. Energy is assigned to the difference of the lengths, that is, the difference in curvature radius, of neighboring springs. Our algorithm then minimizes the total energy by an iterative approach. It accepts as inputs (1) an initial polygonal curve (surface), which consists of a set of polygonal segments (faces) and a set of nodes as polygon-vertices, and (2) constraints for controlling the shape. The outputs are polygonal curves (surfaces) with smooth shapes. We also describe a method for improving the performance of our iterative process to obtain a linear execution time. Our algorithm provides a tool for the fair curve and surface design in an interactive environment