Research On Energy Constraints Of Curve And Applications

This paper focuses on the energy minimization problems in curve modeling, which is widely used in computer aided geometric design, computer graphics and manufacture of various kinds of components.A curve with the least strain energy is usually considered as fair curve. Therefore, energy minimization is often used as constraints in curve fairing. But, the curvature expression in the exact energy formula is a rational function, which results in non-linear equation system. Thus, the approximate energy is adopted to avoid non-linear problem. Assuming that the modulus length of the first derivative vector of the curve is a constant, and that arc-length parameters are used, we get an approximate energy model, which is expressed by integral of the squared second derivative of the curve. Because of the simple computation and good performance in practical application, the approximate energy model has been widely used. Wang gives several approximate energy models, all of which are derived from the exact energy formula through simplification. Without using approximate energy model, Zhou solves the exact energy problem using Newton iterative method, which is a good approximation to the exact energy modle.Many solutions to energy minimization constraint problem have their own disadvantages. For example, the integral of squared second derivative, which is often used as an approximate energy, may result in very bad solutions in some special cases. Lee gives some counterexamples that the curve becomes far away from fairing while the approximate energy value gets smaller. Wang gives a comparison between the exact energy model and three kinds of approximate energy model. The result showed that curves using the three approximate energy models would result in sharp corner and plane region more frequently. Therefore, they are not good approximation to the exact energy model. The author suggested that use the exact energy model, whose computing cost is a disadvantage. Zhou's method needs to compute the first order and second order derivatives. Because the curvature formula is a rational function, it would be very complex to solve the problem when the expressions of the curves are complex and the number of the unknowns is large.Aiming at the above problems, this paper presents an approximation algorithm for solving energy minimization problem. The algorithm replaces the first order derivative of the exact energy formula by an initial approximate value. Therefore, the nonlinear problem is changed into the linear problem to solve the unknowns in the object function. It's an iterative process to solve the unknowns. In each step, we substitute the old first order derivative vectors for new ones, until the difference between the new result and its previous one is smaller than a given error bound. Compared with existing methods, our new method is simple to use, and avoids high compute complexity and long compute time at a certain extent.Finally, the new method is used to fair curves. Compared with existing approximate method, our method can produce fair curves.