Study on Iterative Methods in Geometric Design
|Keywords||progressive-iterative approximation data fitting geometric design triangle Bernstein-Bézier patches convergence local property shape control adaptive fitting fairness variational method energy minimization B-spline interpolation|
Recently, progressive-iterative approximation (abbr. PIA) is presented for data fitting. PIA generates a sequence of curves or patches, whose limit interpolates the given data points. It has been shown that the blending curves and tensor product blending patches with normalized totally positive basis have the PIA property. Because of its so many advantages, PIA has wide applications in geometric design.In this paper, we study the applicability and local property of PIA. First, we prove that, the quadratic, cubic and quartic non-uniform triangular Bernstein-Bezier patches have the PIA property. Since the most often employed in geometric design are the low degree curves or patches, especially the cubic curves or bi-cubic patches, the result shown in this paper has practical significance for geometric design.Additionally, this paper presents a proof of local PIA property for Loop, Doo-Sabin and Catmull-Clark subdivision. This property brings more flexibility for shape controlling, and also makes the adaptive fitting possible.Moreover, fairing curve and surface generation is an important topic in geometric design. However, the conventional method for generating the fairing curve and surface, which fit the given data points, is hard to control the fitting precision, because it is a minimization problem where the objecti.ve function is the weighted sum of a fitting term and a fairness term. In this paper, we develop the variational PIA method for fitting a data point sequence. While the variational PIA is easy to control the fitting precision, the generated fitting curve or surface is the most fairing one in some scope.Finally, the methods for B-spline curve and surface modification usually involve solving a constrained energy optimization problem, which costs lots of time when the control points of curves or surfaces are in a large amount. So it hardly achieves real-time response in the modification. In this paper, we propose a real time surface modification method based on PIA. Due to the advantage of PIA, the modification method developed in this paper only needs to adjust some control points iteratively, without solving the constrained optimization problem. It makes our method have real-time response, much faster than the optimization method. Meanwhile, the quality of the surface generated by our method is similar to that by solving the constrained energy optimization problem.