参数曲线的根式弧角参数化

This project will apply the theory of symbolic computation in polynomial algebra and classic methods from optimization theory to compute the radical arc-angle reparameterization for parametric curves. As a generalization of arc-length and arc-angle reparameterization of rational curves, the problem of radical arc-angle reparameterization will focus on three aspects. First, for rational curves whose angular speed may become zero, compute a radical approximation of its arc-angle reparameterization. Second, for generic radical curves, study the completeness of radical arc-angle parameterization, i.e., whether any given radical curve has a radical arc-angle reparameterization. If not, present criteria for radical curves to have radical arc-angle reparameterizations. Third, for radical curves having no radical arc-angle reparameterization, compute a radical approximation of the arc-angle reparameterization. It is expected that the results of research in this project will lead to new methods for reparameterization of parametric curves and have application values in CAGD and CAM.

本课题拟将多项式代数中的符号计算理论和最优化理论的经典方法用于计算参数曲线的根式弧角重新参数化。作为有理曲线的弧角和弧长重新参数化方法的推广，根式弧角重新参数化将研究三方面的问题：（1）对于角速度包含零点的有理曲线，计算其弧角参数化的根式近似；（2）对于一般根式曲线，研究根式弧角参数化的完备性，即根式曲线是否都有根式弧角参数化？若否，则给出曲线具有根式弧角参数化的判别准则；（3）对于没有根式弧角参数化的根式曲线，计算其弧角参数化的根式近似。预计该项目的研究成果将产生曲线重新参数化的新方法，并在计算机辅助设计和制造中具有一定的应用价值。

本课题主要研究角速度函数在[0,1]上存在零点的有理参数曲线的弧角重新参数化问题。通过将分段有理变换推广为分段根式变换，并利用有理曲线的代数信息，构造特定形式的分段根式变换，使得在该变换下得到的重新参数化其角速度函数在[0,1]上不存在零点。采用多种策略对根式变换进行优化，给出最优参数的计算公式。该项工作是对具有非零角速度的有理曲线的弧角重新参数化方法的推广，可用于提高计算机软件绘制曲线的质量，在计算机辅助设计和制造中具有一定的应用价值。