有理参数曲线的近似恰当化

Approximate Proper Reparametrization of Rational Curves

有理参数曲线的恰当性是曲线的基本性质,虽然其在有理系数情况下已经有完备的结果,但在工程和CAGD应用中常常得到带误差浮点系数的有理表示形式.为此,讨论了这类有误差的有理参数曲线,定义了近似非恰当参数形式和近似非恰当指数,并通过半代数系统计算近似非恰当指数;在给出近似非恰当指数的同时,得到近似最大公因子.最后基于最小二乘法给出近似参数有理变换表示,计算出曲线恰当的近似有理参数表示.

Properness of a parameterization is a basic property of rational parametric curves and it has been well studied when the coefficients of the parameterization are rational. However, such coefficients in engineering and CAGD are often represented with float numbers with errors. This work gives an algorithm to simplify the rational curves in this situation. After the approximate proper index is defined, it is computed by solving the semi-algebra system. The greatest common divisor is obtained according to the computing process of approximate proper index. Finally, the approximate proper reparameterization is obtained by the least squares method.