有理曲线的均匀区间隐式化

Uniform Interval Implicitization of Rational Surfaces

参数式曲线与隐式曲线是CAGD中常用的两种曲线形式,因此需要建立起二者之间相互转换的体制.长期以来,许多工作都集中在利用结式思想,将一个参数式曲线精确转化为一个隐式曲线上,而事实上用隐式曲线精确表示一条参数式曲线不仅非常麻烦,而且往往也没有必要.故此提出了参数式有理曲线均匀区间隐式化的一种新方法,利用区间算术和空间重心坐标的定义,可以用一个低阶区间多项式隐式曲线来逼近所给的参数式有理曲线,同时使一些目标函数最小化,达到用隐式多项式曲线来逼近参数式有理曲线的很好效果,并提供了一些算法和实例.

Parametric curves/surfaces and algebraic curves/surfaces are two common types of representations of geometric objects in computer aided design and geometric modeling. Both of these representations have their own advantages and disadvantages. Thus it is important to have both representations at the same time. In this paper, a new concept called uniform interval implicitization of rational curves is proposed, which is finding an uniform interval curve with lower degree bounding a given rational curve and minimizing some objective function. An algorithm and some examples are provided to demonstrate the theory.