空间有理曲线的μ基及曲线隐式化

摘要

空间有理曲线是计算机辅助几何设计中常用曲线,以空间曲线为研究对象,讨论其μ基形式,并用μ基方法快速隐式化一类曲线.首先给出μ基定义和一些基本性质,之后基于μ基方法分析给出空间曲线μ基的符号公式,进一步讨论了用μ基方法隐式化空间有理曲线.通过对结式性质的细致分析,得出一类空间曲线快速隐式化的结果,这类曲线隐式的仿射簇可以用μ基的两两单变量结式表示,对不满足条件的情况给出了反例.%Rational space curves are widely used in geometric modeling, this paper studies the μ-basis of the space curves and computes the algebraic varieties based on the μ-basis. The μ-basis plays a bridging role between the parametric expression and implicit expression. After the definition μ-basis is given, the paper presents the symbolic formula to compute the μ-basis. In addition, based on the discussion of resultants, we obtain a class of space curves which can be implicitized by univariate resultant computations on the μ-basis. A typical counterexample is also proposed when the restriction is unsatisfied.