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首页> 外文期刊>Computer Aided Geometric Design >Using implicit equations of parametric curves and surfaces without computing them: Polynomial algebra by values
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Using implicit equations of parametric curves and surfaces without computing them: Polynomial algebra by values

机译：使用参数曲线和曲面的隐式方程式而不计算它们：多项式代数的值

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摘要

The availability of the implicit equation of a plane curve or of a 3D surface can be very useful in order to solve many geometric problems involving the considered curve or surface: for example, when dealing with the point position problem or answering intersection questions. On the other hand, it is well known that in most cases, even for moderate degrees, the implicit equation is either difficult to compute or, if computed, the high degree and the big size of the coefficients makes extremely difficult its use in practice. We will show that, for several problems involving plane curves, 3D surfaces and some of their constructions (for example, offsets), it is possible to use the implicit equation (or, more precisely, its properties) without needing to explicitly determine it. We replace the computation of the implicit equation with the evaluation of the considered parameterizations in a set of points. We then translate the geometric problem in hand, into one or several generalized eigenvalue problems on matrix pencils (depending again on several evaluations of the considered parameterizations). This is the so-called "polynomial algebra by values" approach where the huge polynomial equations coming from Elimination Theory (e.g., using resultants) are replaced by big structured and sparse numerical matrices. For these matrices there are well-known numerical techniques allowing to provide the results we need to answer the geometric questions on the considered curves and surfaces.
机译：平面曲线或3D曲面的隐式方程的可用性对于解决涉及所考虑的曲线或曲面的许多几何问题非常有用：例如，在处理点位置问题或回答相交问题时。另一方面，众所周知，在大多数情况下，即使对于中等度数，隐式方程要么难以计算，要么如果被计算，则系数的高阶和大尺寸使其在实践中极难使用。我们将显示，对于涉及平面曲线，3D曲面及其某些构造（例如，偏移）的若干问题，可以使用隐式方程（或更准确地说，其属性）而无需明确确定它。我们用一组点中考虑的参数化的评估来代替隐式方程的计算。然后，我们将手中的几何问题转换为矩阵铅笔上的一个或几个广义特征值问题（再次取决于所考虑的参数化的几次评估）。这是所谓的“按值的多项式代数”方法，其中将消除理论（例如，使用结果）产生的庞大的多项式方程式替换为大型的结构化且稀疏的数值矩阵。对于这些矩阵，有众所周知的数值技术可以提供我们需要回答所考虑的曲线和曲面上的几何问题的结果。

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