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首页> 外文期刊>Computer Aided Geometric Design >Reduced curvature formulae for surfaces, offset surfaces, curves on a surface and surface intersections
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Reduced curvature formulae for surfaces, offset surfaces, curves on a surface and surface intersections

机译：曲面，偏移曲面，曲面上的曲线和曲面相交的简化曲率公式

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摘要

We introduce the concept of reduced curvature formulae for 3-D space entities (surfaces, curves). A reduced formula entails only derivatives of the functions involved in the entity's representation and admits no further algebraic simplifications. Although not always the most compact, reduced curvature formulae entail only basic arithmetic operators and are more efficient computationally compared to alternative unreduced formulae. Reduced formulae are presented for the normal, mean and Gaussian curvatures of a surface and the curvature of curves on a surface, where each surface or curve on a surface may be defined parametrically or implicitly. Reduced formulae are also presented for the curvature of surface intersection curves, where each of the intersecting surfaces may be a given surface or an offset of a given surface and each given surface may be defined parametrically or implicitly. Known formulae are cited, without derivation, to form a collection, in one place, of new and of known results scattered in the literature. Each curve curvature formula is presented together with a formula for the respective binormal vector, from which formulae for the Frenet frame and torsion of the curve can be derived.
机译：我们介绍了3-D空间实体（曲面，曲线）的降低曲率公式的概念。简化的公式仅包含实体表示中涉及的函数的导数，并且不接受进一步的代数简化。尽管并非总是最紧凑，但减小曲率公式仅需要基本算术运算符，并且与备选未归约公式相比，计算效率更高。为表面的法线，均值和高斯曲率以及表面上的曲线曲率提供了简化的公式，其中，可以参数化或隐式定义表面上的每个表面或曲线。还为表面相交曲线的曲率提供了简化的公式，其中每个相交的表面可以是给定的表面或给定表面的偏移，并且每个给定的表面可以参数或隐式定义。引用已知公式（不推导）可在一处形成散布在文献中的新结果和已知结果的集合。每个曲线曲率公式与相应的双法线矢量的公式一起显示，由此可以得出Frenet框架和曲线扭转的公式。

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