دانلود رایگان مقاله فرمول انحنای کاهش یافته برای سطوح، سطوح افست و سطح تقاطع

Abstract

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.

7. Concluding remarks

The following properties of reduced curvature formulae can now be stated: a) They are closed formulae, entailing only basic arithmetic operators (addition, subtraction, multiplication, division) and square root operators. They are thus suitable for casual users, whose skills do not extend beyond basic algebra and the extraction of function derivatives. b) They are more efficient compared to alternative unreduced formulae (see Appendix A). Although we have presented several reduced formulae, we have not exhausted all cases that may arise. We have not, for example, dealt with the curvature of curves defined by differential equations. Curves on offset surfaces may also arise in ways other than as intersections with other surfaces. An open problem is developing a curvature formula for the curve Cp traced by a point Pp on a surface Sp, offset of a given surface S, when its corresponding point P traces on S a given curve C. In this case, C and Cp do not have the same directions at corresponding points, so Euler’s formula (Equ. (41a)) cannot be used to relate the curvatures of C, Cp. The solution of this problem would be of practical interest in the isoparametric 3-D machining of a surface patch S, with a ball-end cutter.