利用空间曲线的一般方程计算其曲率和挠
率
利用空间曲线的一般方程计算其曲率和挠率
殷璞
(西北师范大学数学与信息科学学院甘肃兰州 730070)
摘要空间曲线由一般方程由
«Skip Record If...»
给出时，本文给出了计算曲线曲率和挠率的公式.
关键词曲率挠率曲线的一般方程
Determine the Curvature and Torsion of a Space Curve
by the General Equation
Yin Pu
(College of Mathematics and Information Science, Northwest Normal University,
Lanzhou730070,Gansu)
Abstract : In this paper, give the general equation of a space curve
«Skip Record If...»,
we calculate the formulas of the curvature and torsion.
Key words: curvature; torsion; the general equation of a space curve
曲线的曲率描述的是曲线的切向量对于弧长的旋转速度,即曲线的弯曲程度;曲线的挠率其绝对值描述的是曲线的副法向量(或密切平面)对于弧长的旋转速度,即曲线的扭曲程度.计算曲线的曲率和挠率一般是利用曲线的自然(弧长)参数方程进行推导的,所以曲线的方程由一般方程给出时,首先要改写成参数方程,然后再计算曲线的曲率和挠率.但是有些方程不容易改写成自然参数方程,本文就从曲线的一般方程出发直接推导计算曲线的曲率和挠率的公式.
下面,设曲线«Skip Record If...»是两光滑曲面«Skip Record If...»的交线,且
«Skip Record If...»
是满秩的.
一、计算曲线的曲率
设所求曲线为«Skip Record If...»,其中«Skip Record If...»为弧长.则
«Skip Record If...» . (1) 将方程(1)式中的两式对«Skip Record If...»求导,有
«Skip Record If...» (2)
记向量
«Skip Record If...»,«Skip Record If...» (3)
则曲线«Skip Record If...»的单位切向量«Skip Record If...»的方向平行于«Skip Record
If...»的方向.在局部,选择弧长«Skip Record If...»的增加方向,使得单位切向量«Skip Record
If...»的方向就是«Skip Record If...»的方向,那么,有
«Skip Record If...»«Skip Record If...»«Skip Record If...»«Skip Record If...»
(4)
又由(3)式知
«Skip Record If...»«Skip Record If...»«Skip Record If...». (5) «Skip Record If...»=«Skip Record If...». (6)
且由(4)式和(5)式可知
«Skip Record If...»=«Skip Record If...»«Skip Record If...».
«Skip Record If...»=«Skip Record If...»«Skip Record If...».
«Skip Record If...»=«Skip Record If...»«Skip Record If...». (7)
将(4)式两端对弧长«Skip Record If...»求导,由Frenet公式,有
«Skip Record If...»«Skip Record If...»«Skip Record If...»+«Skip Record If...»«Skip Record
If...»
«Skip Record If...» (8)
这里,«Skip Record If...»是单位主法向量.
将(4)式与(8)式两端分别作外积
«Skip Record If...»«Skip Record If...»«Skip Record If...»
«Skip Record If...» (9)
这里,«Skip Record If...»是单位副法向量.
因为«Skip Record If...»«Skip Record If...»=«Skip Record If...»«Skip Record If...»－
«Skip Record If...»
=－«Skip Record If...»(10)
«Skip Record If...»«Skip Record If...»=«Skip Record If...»«Skip Record If...»«Skip Record If...»
=«Skip Record If...». (11)
将(10)式和(11)式代入(9)式,有
«Skip Record If...»«Skip Record If...»«Skip Record If...»=«Skip Record If...»－«Skip Record If...». (12)
由(3)式知
«Skip Record If...»=«Skip Record If...»«Skip Record If...»
«Skip Record If...»«Skip Record If...». (13) «Skip Record If...»=«Skip Record If...»«Skip Record If...»
«Skip Record If...»«Skip Record If...».(14) 则由(5)式和(14)式,有
«Skip Record If...»
=«Skip Record If...»«Skip Record If...»
=«Skip Record If...»«Skip Record If...»
＋«Skip Record If...»«Skip Record If...»
＋«Skip Record If...»«Skip Record If...» (15)
=«Skip Record If...»«Skip Record If...»＋
«Skip Record If...»«Skip Record If...»＋
«Skip Record If...»«Skip Record If...».
由(5)式和(13)式,有
«Skip Record If...»
=«Skip Record If...»«Skip Record If...»
=«Skip Record If...»«Skip Record If...»
