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<p><span lang=EN-US style='font-family:隶书;color:#CC3300'><a
href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter3/CG_Txt_3_009.htm"><span
style='text-decoration:none;text-underline:none'><!--[if gte vml 1]><v:shape
 id="_x0000_i1036" type="#_x0000_t75" alt="" href="..\CG_Txt_3_009.htm"
 style='width:24.75pt;height:21.75pt' o:button="t">
 <v:imagedata src="./第三章%20几何造型技术1(参数曲线和曲面).files/image004.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/material/CG_Gif_pub_024.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=33 height=29
src="./第三章%20几何造型技术1(参数曲线和曲面).files/image004.gif" v:shapes="_x0000_i1036"><![endif]></span></a></span><span
lang=EN-US style='font-family:隶书;color:#666633'><a
href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter3/CG_Txt_3_009.htm"><span
style='color:#666633'>位置矢量、切矢量、法矢量、曲率和挠率</span></a><o:p></o:p></span></p>

<p style='line-height:200%'>一条用参数表示的三维曲线是一个有界点集，可写成一个带参数的、连续的、单值的数学函数，其形式为：</p>

<p align=center style='margin-right:36.0pt;margin-left:36.0pt;text-align:center;
line-height:200%'><i><span lang=EN-US style='font-family:"Times New Roman"'>x</span></i><span
lang=EN-US>=</span><i><span lang=EN-US style='font-family:"Times New Roman"'>x</span></i><span
lang=EN-US>(</span><i><span lang=EN-US style='font-family:"Times New Roman"'>t</span></i><span
lang=EN-US>)，</span><i><span lang=EN-US style='font-family:"Times New Roman"'>y</span></i><span
lang=EN-US>=</span><i><span lang=EN-US style='font-family:"Times New Roman"'>y</span></i><span
lang=EN-US>(</span><i><span lang=EN-US style='font-family:"Times New Roman"'>t</span></i><span
lang=EN-US>)，</span><i><span lang=EN-US style='font-family:"Times New Roman"'>z</span></i><span
lang=EN-US>=</span><i><span lang=EN-US style='font-family:"Times New Roman"'>z</span></i><span
lang=EN-US>(</span><i><span lang=EN-US style='font-family:"Times New Roman"'>t</span></i><span
lang=EN-US>)，0≤</span><i><span lang=EN-US style='font-family:"Times New Roman"'>t</span></i><span
lang=EN-US>≤1；</span></p>

<p style='margin-left:36.0pt;text-indent:-18.0pt;line-height:200%;mso-list:
l5 level1 lfo6;tab-stops:list 36.0pt'><![if !supportLists]><span lang=EN-US>1.<span
style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; </span></span><![endif]>位置矢量<span
lang=EN-US> </span></p>

<p style='margin-left:36.0pt;line-height:200%'>如图<span lang=EN-US>3.1.1所示，曲线上任一点的位置矢量可表示为：</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>P</span></i><span lang=EN-US>(</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>t</span></i><span lang=EN-US>)=[</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>x</span></i><span lang=EN-US>(</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>t</span></i><span lang=EN-US>),
</span><i><span lang=EN-US style='font-family:"Times New Roman"'>y</span></i><span
lang=EN-US>(</span><i><span lang=EN-US style='font-family:"Times New Roman"'>t</span></i><span
lang=EN-US>), </span><i><span lang=EN-US style='font-family:"Times New Roman"'>z</span></i><span
lang=EN-US>(</span><i><span lang=EN-US style='font-family:"Times New Roman"'>t</span></i><span
lang=EN-US>)]；其一阶、二阶和</span><i><span lang=EN-US style='font-family:"Times New Roman"'>k</span></i>阶导数矢量（如果存在的话）可分别表示为：<span
lang=EN-US> </span></p>

<p align=center style='margin-left:36.0pt;text-align:center;line-height:200%'><span
lang=EN-US><!--[if gte vml 1]><v:shape id="_x0000_i1037" type="#_x0000_t75"
 alt="" style='width:84pt;height:143.25pt'>
 <v:imagedata src="./第三章%20几何造型技术1(参数曲线和曲面).files/image009.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter3/CG_Gif_3_205.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=112 height=191
src="./第三章%20几何造型技术1(参数曲线和曲面).files/image009.gif" v:shapes="_x0000_i1037"><![endif]></span></p>

<p align=center style='margin-left:36.0pt;text-align:center;line-height:200%'><span
lang=EN-US><!--[if gte vml 1]><v:shape id="_x0000_i1038" type="#_x0000_t75"
 alt="" style='width:214.5pt;height:155.25pt'>
 <v:imagedata src="./第三章%20几何造型技术1(参数曲线和曲面).files/image010.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter3/31img/CG_Gif_3_001.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=286 height=207
src="./第三章%20几何造型技术1(参数曲线和曲面).files/image010.gif" v:shapes="_x0000_i1038"><![endif]></span></p>

<p align=center style='margin-left:36.0pt;text-align:center;line-height:200%'><span
style='font-family:隶书;color:#FF9900'>图<span lang=EN-US>3.1.1 表示一条参数曲线的有关矢量</span></span></p>

<p style='margin-left:36.0pt;text-indent:-18.0pt;line-height:200%;mso-list:
l5 level1 lfo6;tab-stops:list 36.0pt'><![if !supportLists]><span lang=EN-US>2.<span
style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; </span></span><![endif]>切矢量<span
lang=EN-US> </span></p>

<p style='margin-left:36.0pt;line-height:200%'><span lang=EN-US>&nbsp;&nbsp;&nbsp;
若曲线上</span><i><span lang=EN-US style='font-family:"Times New Roman"'>R</span></i>、<i><span
lang=EN-US style='font-family:"Times New Roman"'>Q</span></i>两点的参数分别是<i><span
lang=EN-US style='font-family:"Times New Roman"'>t</span></i>和<i><span
lang=EN-US style='font-family:"Times New Roman"'>t</span></i><span lang=EN-US>+</span><span
lang=EN-US style='font-size:10.0pt'>△</span><i><span lang=EN-US
style='font-family:"Times New Roman"'>t</span></i>，矢量<span lang=EN-US
style='font-size:10.0pt'>△</span><i><span lang=EN-US style='font-family:"Times New Roman"'>P</span></i><span
lang=EN-US>=</span><i><span lang=EN-US style='font-family:"Times New Roman"'>P</span></i><span
lang=EN-US>(</span><i><span lang=EN-US style='font-family:"Times New Roman"'>t</span></i><span
lang=EN-US>+</span><span lang=EN-US style='font-size:10.0pt'>△</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>t</span></i><span lang=EN-US>)-</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>P</span></i><span lang=EN-US>(</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>t</span></i><span lang=EN-US>)，其大小以连接</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>RQ</span></i>的弦长表示。如果在<span
lang=EN-US>R处有确定的切线，则当</span><i><span lang=EN-US style='font-family:"Times New Roman"'>Q</span></i>趋向<span
lang=EN-US><!--[if gte vml 1]><v:shape id="_x0000_i1039" type="#_x0000_t75"
 alt="" style='width:378.75pt;height:72.75pt'>
 <v:imagedata src="./第三章%20几何造型技术1(参数曲线和曲面).files/image011.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter3/CG_Gif_3_206.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=505 height=97
src="./第三章%20几何造型技术1(参数曲线和曲面).files/image011.gif" v:shapes="_x0000_i1039"><![endif]></span></p>

<p align=center style='margin-left:36.0pt;text-align:center;line-height:200%'><i><span
lang=EN-US style='font-family:"Times New Roman"'>(ds)<sup>2</sup>=(dx)<sup>2</sup>+(dy)<sup>2</sup>+(dz)<sup>2</sup></span></i></p>

<p style='margin-left:36.0pt;line-height:200%'>引入参数<i><span lang=EN-US
style='font-family:"Times New Roman"'>t</span></i>，上式可改写为：</p>

<p align=center style='margin-left:36.0pt;text-align:center;line-height:200%'><span
lang=EN-US><!--[if gte vml 1]><v:shape id="_x0000_i1040" type="#_x0000_t75"
 alt="" style='width:283.5pt;height:26.25pt'>
 <v:imagedata src="./第三章%20几何造型技术1(参数曲线和曲面).files/image012.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter3/CG_Gif_3_207.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=378 height=35
src="./第三章%20几何造型技术1(参数曲线和曲面).files/image012.gif" v:shapes="_x0000_i1040"><![endif]></span></p>

<p style='margin-left:36.0pt;line-height:200%'>考虑到矢量的模非负，所以：</p>

<p align=center style='margin-left:36.0pt;text-align:center;line-height:200%'><span
lang=EN-US><!--[if gte vml 1]><v:shape id="_x0000_i1041" type="#_x0000_t75"
 alt="" style='width:93.75pt;height:35.25pt'>
 <v:imagedata src="./第三章%20几何造型技术1(参数曲线和曲面).files/image013.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter3/CG_Gif_3_208.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=125 height=47
src="./第三章%20几何造型技术1(参数曲线和曲面).files/image013.gif" v:shapes="_x0000_i1041"><![endif]></span></p>

<p style='margin-left:36.0pt;line-height:200%'>故弧长<i><span lang=EN-US
style='font-family:"Times New Roman"'>s</span></i>是<i><span lang=EN-US
style='font-family:"Times New Roman"'>t</span></i>的单调增函数，其反函数<i><span
lang=EN-US style='font-family:"Times New Roman"'>t</span></i><span lang=EN-US>(</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>s</span></i><span lang=EN-US>)存在，且一一对应，得</span></p>

<p align=center style='margin-left:36.0pt;text-align:center;line-height:200%'><i><span
lang=EN-US style='font-family:"Times New Roman"'>P(t)</span></i><span
lang=EN-US style='font-family:"Times New Roman"'>=<i>P(t(s))</i>=<i>P(s)</i></span></p>

<p style='margin-left:36.0pt;line-height:200%'><span lang=EN-US>&nbsp;&nbsp;&nbsp;
于是：</span></p>

<p align=center style='margin-left:36.0pt;text-align:center;line-height:200%'><span
lang=EN-US><!--[if gte vml 1]><v:shape id="_x0000_i1042" type="#_x0000_t75"
 alt="" style='width:129pt;height:39.75pt'>
 <v:imagedata src="./第三章%20几何造型技术1(参数曲线和曲面).files/image014.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter3/CG_Gif_3_209.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=172 height=53
src="./第三章%20几何造型技术1(参数曲线和曲面).files/image014.gif" v:shapes="_x0000_i1042"><![endif]></span></p>

<p style='margin-left:36.0pt;line-height:200%'>即<i><span lang=EN-US
style='font-family:"Times New Roman"'>T</span></i>是单位切矢量。</p>

<p style='margin-left:36.0pt;text-indent:-18.0pt;line-height:200%;mso-list:
l5 level1 lfo6;tab-stops:list 36.0pt'><![if !supportLists]><span lang=EN-US>3.<span
style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; </span></span><![endif]>法矢量<span
lang=EN-US> </span></p>

<p style='margin-left:36.0pt;line-height:200%'><span lang=EN-US>&nbsp;&nbsp;&nbsp;
对于空间参数曲线任意一点，所有垂直切矢量</span><i><span lang=EN-US style='font-family:"Times New Roman"'>T</span></i>的矢量有一束，且位于同一平面上，该平面称为法平面，如图<span
lang=EN-US style='font-family:"Times New Roman"'>3</span><span lang=EN-US>.</span><span
lang=EN-US style='font-family:"Times New Roman"'>1</span><span lang=EN-US>.</span><span
lang=EN-US style='font-family:"Times New Roman"'>2</span>所示。</p>

<p align=center style='margin-left:36.0pt;text-align:center;line-height:200%'><span
lang=EN-US><!--[if gte vml 1]><v:shape id="_x0000_i1043" type="#_x0000_t75"
 alt="" style='width:208.5pt;height:132pt'>
 <v:imagedata src="./第三章%20几何造型技术1(参数曲线和曲面).files/image015.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter3/31img/CG_Gif_3_002.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=278 height=176
src="./第三章%20几何造型技术1(参数曲线和曲面).files/image015.gif" v:shapes="_x0000_i1043"><![endif]></span></p>

<p align=center style='margin-left:36.0pt;text-align:center;line-height:200%'><span
style='font-family:隶书;color:#FF9900'>图<span lang=EN-US>3.1.2 曲线的法矢</span></span></p>

<p style='margin-left:36.0pt;line-height:200%'><span lang=EN-US>&nbsp;&nbsp;&nbsp;
若对曲线上任一点的单位切矢为T，因为</span><span lang=EN-US style='font-family:"Times New Roman"'>[<i>T</i>(<i>s</i>)]<i><sup>2</sup></i>=1</span>，两边对<i><span
lang=EN-US style='font-family:"Times New Roman"'>s</span></i>求导矢<span
lang=EN-US><!--[if gte vml 1]><v:shape id="_x0000_i1044" type="#_x0000_t75"
 alt="" style='width:385.5pt;height:73.5pt'>
 <v:imagedata src="./第三章%20几何造型技术1(参数曲线和曲面).files/image016.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter3/CG_Gif_3_210.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=514 height=98
src="./第三章%20几何造型技术1(参数曲线和曲面).files/image016.gif" v:shapes="_x0000_i1044"><![endif]>
于矢量B的法矢称为曲线在该点的副法矢，B则称为单位副法失量。</span></p>

<p style='margin-left:36.0pt;line-height:200%'><span lang=EN-US>&nbsp;&nbsp;&nbsp;
对于一般参数t，我们可以推导出：</span></p>

<p align=center style='margin-left:36.0pt;text-align:center;line-height:200%'><span
lang=EN-US><!--[if gte vml 1]><v:shape id="_x0000_i1045" type="#_x0000_t75"
 alt="" style='width:192.75pt;height:78pt'>
 <v:imagedata src="./第三章%20几何造型技术1(参数曲线和曲面).files/image017.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter3/CG_Gif_3_211.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=257 height=104
src="./第三章%20几何造型技术1(参数曲线和曲面).files/image017.gif" v:shapes="_x0000_i1045"><![endif]></span></p>

<p style='margin-left:36.0pt;line-height:200%'><i><span lang=EN-US
style='font-family:"Times New Roman"'>T</span></i><span lang=EN-US>(切矢)、</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>N</span></i><span lang=EN-US>(主法矢)和</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>B</span></i><span lang=EN-US>(副法矢)构成了曲线上的活动坐标架，且</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>N</span></i>、<i><span
lang=EN-US style='font-family:"Times New Roman"'>B</span></i>构成的平面称为法平面，<i><span
lang=EN-US style='font-family:"Times New Roman"'>N</span></i>、<i><span
lang=EN-US style='font-family:"Times New Roman"'>T</span></i>构成的平面称为密切平面，<i><span
lang=EN-US style='font-family:"Times New Roman"'>B</span></i>、<i><span
lang=EN-US style='font-family:"Times New Roman"'>T</span></i>构成的平面称为从切平面。</p>

<p style='margin-left:36.0pt;text-indent:-18.0pt;line-height:200%;mso-list:
l5 level1 lfo6;tab-stops:list 36.0pt'><![if !supportLists]><span lang=EN-US>4.<span
style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; </span></span><![endif]>曲率和挠率<span
lang=EN-US> </span></p>

<p style='line-height:200%'><span lang=EN-US><!--[if gte vml 1]><v:shape id="_x0000_i1046"
 type="#_x0000_t75" alt="" style='width:402pt;height:128.25pt'>
 <v:imagedata src="./第三章%20几何造型技术1(参数曲线和曲面).files/image018.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter3/CG_Gif_3_212.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=536 height=171
src="./第三章%20几何造型技术1(参数曲线和曲面).files/image018.gif" v:shapes="_x0000_i1046"><![endif]></span></p>

<p style='line-height:200%'><span lang=EN-US><!--[if gte vml 1]><v:shape id="_x0000_i1047"
 type="#_x0000_t75" alt="" style='width:401.25pt;height:62.25pt'>
 <v:imagedata src="./第三章%20几何造型技术1(参数曲线和曲面).files/image019.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter3/CG_Gif_3_213.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=535 height=83
src="./第三章%20几何造型技术1(参数曲线和曲面).files/image019.gif" v:shapes="_x0000_i1047"><![endif]></span></p>

<p style='line-height:200%'><span lang=EN-US><!--[if gte vml 1]><v:shape id="_x0000_i1048"
 type="#_x0000_t75" alt="" style='width:397.5pt;height:96pt'>
 <v:imagedata src="./第三章%20几何造型技术1(参数曲线和曲面).files/image020.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter3/CG_Gif_3_214.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=530 height=128
src="./第三章%20几何造型技术1(参数曲线和曲面).files/image020.gif" v:shapes="_x0000_i1048"><![endif]>长的转动率（如图3.1.3(b)）。挠率</span><span
lang=EN-US style='font-family:Symbol'>?</span>大于<span lang=EN-US>0、等于0和小于0分别表示曲线为右旋空间曲线、平面曲线和左旋空间曲线。</span></p>

<p style='line-height:200%'><span lang=EN-US><!--[if gte vml 1]><v:shape id="_x0000_i1049"
 type="#_x0000_t75" alt="" style='width:307.5pt;height:21pt'>
 <v:imagedata src="./第三章%20几何造型技术1(参数曲线和曲面).files/image021.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter3/CG_Gif_3_215.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=410 height=28
src="./第三章%20几何造型技术1(参数曲线和曲面).files/image021.gif" v:shapes="_x0000_i1049"><![endif]></span></p>

<p align=center style='text-align:center;line-height:200%'><span lang=EN-US><!--[if gte vml 1]><v:shape
 id="_x0000_i1050" type="#_x0000_t75" alt="" style='width:134.25pt;height:19.5pt'>
 <v:imagedata src="./第三章%20几何造型技术1(参数曲线和曲面).files/image022.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter3/CG_Gif_3_216.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=179 height=26
src="./第三章%20几何造型技术1(参数曲线和曲面).files/image022.gif" v:shapes="_x0000_i1050"><![endif]></span></p>

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