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📁 计算机图形学网络课程
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 <v:imagedata src="./第六章%20附录——图形变换.files/image033.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter6/CG_Gif_6_279.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=253 height=115
src="./第六章%20附录——图形变换.files/image033.gif" v:shapes="_x0000_i1060"><![endif]></span></p>

<p style='line-height:200%'><span lang=EN-US>&nbsp;&nbsp;&nbsp; 可将其表示为矩阵形式：</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>AX</span></i><span lang=EN-US>=</span><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"'>A</span></i>为系数矩阵。该方程有唯一解的条件是<i><span
lang=EN-US style='font-family:"Times New Roman"'>A</span></i><span lang=EN-US> 是非奇异矩阵，则方程的解为：</span></p>

<p align=center style='text-align:center'><i><span lang=EN-US><!--[if gte vml 1]><v:shape
 id="_x0000_i1061" type="#_x0000_t75" alt="" style='width:43.5pt;height:15pt'>
 <v:imagedata src="./第六章%20附录——图形变换.files/image034.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter6/CG_Gif_6_280.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=58 height=20
src="./第六章%20附录——图形变换.files/image034.gif" v:shapes="_x0000_i1061"><![endif]></span></i></p>

<p align=center style='text-align:center'><b><span lang=EN-US style='font-size:
18.0pt;color:green'>6.2</span><span lang=EN-US> </span></b><b><span
style='font-size:18.0pt;font-family:仿宋_GB2312;color:green'>几何变换</span></b></p>

<p><span lang=EN-US style='mso-ascii-font-family:楷体_GB2312;mso-fareast-font-family:
楷体_GB2312'>&nbsp;&nbsp;&nbsp;</span><span lang=EN-US style='font-family:楷体_GB2312'>
这一节里，我们将给出几何变换的一些基本公式。</span></p>

<p><b><span lang=EN-US style='font-size:10.0pt;font-family:幼圆;color:gray'><a
href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter6/CG_Txt_6_010.htm"><span
style='text-decoration:none;text-underline:none'><!--[if gte vml 1]><v:shape
 id="_x0000_i1062" type="#_x0000_t75" alt="" href="..\CG_Txt_6_010.htm"
 style='width:21pt;height:37.5pt' o:button="t">
 <v:imagedata src="./第六章%20附录——图形变换.files/image001.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/material/CG_Gif_pub_021.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=28 height=50
src="./第六章%20附录——图形变换.files/image001.gif" v:shapes="_x0000_i1062"><![endif]></span></a></span></b><b><span
lang=EN-US style='font-family:幼圆;color:gray'><a
href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter6/CG_Txt_6_010.htm"><span
style='color:gray'>窗口区到视图区的坐标变换</span></a><o:p></o:p></span></b></p>

<p style='line-height:200%'>实际的窗口区与视图区往往不一样大小，要在视图区正确地显示形体的，必须将其从窗口区变换到视图区。<span
lang=EN-US> </span></p>

<p align=center style='text-align:center'><span lang=EN-US><br>
<!--[if gte vml 1]><v:shape id="_x0000_i1063" type="#_x0000_t75" alt=""
 style='width:366pt;height:149.25pt'>
 <v:imagedata src="./第六章%20附录——图形变换.files/image035.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter6/CG_Gif_6_004.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=488 height=199
src="./第六章%20附录——图形变换.files/image035.gif" v:shapes="_x0000_i1063"><![endif]>&nbsp;
</span></p>

<p>比例关系，两者的变换公式为：<span lang=EN-US> </span></p>

<p align=center style='text-align:center'><span lang=EN-US><!--[if gte vml 1]><v:shape
 id="_x0000_i1064" type="#_x0000_t75" alt="" style='width:173.25pt;height:37.5pt'>
 <v:imagedata src="./第六章%20附录——图形变换.files/image036.png" o:href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter6/CG_Gif_6_216.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=231 height=50
src="./第六章%20附录——图形变换.files/image037.gif" v:shapes="_x0000_i1064"><![endif]></span></p>

<p align=center style='text-align:center'><span lang=EN-US>&nbsp;&nbsp; <!--[if gte vml 1]><v:shape
 id="_x0000_i1065" type="#_x0000_t75" alt="" style='width:190.5pt;height:37.5pt'>
 <v:imagedata src="./第六章%20附录——图形变换.files/image038.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter6/CG_Gif_6_217.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=254 height=50
src="./第六章%20附录——图形变换.files/image038.gif" v:shapes="_x0000_i1065"><![endif]></span></p>

<p>可以简单地将两者的关系表示为：</p>

<p align=center style='text-align:center'><span lang=EN-US>&nbsp; <!--[if gte vml 1]><v:shape
 id="_x0000_i1066" type="#_x0000_t75" alt="" style='width:133.5pt;height:144.75pt'>
 <v:imagedata src="./第六章%20附录——图形变换.files/image039.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter6/CG_Gif_6_219.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=178 height=193
src="./第六章%20附录——图形变换.files/image039.gif" v:shapes="_x0000_i1066"><![endif]>&nbsp;</span></p>

<p><span lang=EN-US><!--[if gte vml 1]><v:shape id="_x0000_i1067" type="#_x0000_t75"
 alt="" style='width:310.5pt;height:49.5pt'>
 <v:imagedata src="./第六章%20附录——图形变换.files/image040.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter6/CG_Gif_6_220.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=414 height=66
src="./第六章%20附录——图形变换.files/image040.gif" v:shapes="_x0000_i1067"><![endif]>&nbsp;
</span></p>

<p><span lang=EN-US><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></span></p>

<p><b><span lang=EN-US style='font-size:10.0pt;font-family:幼圆;color:gray'><a
href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter6/CG_Txt_6_011.htm"><span
style='text-decoration:none;text-underline:none'><!--[if gte vml 1]><v:shape
 id="_x0000_i1068" type="#_x0000_t75" alt="" href="..\CG_Txt_6_011.htm"
 style='width:21pt;height:37.5pt' o:button="t">
 <v:imagedata src="./第六章%20附录——图形变换.files/image001.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/material/CG_Gif_pub_021.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=28 height=50
src="./第六章%20附录——图形变换.files/image001.gif" v:shapes="_x0000_i1068"><![endif]></span></a></span></b><b><span
lang=EN-US style='font-family:幼圆;color:gray'><a
href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter6/CG_Txt_6_011.htm"><span
style='color:gray'>二维图形的几何变换</span></a><o:p></o:p></span></b></p>

<p style='line-height:200%'>正如我们在附录中提到的那样，用齐次坐标表示点的变换将非常方便，因此在本节中所有的几何变换都将采用齐次坐标进行运算。二维齐次坐标变换的矩阵的形式是：<span
lang=EN-US> <br>
<!--[if gte vml 1]><v:shape id="_x0000_i1069" type="#_x0000_t75" alt=""
 style='width:68.25pt;height:53.25pt'>
 <v:imagedata src="./第六章%20附录——图形变换.files/image041.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter6/CG_Gif_6_256.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=91 height=71
src="./第六章%20附录——图形变换.files/image041.gif" v:shapes="_x0000_i1069"><![endif]>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<br>
这个矩阵每一个元素都是有特殊含义的。<br>
<!--[if gte vml 1]><v:shape id="_x0000_i1070" type="#_x0000_t75" alt=""
 style='width:401.25pt;height:30.75pt'>
 <v:imagedata src="./第六章%20附录——图形变换.files/image042.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter5/CG_Gif_5_209.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=535 height=41
src="./第六章%20附录——图形变换.files/image042.gif" v:shapes="_x0000_i1070"><![endif]>形进行平移变换；[</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>g</span></i><span lang=EN-US
style='font-family:"Times New Roman"'> <i>h</i></span><span lang=EN-US>]是对图形作投影变换；[</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>i</span></i><span lang=EN-US>]则是对图形整体进行缩放变换。<br>
1）平移变换 <br>
<!--[if gte vml 1]><v:shape id="_x0000_i1071" type="#_x0000_t75" alt=""
 style='width:269.25pt;height:60.75pt'>
 <v:imagedata src="./第六章%20附录——图形变换.files/image043.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter6/CG_Gif_6_253.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=359 height=81
src="./第六章%20附录——图形变换.files/image043.gif" v:shapes="_x0000_i1071"><![endif]><br>
<br>
2）缩放变换 <br>
<!--[if gte vml 1]><v:shape id="_x0000_i1072" type="#_x0000_t75" alt=""
 style='width:276.75pt;height:60.75pt'>
 <v:imagedata src="./第六章%20附录——图形变换.files/image044.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter6/CG_Gif_6_254.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=369 height=81
src="./第六章%20附录——图形变换.files/image044.gif" v:shapes="_x0000_i1072"><![endif]><br>
<br>
3）旋转变换<!--[if gte vml 1]><v:shape id="_x0000_i1073" type="#_x0000_t75" alt=""
 style='width:376.5pt;height:21pt'>
 <v:imagedata src="./第六章%20附录——图形变换.files/image045.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter6/CG_Gif_6_291.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=502 height=28
src="./第六章%20附录——图形变换.files/image045.gif" v:shapes="_x0000_i1073"><![endif]><br>
<!--[if gte vml 1]><v:shape id="_x0000_i1074" type="#_x0000_t75" alt=""
 style='width:315.75pt;height:49.5pt'>
 <v:imagedata src="./第六章%20附录——图形变换.files/image046.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter6/CG_Gif_6_255.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=421 height=66
src="./第六章%20附录——图形变换.files/image046.gif" v:shapes="_x0000_i1074"><![endif]><br>
<br>
<!--[if gte vml 1]><v:shape id="_x0000_i1075" type="#_x0000_t75" alt=""
 style='width:252pt;height:21pt'>
 <v:imagedata src="./第六章%20附录——图形变换.files/image047.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter6/CG_Gif_6_292.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=336 height=28
src="./第六章%20附录——图形变换.files/image047.gif" v:shapes="_x0000_i1075"><![endif]><br>
4）对称变换<br>
<i><!--[if gte vml 1]><v:shape id="_x0000_i1076" type="#_x0000_t75" alt=""
 style='width:214.5pt;height:58.5pt'>
 <v:imagedata src="./第六章%20附录——图形变换.files/image048.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter6/CG_Gif_6_222.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=286 height=78
src="./第六章%20附录——图形变换.files/image048.gif" v:shapes="_x0000_i1076"><![endif]></i></span></p>

<p>对称变换其实只是<i><span lang=EN-US style='font-family:"Times New Roman"'>a</span></i>、<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"'>d</span></i>、<i><span
lang=EN-US style='font-family:"Times New Roman"'>e</span></i>取<span lang=EN-US>0、1等特殊值产生的一些特殊效果。例如：
</span></p>

<p style='margin-left:36.0pt;text-indent:-18.0pt;line-height:200%;mso-list:
l4 level1 lfo18;tab-stops:list 36.0pt'><![if !supportLists]><span lang=EN-US>A.<span
style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; </span></span><![endif]>当<i><span
lang=EN-US style='font-family:"Times New Roman"'>b</span></i><span lang=EN-US>=</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>d</span></i><span lang=EN-US>=0，</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>a</span></i><span lang=EN-US>=-1，</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>e</span></i><span lang=EN-US>=1时有</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>x</span></i><sup><span
lang=EN-US>&acute;</span></sup><span lang=EN-US>=-</span><i><span lang=EN-US
style='font-family:"Times New Roman"'>x</span></i>，<i><span lang=EN-US
style='font-family:"Times New Roman"'>y</span></i><sup><span lang=EN-US>&acute;</span></sup><span
lang=EN-US>=</span><i><span lang=EN-US style='font-family:"Times New Roman"'>y</span></i>，产生与<i><span
lang=EN-US style='font-family:"Times New Roman"'>y</span></i>轴对称的图形<i>。</i></p>

<p style='margin-left:36.0pt;text-indent:-18.0pt;line-height:200%;mso-list:
l4 level1 lfo18;tab-stops:list 36.0pt'><![if !supportLists]><span lang=EN-US>B.<span
style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; </span></span><![endif]>当<i><span
lang=EN-US style='font-family:"Times New Roman"'>b</span></i><span lang=EN-US>=</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>d</span></i><span lang=EN-US>=0，</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>a</span></i><span lang=EN-US>=-1，</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>e</span></i><span lang=EN-US>=-1时有</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>x</span></i><sup><span
lang=EN-US>&acute;</span></sup><span lang=EN-US>=</span><i><span lang=EN-US
style='font-family:"Times New Roman"'>x</span></i>，<i><span lang=EN-US
style='font-family:"Times New Roman"'>y</span></i><sup><span lang=EN-US>&acute;</span></sup><span
lang=EN-US>=-</span><i><span lang=EN-US style='font-family:"Times New Roman"'>y</span></i>，产生与<i><span
lang=EN-US style='font-family:"Times New Roman"'>x</span></i>轴对称的图形。</p>

<p style='margin-left:36.0pt;text-indent:-18.0pt;line-height:200%;mso-list:
l4 level1 lfo18;tab-stops:list 36.0pt'><![if !supportLists]><span lang=EN-US>C.<span
style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; </span></span><![endif]>当<i><span
lang=EN-US style='font-family:"Times New Roman"'>b</span></i><span lang=EN-US>=</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>d</span></i><span lang=EN-US>=0，</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>a</span></i><span lang=EN-US>=</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>e</span></i><span lang=EN-US>=-1时有</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>x</span></i><sup><span
lang=EN-US>&acute;</span></sup><span lang=EN-US>=-</span><i><span lang=EN-US
style='font-family:"Times New Roman"'>x</span></i>，<i><span lang=EN-US
style='font-family:"Times New Roman"'>y</span></i><sup><span lang=EN-US>&acute;</span></sup><span
lang=EN-US>=-</span><i><span lang=EN-US style='font-family:"Times New Roman"'>y</span></i>，产生与原点对称的图形。</p>

<p style='margin-left:36.0pt;text-indent:-18.0pt;line-height:200%;mso-list:
l4 level1 lfo18;tab-stops:list 36.0pt'><![if !supportLists]><span lang=EN-US>D.<span
style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; </span></span><![endif]>当<i><span
lang=EN-US style='font-family:"Times New Roman"'>b</span></i><span lang=EN-US>=</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>d</span></i><span lang=EN-US>=1，</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>a</span></i><span lang=EN-US>=</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>e</span></i><span lang=EN-US>=0时有</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>x</span></i><sup><span
lang=EN-US>&acute;</span></sup><span lang=EN-US>=</span><i><span lang=EN-US
style='font-family:"Times New Roman"'>y</span></i>，<i><span lang=EN-US
style='font-family:"Times New Roman"'>y</span></i><sup><span lang=EN-US>&acute;</span></sup><span
lang=EN-US>=</span><i><span lang=EN-US style='font-family:"Times New Roman"'>x</span></i>，产生与直线<span
lang=EN-US>y=x对称的图形。</span></p>

<p style='margin-left:36.0pt;text-indent:-18.0pt;line-height:200%;mso-list:
l4 level1 lfo18;tab-stops:list 36.0pt'><![if !supportLists]><span lang=EN-US>E.<span
style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; </span></span><![endif]>当<i><span
lang=EN-US style='font-family:"Times New Roman"'>b</span></i><span lang=EN-US>=</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>d</span></i><span lang=EN-US>=-1，</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>a</span></i><span lang=EN-US>=</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>e</span></i><span lang=EN-US>=0时有</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>x</span></i><sup><span
lang=EN-US>&acute;</span></sup><span lang=EN-US>=-</span><i><span lang=EN-US
style='font-family:"Times New Roman"'>y</span></i>，<i><span lang=EN-US
style='font-family:"Times New Roman"'>y</span></i><sup><span lang=EN-US>&acute;</span></sup><span
lang=EN-US>=-</span><i><span lang=EN-US style='font-family:"Times New Roman"'>x</span></i>，产生与直线<i><span
lang=EN-US style='font-family:"Times New Roman"'>y</span></i><span la
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