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<h2 style='text-align:justify;text-justify:inter-ideograph'><span
style='font-family:"Times New Roman"'>2.1 </span><span lang=ZH-CN
style='mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"'>平移</span><span
style='font-family:"Times New Roman"'><o:p></o:p></span></h2>
<p style='margin:0cm;margin-bottom:.0001pt;text-align:justify;text-justify:
inter-ideograph;line-height:18.0pt'><span lang=ZH-CN style='font-size:10.5pt'>平移</span><span
style='font-size:10.5pt;font-family:"Times New Roman"'>(translation)</span><span
lang=ZH-CN style='font-size:10.5pt'>变换大概是几何变换中最简单的一种了。</span><span
style='font-size:10.5pt;font-family:"Times New Roman"'><o:p></o:p></span></p>
<p style='margin:0cm;margin-bottom:.0001pt;text-align:justify;text-justify:
inter-ideograph;line-height:18.0pt'><span lang=ZH-CN style='font-size:10.5pt'>如图</span><span
style='font-size:10.5pt;font-family:"Times New Roman"'>2.1</span><span
lang=ZH-CN style='font-size:10.5pt'>所示，初始坐标为</span><span style='font-size:10.5pt;
font-family:"Times New Roman"'>(x<sub>0</sub>,y<sub>0</sub>)</span><span
lang=ZH-CN style='font-size:10.5pt'>的点经过平移</span><span style='font-size:10.5pt;
font-family:"Times New Roman"'>(t<sub>x</sub>,t<sub>y</sub>)(</span><span
lang=ZH-CN style='font-size:10.5pt'>以向右，向下为正方向</span><span style='font-size:
10.5pt;font-family:"Times New Roman"'>)</span><span lang=ZH-CN
style='font-size:10.5pt'>后，坐标变为</span><span style='font-size:10.5pt;font-family:
"Times New Roman"'>(x<sub>1</sub>,y<sub>1</sub>)</span><span lang=ZH-CN
style='font-size:10.5pt'>。这两点之间的关系是</span><span style='font-size:10.5pt;
font-family:"Times New Roman"'>x<sub>1</sub>=x<sub>0</sub>+t<sub>x </sub></span><span
lang=ZH-CN style='font-size:10.5pt'>，</span><span style='font-size:10.5pt;
font-family:"Times New Roman"'>y<sub>1</sub>=y<sub>0</sub>+t<sub>y</sub></span><span
lang=ZH-CN style='font-size:10.5pt'>。</span><span style='font-size:10.5pt;
font-family:"Times New Roman"'><o:p></o:p></span></p>
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<p align=center style='margin:0cm;margin-bottom:.0001pt;text-align:center'><b><span
lang=ZH-CN style='font-size:10.5pt'>图</span></b><b><span style='font-size:10.5pt;
font-family:"Times New Roman"'>2.1 </span></b><b><span
lang=ZH-CN style='font-size:10.5pt'>平移的示意图</span></b><span style='font-size:
10.5pt;font-family:"Times New Roman"'><o:p></o:p></span></p>
<p style='margin:0cm;margin-bottom:.0001pt;text-align:justify;text-justify:
inter-ideograph;line-height:18.0pt'><span lang=ZH-CN style='font-size:10.5pt'>以矩阵的形式表示为</span><span
style='font-size:10.5pt;font-family:"Times New Roman"'><o:p></o:p></span></p>
<p align=center style='margin:0cm;margin-bottom:.0001pt;text-align:center;
line-height:18.0pt'><sub><span style='font-size:10.5pt;font-family:"Times New Roman"'><!--[if gte vml 1]><v:shape
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src="./chp2.files/image002.gif" v:shapes="_x0000_i1026"><![endif]></span></sub><span
style='font-size:10.5pt;font-family:"Times New Roman"'><o:p></o:p></span></p>
<p align=right style='margin:0cm;margin-bottom:.0001pt;text-align:right;
line-height:18.0pt'><span style='font-size:10.5pt;font-family:"Times New Roman"'>
(2.1)<o:p></o:p></span></p>
<p style='margin:0cm;margin-bottom:.0001pt;text-align:justify;text-justify:
inter-ideograph;line-height:18.0pt'><span lang=ZH-CN style='font-size:10.5pt'>我们更关心的是它的逆变换：</span><span
style='font-size:10.5pt;font-family:"Times New Roman"'><o:p></o:p></span></p>
<p align=center style='margin:0cm;margin-bottom:.0001pt;text-align:center;
line-height:18.0pt'><sub><span style='font-size:10.5pt;font-family:"Times New Roman"'><!--[if gte vml 1]><v:shape
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src="./chp2.files/image003.gif" v:shapes="_x0000_i1027"><![endif]></span></sub><span
style='font-size:10.5pt;font-family:"Times New Roman"'><o:p></o:p></span></p>
<p align=right style='margin:0cm;margin-bottom:.0001pt;text-align:right;
line-height:18.0pt'><span style='font-size:10.5pt;font-family:"Times New Roman"'>(2.2)<o:p></o:p></span></p>
<p style='margin:0cm;margin-bottom:.0001pt;text-align:justify;text-justify:
inter-ideograph;line-height:18.0pt'><span lang=ZH-CN style='font-size:10.5pt'>这是因为：我们想知道的是平移后的图象中每个象素的颜色。例如我们想知道，新图中左上角点的</span><span
style='font-size:10.5pt;font-family:"Times New Roman"'>RGB</span><span
lang=ZH-CN style='font-size:10.5pt'>值是多少？很显然，该点是原图的某点经过平移后得到的，这两点的颜色肯定是一样的，所以只要知道了原图那点的</span><span
style='font-size:10.5pt;font-family:"Times New Roman"'>RGB</span><span
lang=ZH-CN style='font-size:10.5pt'>值即可。那么到底新图中的左上角点对应原图中的哪一点呢？将左上角点的坐标</span><span
style='font-size:10.5pt;font-family:"Times New Roman"'>(0,0)</span><span
lang=ZH-CN style='font-size:10.5pt'>入公式</span><span style='font-size:10.5pt;
font-family:"Times New Roman"'>(2.2)</span><span lang=ZH-CN style='font-size:
10.5pt'>，得到</span><span style='font-size:10.5pt;font-family:"Times New Roman"'>x<sub>0=</sub>-t<sub>x
</sub></span><span lang=ZH-CN style='font-size:10.5pt'>，</span><span
style='font-size:10.5pt;font-family:"Times New Roman"'>y<sub>0</sub>=-t<sub>y</sub></span><span
lang=ZH-CN style='font-size:10.5pt'>；所以新图中的</span><span style='font-size:10.5pt;
font-family:"Times New Roman"'>(0,0)</span><span lang=ZH-CN style='font-size:
10.5pt'>点的颜色和原图中</span><span style='font-size:10.5pt;font-family:"Times New Roman"'>(-t<sub>x</sub>
, -t<sub>y</sub>)</span><span lang=ZH-CN style='font-size:10.5pt'>的一样。</span><span
style='font-size:10.5pt;font-family:"Times New Roman"'><o:p></o:p></span></p>
<p style='margin:0cm;margin-bottom:.0001pt;text-align:justify;text-justify:
inter-ideograph;line-height:18.0pt'><span lang=ZH-CN style='font-size:10.5pt'>这样就存在一个问题：如果新图中有一点</span><span
style='font-size:10.5pt;font-family:"Times New Roman"'>(x<sub>1</sub>,y<sub>1</sub>)</span><span
lang=ZH-CN style='font-size:10.5pt'>，按照公式</span><span style='font-size:10.5pt;
font-family:"Times New Roman"'>(2.2)</span><span lang=ZH-CN style='font-size:
10.5pt'>得到的</span><span style='font-size:10.5pt;font-family:"Times New Roman"'>(x<sub>0</sub>,y<sub>0</sub>)</span><span
lang=ZH-CN style='font-size:10.5pt'>不在原图中该怎么办？通常的做法是，把该点的</span><span
style='font-size:10.5pt;font-family:"Times New Roman"'>RGB</span><span
lang=ZH-CN style='font-size:10.5pt'>值统一设成</span><span style='font-size:10.5pt;
font-family:"Times New Roman"'>(0,0,0)</span><span lang=ZH-CN style='font-size:
10.5pt'>或者</span><span style='font-size:10.5pt;font-family:"Times New Roman"'>(255,255,255)</span><span
lang=ZH-CN style='font-size:10.5pt'>。</span><span style='font-size:10.5pt;
font-family:"Times New Roman"'><o:p></o:p></span></p>
<p style='margin:0cm;margin-bottom:.0001pt;text-align:justify;text-justify:
inter-ideograph;line-height:18.0pt'><span lang=ZH-CN style='font-size:10.5pt'>另一个问题是：平移后的图象是否要放大？一种做法是不放大，移出的部分被截断。例如，图</span><span
style='font-size:10.5pt;font-family:"Times New Roman"'>2.2</span><span
lang=ZH-CN style='font-size:10.5pt'>为原图，图</span><span style='font-size:10.5pt;
font-family:"Times New Roman"'>2.3</span><span lang=ZH-CN style='font-size:
10.5pt'>为移动后的图。这种处理，文件大小不会改变。</span><span style='font-size:10.5pt;font-family:
"Times New Roman"'><o:p></o:p></span></p>
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<p align=center style='margin:0cm;margin-bottom:.0001pt;text-align:center;
line-height:18.0pt'><b><span lang=ZH-CN style='font-size:10.5pt'>图</span></b><b><span
style='font-size:10.5pt;font-family:"Times New Roman"'>2.2
</span></b><b><span lang=ZH-CN style='font-size:10.5pt'>移动前的图</span></b><span
style='font-size:10.5pt;font-family:"Times New Roman"'><o:p></o:p></span></p>
<p align=center style='margin:0cm;margin-bottom:.0001pt;text-align:center;
line-height:18.0pt'><b><span style='font-size:10.5pt;font-family:"Times New Roman"'><!--[if gte vml 1]><v:shape
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style='font-size:10.5pt;font-family:"Times New Roman"'><o:p></o:p></span></p>
<p align=center style='margin:0cm;margin-bottom:.0001pt;text-align:center;
line-height:18.0pt'><b><span lang=ZH-CN style='font-size:10.5pt'>图</span></b><b><span
style='font-size:10.5pt;font-family:"Times New Roman"'>2.3
</span></b><b><span lang=ZH-CN style='font-size:10.5pt'>移动后的图</span></b><span
style='font-size:10.5pt;font-family:"Times New Roman"'><o:p></o:p></span></p>
<p style='margin:0cm;margin-bottom:.0001pt;text-align:justify;text-justify:
inter-ideograph;line-height:18.0pt'><span lang=ZH-CN style='font-size:10.5pt'>还有一种做法是：将图象放大，使得能够显示下所有部分，如图</span><span
style='font-size:10.5pt;font-family:"Times New Roman"'>2.4</span><span
lang=ZH-CN style='font-size:10.5pt'>所示。</span><span style='font-size:10.5pt;
font-family:"Times New Roman"'><o:p></o:p></span></p>
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<p class=a style='margin:0cm;margin-bottom:.0001pt;line-height:18.0pt'><b><span
lang=ZH-CN style='font-family:宋体'>图</span>2.4 </b><b><span
lang=ZH-CN style='font-family:宋体'>移动后图象被放大</span></b></p>
<p style='margin:0cm;margin-bottom:.0001pt;text-align:justify;text-justify:
inter-ideograph;line-height:18.0pt'><span lang=ZH-CN style='font-size:10.5pt'>这种处理，文件大小要改变。设原图的宽和高分别是</span><span
style='font-size:10.5pt;font-family:"Times New Roman"'>w<sub>1</sub>,h<sub>1</sub></span><span
lang=ZH-CN style='font-size:10.5pt'>则新图的宽和高变为</span><span style='font-size:
10.5pt;font-family:"Times New Roman"'>w<sub>1</sub>+|t<sub>x</sub>|</span><span
lang=ZH-CN style='font-size:10.5pt'>和</span><span style='font-size:10.5pt;
font-family:"Times New Roman"'>h<sub>1</sub>+|t<sub>y</sub>|</span><span
lang=ZH-CN style='font-size:10.5pt'>，加绝对值符号是因为</span><span style='font-size:
10.5pt;font-family:"Times New Roman"'>t<sub>x</sub>, t<sub>y</sub></span><span
lang=ZH-CN style='font-size:10.5pt'>有可能为负</span><span style='font-size:10.5pt;
font-family:"Times New Roman"'>(</span><span lang=ZH-CN style='font-size:10.5pt'>即向左，向上移动</span><span
style='font-size:10.5pt;font-family:"Times New Roman"'>)</span><span
lang=ZH-CN style='font-size:10.5pt'>。</span><span style='font-size:10.5pt;
font-family:"Times New Roman"'><o:p></o:p></span></p>
<p style='margin:0cm;margin-bottom:.0001pt;text-align:justify;text-justify:
inter-ideograph;line-height:18.0pt'><span lang=ZH-CN style='font-size:10.5pt'>下面的函数</span><span
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