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<p align=center style='text-align:center;line-height:18.0pt'><b><span
style='font-family:宋体;"Times New Roman"'>图</span>2.8
</b><b><span style='font-family:
宋体;"Times New Roman"'>旋转后的图</span><span
lang=EN-US></span></b></p>
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<p style='line-height:18.0pt'><span
style='font-family:宋体;"Times New Roman"'>可以看出，旋转后图象变大了。另一种做法是不让图象变大，转出的部分被裁剪掉。如图</span><span
lang=EN-US>2.9</span><span style='font-family:宋体;
"Times New Roman"'>所示。</span></p>
<p style='line-height:18.0pt'><span
style='font-family:宋体;"Times New Roman"'>我们采用第一种做法，首先给出变换矩阵。在我们熟悉的坐标系中，将一个点顺时针旋转</span><span
lang=EN-US>a</span><span style='font-family:宋体;
"Times New Roman"'>角后的坐标变换公式，如图</span><span lang=EN-US>2.10</span><span
style='font-family:宋体;"Times New Roman"'>所示，</span><span lang=EN-US>r</span><span style='font-family:
宋体;"Times New Roman"'>为该点到原点的距离，在旋转过程中，</span><span
lang=EN-US>r</span><span style='font-family:宋体;
"Times New Roman"'>保持不变；</span><span lang=EN-US>b</span><span
style='font-family:宋体;"Times New Roman"'>为</span><span lang=EN-US>r</span><span style='font-family:
宋体;"Times New Roman"'>与</span><span
lang=EN-US>x</span><span style='font-family:宋体;
"Times New Roman"'>轴之间的夹角。</span></p>
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<p class=a style='line-height:18.0pt'><span lang=EN-US> <img width=138 height=106
src="chap02.files/image025.jpg" v:shapes="_x0000_i1029"> </span></p>
<p align=center style='text-align:center;line-height:18.0pt'><b><span
style='font-family:宋体;"Times New Roman"'>图</span><span lang=EN-US>2.9 </span></b><b><span
style='font-family:宋体;"Times New Roman"'>旋转后保持原图大小，</span><span lang=EN-US></span></b></p>
<p align=center style='text-align:center;line-height:18.0pt'><b><span
style='font-family:宋体;"Times New Roman"'>转出的部分被裁掉</span></b></p>
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<p style='line-height:18.0pt'><span lang=EN-US> <img width=235 height=208
src="chap02.files/image027.jpg" v:shapes="_x0000_i1030"> </span></p>
<p align=center style='text-align:center;line-height:18.0pt'><b><span
style='font-family:宋体;"Times New Roman"'>图</span>2.10 </b><b><span style='font-family:宋体;"Times New Roman";"Times New Roman"'>旋转示意图</span><span
lang=EN-US></span></b></p>
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<p style='line-height:18.0pt'><span
style='font-family:宋体;"Times New Roman"'>旋转前：</span><span lang=EN-US>x<sub>0</sub>=rcosb</span><span
style='font-family:宋体;"Times New Roman"'>；</span><span lang=EN-US>y<sub>0</sub>=rsinb</span></p>
<p style='line-height:18.0pt'><span
style='font-family:宋体;"Times New Roman"'>旋转</span><span lang=EN-US>a</span><span style='font-family:
宋体;"Times New Roman"'>角度后：</span></p>
<p style='line-height:18.0pt'><span
lang=EN-US>x<sub>1</sub>=rcos(b-a)=rcosbcosa+rsinbsina=x<sub>0</sub>cosa+y<sub>0</sub>sina</span><span
style='font-family:宋体;"Times New Roman"'>；</span></p>
<p style='line-height:18.0pt'><span
lang=EN-US>y<sub>1</sub>=rsin(b-a)=rsinbcosa-rcosbsina=-x<sub>0</sub>sina+y<sub>0</sub>cosa</span><span
style='font-family:宋体;"Times New Roman"'>；</span></p>
<p style='line-height:18.0pt'><span
style='font-family:宋体;"Times New Roman"'>以矩阵的形式表示：</span></p>
<p align=center style='text-align:center;line-height:18.0pt'><span
lang=EN-US><sub> <img width=292 height=75
src="chap02.files/image029.gif" v:shapes="_x0000_i1031"> </sub> </span></p>
<p align=right style='text-align:right;line-height:18.0pt'><span
lang=EN-US>(2.5)</span></p>
<p style='line-height:18.0pt'><span
style='font-family:宋体;"Times New Roman"'>上面的公式中，坐标系</span><span lang=EN-US>xoy</span><span
style='font-family:宋体;"Times New Roman"'>是以图象的中心为原点，向右为</span><span lang=EN-US>x</span><span
style='font-family:宋体;"Times New Roman"'>轴正方向，向上为</span><span lang=EN-US>y</span><span
style='font-family:宋体;"Times New Roman"'>轴正方向。它和以图象左上角点为原点</span><span lang=EN-US>o’</span><span
style='font-family:宋体;"Times New Roman"'>，向右为</span><span lang=EN-US>x’</span><span style='font-family:
宋体;"Times New Roman"'>轴正方向，向下为</span><span
lang=EN-US>y’</span><span style='font-family:宋体;
"Times New Roman"'>轴正方向的坐标系</span><span lang=EN-US>x’o’y’</span><span
style='font-family:宋体;"Times New Roman"'>之间的转换关系如何呢？如图</span><span lang=EN-US>2.11</span><span
style='font-family:宋体;"Times New Roman"'>所示。</span></p>
<p class=a style='line-height:18.0pt'><span lang=EN-US> <img width=207 height=160
src="chap02.files/image031.jpg" v:shapes="_x0000_i1032"> </span></p>
<p align=center style='text-align:center;line-height:18.0pt'><b><span
style='font-family:宋体;"Times New Roman"'>图</span>2.11 </b><b><span style='font-family:宋体;
"Times New Roman"'>两种坐标系间的转换关系</span><span
lang=EN-US></span></b></p>
<p style='line-height:18.0pt'><span
style='font-family:宋体;"Times New Roman"'>设图象的宽为</span><span lang=EN-US>w</span><span
style='font-family:宋体;"Times New Roman"'>，高为</span><span lang=EN-US>h</span><span style='font-family:
宋体;"Times New Roman"'>，容易得到：</span></p>
<p align=center style='text-align:center;line-height:18.0pt'><span
lang=EN-US><sub> <img width=296 height=75
src="chap02.files/image033.gif" v:shapes="_x0000_i1033"> </sub> </span></p>
<p align=right style='text-align:right;line-height:18.0pt'><span
lang=EN-US>(2.6)</span></p>
<p style='line-height:18.0pt'><span
style='font-family:宋体;"Times New Roman"'>逆变换为：</span></p>
<p align=center style='text-align:center;line-height:18.0pt'><span
lang=EN-US><sub> <img width=284 height=75
src="chap02.files/image035.gif" v:shapes="_x0000_i1034"> </sub> </span></p>
<p align=right style='text-align:right;line-height:18.0pt'><span
lang=EN-US>(2.7)</span></p>
<p style='line-height:18.0pt'><span
style='font-family:宋体;"Times New Roman"'>有了上面的公式，我们可以把变换分成三步：</span></p>
<p style='line-height:18.0pt'><span
lang=EN-US>1.</span><span style='font-family:宋体;
"Times New Roman"'>将坐标系</span><span lang=EN-US>o’</span><span
style='font-family:宋体;"Times New Roman"'>变成</span><span lang=EN-US>o</span><span style='font-family:
宋体;"Times New Roman"'>；</span></p>
<p style='line-height:18.0pt'><span
lang=EN-US>2.</span><span style='font-family:宋体;
"Times New Roman"'>将该点顺时针旋转</span><span lang=EN-US>a</span><span
style='font-family:宋体;"Times New Roman"'>角；</span></p>
<p style='line-height:18.0pt'><span
lang=EN-US>3.</span><span style='font-family:宋体;
"Times New Roman"'>将坐标系</span><span lang=EN-US>o</span><span
style='font-family:宋体;"Times New Roman"'>变回</span><span lang=EN-US>o’</span><span style='font-family:
宋体;"Times New Roman"'>，这样，我们就得到了变换矩阵，是上面三个矩阵的级联。</span></p>
<p align=center style='text-align:center;line-height:18.0pt'><span
lang=EN-US><sub> <img width=529 height=125
src="chap02.files/image037.gif" v:shapes="_x0000_i1040"> </sub> </span></p>
<p align=right style='text-align:right;line-height:18.0pt'><span
lang=EN-US>(2.8)</span></p>
<p style='line-height:18.0pt'><span
style='font-family:宋体;"Times New Roman"'>要注意的是，因为新图变大，所以上面公式中出现了</span><span lang=EN-US>w<sub>old</sub></span><span
style='font-family:宋体;"Times New Roman"'>，</span><span lang=EN-US>h<sub>old</sub></span><span
style='font-family:宋体;"Times New Roman"'>，</span><span lang=EN-US>w<sub>new</sub></span><span
style='font-family:宋体;"Times New Roman"'>，</span><span lang=EN-US>h<sub>new</sub></span><span
style='font-family:宋体;"Times New Roman"'>，它们分别表示原图</span><span lang=EN-US>(old)</span><span
style='font-family:宋体;"Times New Roman"'>和新图</span><span lang=EN-US>(new)</span><span
style='font-family:宋体;"Times New Roman"'>的宽、高。我们从图</span><span lang=EN-US>2.8</span><span
style='font-family:宋体;"Times New Roman"'>中容易看出：</span><span lang=EN-US>w<sub>new</sub>=max(|x<sub>4</sub>-x<sub>1</sub>|,|x<sub>3</sub>-x<sub>2</sub>|)
</span><span style='font-family:宋体;
"Times New Roman"'>；</span><span lang=EN-US>h<sub>new</sub>=max(|y<sub>4</sub>-y<sub>1</sub>|,|y<sub>3</sub>-y<sub>2</sub>|)</span><span
style='font-family:宋体;"Times New Roman"'>。</span></p>
<p style='line-height:18.0pt'><span
lang=EN-US>(2.8)</span><span style='font-family:宋体;
"Times New Roman"'>的逆变换为</span></p>
<p align=center style='text-align:center;line-height:18.0pt'><span
lang=EN-US><sub> <img width=526 height=125
src="chap02.files/image039.gif" v:shapes="_x0000_i1035"> </sub> </span></p>
<p align=right style='text-align:right;line-height:18.0pt'><span
lang=EN-US>(2.9)</span></p>
<p style='line-height:18.0pt'><span
style='font-family:宋体;"Times New Roman"'>这样，对于新图中的每一点，我们就可以根据公式</span><span lang=EN-US>(2.9)</span><span
style='font-family:宋体;"Times New Roman"'>求出对应原图中的点，得到它的灰度。如果超出原图范围，则填成白色。要注意的是，由于有浮点运算，计算出来点的坐标可能不是整数，采用取整处理，即找最接近的点，这样会带来一些误差</span><span
lang=EN-US>(</span><span style='font-family:宋体;
"Times New Roman"'>图象可能会出现锯齿</span><span lang=EN-US>)</span><span
style='font-family:宋体;"Times New Roman"'>。更精确的方法是采用插值，将在图象缩放时介绍。</span></p>
<p style='line-height:18.0pt'><span
style='font-family:宋体;"Times New Roman"'>源程序如下：</span></p>
<p style='line-height:18.0pt'><span lang=EN-US>#define PI 3.1415926535</span></p>
<p style='line-height:18.0pt'><span lang=EN-US>#define RADIAN(angle) ((angle)*PI/180.0)
//</span><span style='font-family:宋体;
"Times New Roman"'>角度到弧度转化的宏</span></p>
<p style='line-height:18.0pt'><span lang=EN-US>BOOL Rotation(HWND hWnd)</span></p>
<p style='line-height:18.0pt'><span lang=EN-US>{</span></p>
<p style='line-height:18.0pt'>DLGPROC
dlgInputBox = NULL;</p>
<p style='line-height:18.0pt'>DWORD
OffBits,SrcBufSize,DstBufSize,DstLineBytes;</p>
<p style='line-height:18.0pt'>LPBITMAPINFOHEADER lpImgData;</p>
<p style='line-height:18.0pt'>LPSTR
lpPtr;</p>
<p style='line-height:18.0pt'>HLOCAL
hTempImgData;</p>
<p style='line-height:18.0pt'>LPBITMAPINFOHEADER lpTempImgData;</p>
<p style='line-height:18.0pt'>LPSTR
lpTempPtr;</p>
<p style='line-height:18.0pt'>float
SrcX1,SrcY1,SrcX2,SrcY2;</p>
<p style='line-height:18.0pt'>float
SrcX3,SrcY3,SrcX4,SrcY4;</p>
<p style='line-height:18.0pt'>float
DstX1,DstY1,DstX2,DstY2;</p>
<p style='line-height:18.0pt'>float
DstX3,DstY3,DstX4,DstY4;</p>
<p style='line-height:18.0pt'>DWORD
Wold,Hold,Wnew,Hnew;</p>
<p style='line-height:18.0pt'>HDC &nb⌨️ 快捷键说明
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