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<h1 class="post-title p-name">3D基本变换和观测变换(viewing transform)</h1>
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<h2 id="3d基本变换"><a href="#3d基本变换" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>3D基本变换</h2>
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<h3 id="平移矩阵"><a href="#平移矩阵" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>平移矩阵</h3>
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<p>通常处理三维中对模型进行平移的行为</p>
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<p><br>$T(t_x,t_y,t_z) = \begin{pmatrix} 1 &0 &0 &t_x \\ 0 &1 &0 &t_y \\ 0 &0 &0 &t_z \\ 0 &0 &0 &1 \end{pmatrix} $<br></p>
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<p>$t_x,t_y,t_z$通常表示对应轴上平移的距离</p>
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<hr>
|
||
<h3 id="缩放矩阵"><a href="#缩放矩阵" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>缩放矩阵</h3>
|
||
<p>通常处理三维中对模型进行缩放的行为</p>
|
||
<p>$S(s_x,s_y,s_z) = \begin{pmatrix} s_x &0 &0 &0 \\0 &s_y &0 &0 \\ 0 &0 &s_z &0 \\ 0 &0 &0 &1 \end{pmatrix} $</p>
|
||
<p>$s_x,s_y,s_z$通常表示对应xyz轴的缩放比例</p>
|
||
<hr>
|
||
<h3 id="旋转矩阵"><a href="#旋转矩阵" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>旋转矩阵</h3>
|
||
<p>这个矩阵通常处理三维中对模型绕坐标轴进行旋转的行为</p>
|
||
<hr>
|
||
<p>绕x轴旋转的矩阵为
|
||
$\begin{pmatrix} 1 &0 &0 &0 \\ 0 &cos(r) &-sin(r) &0 \\ 0 &sin(r) &cos(r) &0 \\ 0 &0 &0 &1 \end{pmatrix} $</p>
|
||
<hr>
|
||
<p>绕y轴旋转的矩阵为
|
||
$\begin{pmatrix} cos(r) &0 &sin(r) &0 \\ 0 &1 &0 &0 \\ -sin(r) &0 &cos(r) &0 \\ 0 &0 &0 &1 \end{pmatrix} $</p>
|
||
<hr>
|
||
<p>绕z轴旋转的矩阵为
|
||
$\begin{pmatrix} cos(r) &-sin(r) &0 &0 \\ sin(r) &cos(r) &0 &0 \\ 0 &0 &1 &0 \\ 0 &0 &0 &1 \end{pmatrix} $</p>
|
||
<hr>
|
||
<p>对于给定三个旋转角度的旋转，通常使用欧拉角</p>
|
||
<p>$R_{xyz}(\alpha,\beta,\gamma)=R_x(\alpha)R_y(\beta)R_z(\gamma)$</p>
|
||
<p>此时的三个旋转方向将被称为roll,pich,yaw</p>
|
||
<p><img src="../../images/flight_euler_angle.png" alt="flight_euler"></p>
|
||
<hr>
|
||
<p>对于围绕某一特定点进行旋转的行为，则将该点平移至原点处后视为绕特定轴旋转
|
||
<img src="../../images/rotate_around_point.png" alt=""></p>
|
||
<hr>
|
||
<h3 id="视角变换矩阵"><a href="#视角变换矩阵" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>视角变换矩阵</h3>
|
||
<p>这个矩阵通常用来定义相机对应的视角朝向，利用这个矩阵来将相机位置移动到原点，便于后续的模型进行平移旋转等变换</p>
|
||
<div class="highlight"><div class="chroma">
|
||
<div class="table-container"><table class="lntable"><tr><td class="lntd">
|
||
<pre tabindex="0" class="chroma"><code><span class="lnt">1
|
||
</span><span class="lnt">2
|
||
</span><span class="lnt">3
|
||
</span><span class="lnt">4
|
||
</span><span class="lnt">5
|
||
</span><span class="lnt">6
|
||
</span><span class="lnt">7
|
||
</span><span class="lnt">8
|
||
</span><span class="lnt">9
|
||
</span></code></pre></td>
|
||
<td class="lntd">
|
||
<pre tabindex="0" class="chroma"><code class="language-c++" data-lang="c++"><span class="line"><span class="cl"><span class="n">Eigen</span><span class="o">::</span><span class="n">Matrix4f</span> <span class="n">view</span> <span class="o">=</span> <span class="n">Eigen</span><span class="o">::</span><span class="n">Matrix4f</span><span class="o">::</span><span class="n">Identity</span><span class="p">();</span>
|
||
</span></span><span class="line"><span class="cl">
|
||
</span></span><span class="line"><span class="cl"><span class="n">Eigen</span><span class="o">::</span><span class="n">Matrix4f</span> <span class="n">translate</span><span class="p">;</span>
|
||
</span></span><span class="line"><span class="cl"><span class="n">translate</span> <span class="o"><<</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="n">eye_pos</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="n">eye_pos</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="n">eye_pos</span><span class="p">[</span><span class="mi">2</span><span class="p">],</span>
|
||
</span></span><span class="line"><span class="cl"> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">;</span>
|
||
</span></span><span class="line"><span class="cl">
|
||
</span></span><span class="line"><span class="cl"><span class="n">view</span> <span class="o">=</span> <span class="n">translate</span> <span class="o">*</span> <span class="n">view</span><span class="p">;</span><span class="c1">//移动相机位置到顶点
|
||
</span></span></span></code></pre></td></tr></table></div>
|
||
</div>
|
||
</div><p>在上述代码中，eye_pos(x,y,z,1)往往为相机的位置</p>
|
||
<hr>
|
||
<h3 id="正交投影矩阵"><a href="#正交投影矩阵" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>正交/投影矩阵</h3>
|
||
<h4 id="正交矩阵"><a href="#正交矩阵" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>正交矩阵</h4>
|
||
<p>正交矩阵将使摄像头置于坐标系原点，看向-Z轴方向，可以在Y轴上平行移动。
|
||
最终结果将表现为在XY轴平面上的2D图像，让模型坐标归一化到[-1,1]之间</p>
|
||
<hr>
|
||
<p>总体流程</p>
|
||
<p>为了将一个$[l,r]\times[b,t]\times[f,n]$的长方体转换为符合canonical(正则、规范、标准)的正方体，我们需要进行两步操作</p>
|
||
<p>第一步</p>
|
||
<p>平移这个长方体到坐标系的原点</p>
|
||
<p>对应矩阵$M_{translate}=\begin{bmatrix} \frac{2}{r-l} &0 &0 &0 \\ 0 &\frac{2}{t-b} &0 &0 \\ 0 &0 &0 &\frac{2}{n-f} \\ 0 &0 &0 &1 \end{bmatrix}$</p>
|
||
<p>第二步</p>
|
||
<p>缩放这个长方体到符合正则、规范、标准的正方体</p>
|
||
<p>对应矩阵$M_{scale}=\begin{bmatrix} 1 &0 &0 &-\frac{r+l}{2} \\ 0 &1 &0 &-\frac{t+b}{2} \\ 0 &0 &1 &-\frac{n+f}{2} \\ 0 &0 &0 &1 \end{bmatrix}$</p>
|
||
<p>由第一二步可得出</p>
|
||
<p>正交矩阵为$M_{ortho}=\begin{bmatrix} \frac{2}{r-l} &0 &0 &0 \\ 0 &\frac{2}{t-b} &0 &0 \\ 0 &0 &0 &\frac{2}{n-f} \\ 0 &0 &0 &1 \end{bmatrix}\times\begin{bmatrix} 1 &0 &0 &-\frac{r+l}{2} \\ 0 &1 &0 &-\frac{t+b}{2} \\ 0 &0 &1 &-\frac{n+f}{2} \\ 0 &0 &0 &1 \end{bmatrix}$</p>
|
||
<p>图示</p>
|
||
<p><img src="../../images/3I~K5PRW$Y5JLF3%7DRC@RE0P.png" alt="正交矩阵"></p>
|
||
<hr>
|
||
<h4 id="投影矩阵"><a href="#投影矩阵" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>投影矩阵</h4>
|
||
<p>投影矩阵将使模型满足自然界透视效果，如物体近大远小、所有的平行线变得不再平行，总会交于一点</p>
|
||
<hr>
|
||
<p>推导过程</p>
|
||
<p>将远平面与近平面连线形成的梯形“挤压”到成为一个正方体</p>
|
||
<p><img src="../../images/X(%7BM%7BB)24R8DF%5BAB98E9%5D@1.png" alt=""></p>
|
||
<p>挤压的过程对梯形做横切面可知，计算挤压后的坐标点值实质上为计算相似三角形</p>
|
||
<p>如下图，可知挤压后的坐标与之前的坐标存在的数学关系为
|
||
<img src="../../images/3O_1NAGAMER%25%5BEP6T91$LBO.png" alt=""></p>
|
||
<p>$$y^{'}=\frac{n}{z}y$$ $$x^{'}=\frac{n}{z}x$$</p>
|
||
<p>由此可知经过“挤压”后的坐标为
|
||
$$M_{persp->ortho}\times\begin{pmatrix}
|
||
x \\ y \\ z \\ 1
|
||
\end{pmatrix}=\begin{pmatrix}
|
||
nx \\ ny \\ {未知} \\ z
|
||
\end{pmatrix}$$</p>
|
||
<p>所以
|
||
$$M_{persp->ortho}=\begin{pmatrix}
|
||
n &0 &0 &0 \\ 0 &n &0 &0 \\ ? &? &? &? \\ 0 &0 &1 &0
|
||
\end{pmatrix}$$</p>
|
||
<p>又由远平面在被“挤压”后相当于近平面做正交投影得到的远平面，可知</p>
|
||
<ol>
|
||
<li>任何在近平面上的点的坐标在“挤压”的过程中不发生改变</li>
|
||
<li>任何在远平面的点的坐标中的Z值不发生改变,即$$\begin{pmatrix}
|
||
0 \\ 0 \\ f \\ 1
|
||
\end{pmatrix}\rArr\begin{pmatrix}
|
||
0 \\ 0 \\ f \\ 1
|
||
\end{pmatrix}==\begin{pmatrix}
|
||
0 \\ 0 \\ f^2 \\ f
|
||
\end{pmatrix}$$</li>
|
||
</ol>
|
||
<p>在将上述坐标公式中Z的值以n替换之后可得
|
||
$$\begin{pmatrix}
|
||
x \\ y \\ n \\ 1
|
||
\end{pmatrix}\rArr\begin{pmatrix}
|
||
x \\ y \\ n \\ 1
|
||
\end{pmatrix}==\begin{pmatrix}
|
||
nx \\ ny \\ n^2 \\ n
|
||
\end{pmatrix}$$
|
||
据此，可推测第三行未知坐标值符合以下关系
|
||
$$\begin{pmatrix}
|
||
0 &0 &A &B
|
||
\end{pmatrix}\begin{pmatrix}
|
||
x \\ y \\ n \\ 1
|
||
\end{pmatrix}=n^2$$
|
||
因此
|
||
$$An+B=n^2$$
|
||
联立性质2推导的
|
||
$$Af+B=f^2$$
|
||
可得出
|
||
$$A=n+f$$
|
||
$$B=-nf$$</p>
|
||
<p>所以
|
||
$$M_{persp}=M_{ortho}M_{persp->ortho}=$$
|
||
$$\begin{pmatrix} \frac{2}{r-l} &0 &0 &0 \\ 0 &\frac{2}{t-b} &0 &0 \\ 0 &0 &0 &\frac{2}{n-f} \\ 0 &0 &0 &1 \end{pmatrix}$$
|
||
$$\times$$
|
||
$$\begin{pmatrix} 1 &0 &0 &-\frac{r+l}{2} \\ 0 &1 &0 &-\frac{t+b}{2} \\ 0 &0 &1 &-\frac{n+f}{2} \\ 0 &0 &0 &1 \end{pmatrix}$$
|
||
$$\times$$
|
||
$$\begin{pmatrix} n &0 &0 &0 \\ 0 &n &0 &0 \\ 0 &0 &n+f &-nf \\ 0 &0 &1 &0 \end{pmatrix}$$</p>
|
||
<hr>
|
||
<h4 id="涉及fovy的投影矩阵"><a href="#涉及fovy的投影矩阵" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>涉及FovY的投影矩阵</h4>
|
||
<p>FovY表示视域，即摄像机在固定时能看到的最大角度或最低角度的范围</p>
|
||
<p>Aspect ratio 表示纵横比，投影平面的长宽比
|
||
<img src="../../images/fovY.png" alt=""></p>
|
||
<p>对应的相似三角形关系不变，参数改变
|
||
<img src="../../images/triangle.png" alt="">
|
||
可得如下关系
|
||
$$\tan{\frac{fovY}{2}}=\frac{t}{|n|}$$
|
||
$$aspect=\frac{r}{t}$$
|
||
因此
|
||
$$t=near\times tan(\frac{fovY}{2})$$</p>
|
||
<p>$$r=aspect\times near\times tan(\frac{fovY}{2})$$
|
||
$$l=-aspect\times near \times tan(fovY/2)$$
|
||
带入上述由l,b,n,f构成的矩阵可得
|
||
$$M_{persp->ortho}$$
|
||
$$=$$
|
||
$$\begin{pmatrix} \frac{2}{r-l} &0 &0 &0 \\ 0 &\frac{2}{t-b} &0 &0 \\ 0 &0 &\frac{2}{n-f} &0 \\ 0 &0 &0 &1 \end{pmatrix}$$
|
||
$$\times$$
|
||
$$\begin{pmatrix} 1 &0 &0 &-\frac{r+l}{2} \\ 0 &1 &0 &-\frac{t+b}{2} \\ 0 &0 &1 &-\frac{n+f}{2} \\ 0 &0 &0 &1 \end{pmatrix}$$
|
||
$$=$$</p>
|
||
<p>$$\begin{pmatrix}
|
||
\frac{\frac{\cot{FovY}}{2}}{apsect*near} &0 &0 &0 \\ 0 & \frac{\frac{\cot{FovY}}{2}}{near} & 0 & 0 \\ 0 & 0 & \frac{2}{near-far} & 0 \\ 0 & 0 & 0 & 1
|
||
\end{pmatrix}$$</p>
|
||
<p>$$\times$$</p>
|
||
<p>$$\begin{pmatrix}
|
||
1 &0 &0 &0 \\ 0 &1 &0 &0 \\ 0 &0 &1 &-\frac{near+far}{2} \\ 0 &0 &0 &1
|
||
\end{pmatrix}$$</p>
|
||
<p>$$=$$</p>
|
||
<p>$$\begin{pmatrix}
|
||
\frac{\frac{\cot{FovY}}{2}}{apsect * near} &0 &0 &0 \\ 0 &\frac{\frac{\cot{FovY}}{2}}{near} &0 &0 \\ 0 &0 &\frac{2}{near-far} &-\frac{near+far}{near-far} \\ 0 &0 &0 &1
|
||
\end{pmatrix}$$</p>
|
||
<hr>
|
||
<h4 id="视口变换"><a href="#视口变换" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>视口变换</h4>
|
||
<p>经过MVP矩阵计算后得到的一个正则的正方体需要将X轴和Y轴上的坐标映射到屏幕坐标[0,width]$\times$[0,height]</p>
|
||
<p>变换时需要先将[-1,1]缩放到屏幕大小[width,height]，再进行平移使得原点坐标与屏幕原点对齐</p>
|
||
<p>变换矩阵
|
||
$$M_{viewport}=\begin{bmatrix}
|
||
\frac{width}{2} &0 &0 &\frac{width}{2} \\ 0 &\frac{height}{2} &0 &\frac{height}{2} \\ 0 &0 &1 &0 \\ 0 &0 &0 &1
|
||
\end{bmatrix}$$</p>
|
||
|
||
</div>
|
||
|
||
|
||
|
||
|
||
</article>
|
||
|
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|
||
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<div class="updated-badge-container">
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<span title="Updated @ 2023-06-18 21:19:01 CST" style="cursor:help">
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<svg xmlns="http://www.w3.org/2000/svg" width="130" height="20" class="updated-badge"><linearGradient id="b" x2="0" y2="100%"><stop offset="0" stop-color="#bbb" stop-opacity=".1"/><stop offset="1" stop-opacity=".1"/></linearGradient><clipPath id="a"><rect width="130" height="20" rx="3" fill="#fff"/></clipPath><g clip-path="url(#a)"><path class="updated-badge-left" d="M0 0h55v20H0z"/><path class="updated-badge-right" d="M55 0h75v20H55z"/><path fill="url(#b)" d="M0 0h130v20H0z"/></g><g fill="#fff" text-anchor="middle" font-size="110"><text x="285" y="150" fill="#010101" fill-opacity=".3" textLength="450" transform="scale(.1)">updated</text><text x="285" y="140" textLength="450" transform="scale(.1)">updated</text><text x="915" y="150" fill="#010101" fill-opacity=".3" textLength="650" transform="scale(.1)">2023-06-18</text><text x="915" y="140" textLength="650" transform="scale(.1)">2023-06-18</text></g></svg>
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navigator.serviceWorker.register('\/sw.js');
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<link rel="stylesheet" href="https://fastly.jsdelivr.net/npm/katex@0.13.0/dist/katex.min.css" integrity="sha256-gPJfuwTULrEAAcI3X4bALVU/2qBU+QY/TpoD3GO+Exw=" crossorigin="anonymous">
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<script>
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if (typeof renderMathInElement === 'undefined') {
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var getScript = (options) => {
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var script = document.createElement('script');
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script.defer = true;
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script.crossOrigin = 'anonymous';
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Object.keys(options).forEach((key) => {
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script[key] = options[key];
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});
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document.body.appendChild(script);
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};
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getScript({
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src: 'https:\/\/fastly.jsdelivr.net/npm/katex@0.13.0/dist/katex.min.js',
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integrity: 'sha256-YTW9cMncW/ZQMhY69KaUxIa2cPTxV87Uh627Gf5ODUw=',
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onload: () => {
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getScript({
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src: 'https:\/\/fastly.jsdelivr.net/npm/katex@0.13.0/dist/contrib/mhchem.min.js',
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integrity: 'sha256-yzSfYeVsWJ1x+2g8CYHsB/Mn7PcSp8122k5BM4T3Vxw=',
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onload: () => {
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getScript({
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src: 'https:\/\/fastly.jsdelivr.net/npm/katex@0.13.0/dist/contrib/auto-render.min.js',
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integrity: 'sha256-fxJzNV6hpc8tgW8tF0zVobKa71eTCRGTgxFXt1ZpJNM=',
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renderKaTex();
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renderKaTex();
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function renderKaTex() {
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renderMathInElement(
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document.body,
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{left: "$", right: "$", display: false},
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{left: "\\(", right: "\\)", display: false}
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);
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<script src="https://fastly.jsdelivr.net/npm/medium-zoom@latest/dist/medium-zoom.min.js"></script>
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<script>
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let imgNodes = document.querySelectorAll('div.post-body img');
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imgNodes = Array.from(imgNodes).filter(node => node.parentNode.tagName !== "A");
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mediumZoom(imgNodes, {
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background: 'hsla(var(--color-bg-h), var(--color-bg-s), var(--color-bg-l), 0.95)'
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<script src="https://fastly.jsdelivr.net/npm/instant.page@5.1.0/instantpage.min.js" type="module" defer></script>
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</body>
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</html>
|