# 1.4. 图形变换

## 1.4. 图形变换

#### 1.4.1. 平移变换

\begin{eqnarray*}
{T_n} = \left( {\begin{array}{*{20}{l}}1&0& \cdots &0&{{t_1}}\\0&1& \cdots &0&{{t_2}}\\ \vdots & \vdots & \ddots & \vdots & \vdots \\0&0& \cdots &1&{{t_n}}\\0&0& \cdots &0&1\end{array}} \right)
\tag{1.34}
\end{eqnarray*}

#### 1.4.2. 旋转变换

\begin{eqnarray*}
R(\theta ) = \left( {\begin{array}{*{20}{c}}{\cos \theta }&{ – \sin \theta }&0\\{\sin \theta }&{\cos \theta }&0\\0&0&1\end{array}} \right)
\tag{1.35}
\end{eqnarray*}

(1.37)

1. 把轴进行两次旋转变换，即，使得轴轴方向相同；
2. 绕着轴旋转度，即
3. 与第步相反的旋转变换，使得轴恢复到最初方向。

(1.38)

\begin{eqnarray*}
R(\theta ) = \left( {\begin{array}{*{20}{c}}{c + (1 – c)u_x^2}&{(1 – c){u_y}{u_x} – s{u_z}}&{(1 – c){u_z}{u_x} + s{u_y}}&0\\{(1 – c){u_x}{u_y} + s{u_z}}&{c + (1 – c)u_y^2}&{(1 – c){u_z}{u_y} – s{u_x}}&0\\{(1 – c){u_x}{u_z} – s{u_y}}&{(1 – c){u_y}{u_z} + s{u_x}}&{c + (1 – c)u_z^2}&0\\0&0&0&1\end{array}} \right)
\tag{1.39}
\end{eqnarray*}

(1.40)

\begin{eqnarray*}
\begin{array}{l}{u_x} = \frac{{{m_{32}} – {m_{23}}}}{{2\sin \theta }}\\{u_y} = \frac{{{m_{13}} – {m_{31}}}}{{2\sin \theta }}\\{u_z} = \frac{{{m_{21}} – {m_{12}}}}{{2\sin \theta }}\end{array}
\tag{1.41}
\end{eqnarray*}

1. ;
2.  ;

#### 1.4.3. 缩放变换

\begin{eqnarray*}
{S_n} = \left( {\begin{array}{*{20}{l}}{{s_1}}&0& \cdots &0&0\\0&{{s_2}}& \cdots &0&0\\ \vdots & \vdots & \ddots & \vdots & \vdots \\0&0& \cdots &{{s_n}}&0\\0&0& \cdots &0&1\end{array}} \right)
\tag{1.42}
\end{eqnarray*}

\begin{eqnarray*}
S_n^{ – 1} = \left( {\begin{array}{*{20}{l}}{1/{s_1}}&0& \cdots &0&0\\0&{1/{s_2}}& \cdots &0&0\\ \vdots & \vdots & \ddots & \vdots & \vdots \\0&0& \cdots &{1/{s_n}}&0\\0&0& \cdots &0&1\end{array}} \right)
\tag{1.43}
\end{eqnarray*}

#### One comment on “1.4. 图形变换”

1. openlib

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