games101笔记 坐标变换

games101 坐标变换

3D 变换

• \(3D~point = (x, y, z, \textbf{1})^T\)
• \(3D~vector = (x, y, z, \textbf{0})^T\)

​ 一般地 \((x, y, z, w) (w != 0)\) 是一个3D的点：

​ \((x/w, y/w, z/w)\)

3D 旋转

• \(\textbf{Euler angles}\)

  • \(Roll\)：面

  • \(Yaw\)：颈

  • \(Pitch\)：耳

Rodrigues's Rotation Formula

• 通过 \(\alpha\) 角和 \(n\) 轴

\(\R(n, \alpha) = cos(\alpha)I + (1 - cos(\alpha))nn^T + sin(\alpha)\underbrace{\left[ \begin{matrix} 0 & -n_z & n_y\\ n_z & 0 & -n_x\\-n_y & n_x & 0 \end{matrix} \right]}_{\text{N}}\)

后面的矩阵其实也就是 叉乘 的矩阵的形式

\(I\) 是单位矩阵 \(Identity~Matrix\)

• 一些默认
  • 起点在原点上
  • 方向是 \(n\) 的方向

​ 如果要绕任意点旋转，可以先将该点平移到原点，旋转后再平移回去

Viewing/Camera transformation

• 定义相机的位置

  • \(Position~ \vec{e}\)
  • \(look~at/gaze~direction~ \widehat{g}\)
  • \(up~direction~\widehat{t}\)

• 一些约定俗成的事儿

  • up at \(Y\)
  • look at -\(Z\)
  • origin

• M_view in math

  \(M_{view} = R_{view}T_{view}\)

  • Translate \(\vec{e}\) to origin

    \(T_{view} = \left[\begin{matrix}1 & 0 & 0 & -x_e \\ 0 & 1 & 0 & -y_e\\0 & 0 & 1 & -z_e \\ 0 & 0 & 0 & 1\end{matrix}\right]\)

  • Rotates \(\widehat{g}\) to -\(Z\)，\(\widehat{t}\) to \(Y\)， \((\widehat{g} \times \widehat{t})\) to \(X\)

    \(R_{view}^{-1} = \left[\begin{matrix}x_{\widehat{g}\times\widehat{t}} & x_t & x_{-g} & 0\\y_{\widehat{g}\times\widehat{t}} & y_t & y_{-g} & 0\\z_{\widehat{g}\times\widehat{t}} & z_t & z_{-g} & 0\\0 & 0 & 0 & 1\end{matrix}\right]\) \(\Rightarrow\) \(R_{view} = \left[\begin{matrix}x_{\widehat{g}\times\widehat{t}} & y_{\widehat{g}\times\widehat{t}} & z_{\widehat{g}\times\widehat{t}} & 0\\x_t & y_t & z_t & 0 \\x_{-g} & y_{-g} & z_{-g} & 0\\0 & 0 & 0 & 1\end{matrix}\right]\)