计算投影

设有两点坐标分别为$S = (S_x,S_y,S_z)$ , $T =(T_x,T_y,T_z)$ ,

平面 $\Pi : Ax+ By + Cz + D = 0$ .$L_{st}$ 与平面相交于$P$

设由$S,T$连接形成的直线$L_{ST} = \left\{\begin{matrix} x =\lambda t + S_x \\ y = \mu t + S_y \\ z = \nu t + S_z\end{matrix}\right.$ 其中 $\left\{\begin{matrix} \lambda = T_x - S_x\\ \mu = T_y - S_y \\ \nu = T_z - s_z\end{matrix}\right.$

要求出$L_{ST}$ 与平面$\Pi$ 交点$P$

将参数方程带入求解出参数$t$

$A(\lambda t + S_x) + B(\mu t + S_y) + C(\nu t + S_z) + D = 0$

$t= - \frac{AS_x + BS_y + CS_z + D}{A\lambda+ B\mu + C\nu}$

则由参数方程可得$V_{SP} = \begin{bmatrix} \lambda t \\ \mu t \\ \nu t\end{bmatrix}$

然后计算分别沿$x$ 轴旋转$\alpha$ 角和$z$ 轴旋转$\beta$ 角的复合矩阵

记$(sin\theta = S_\theta ,cos\theta = C_\theta)$

$R_z = \begin{bmatrix}C_\alpha&-S_\alpha& 0\\ S_\alpha& C_\alpha & 0\\ 0& 0 & 1\end{bmatrix}$ $R_x =\begin{bmatrix}1 & 0 & 0\\ 0 & C_\beta & -S_\beta \\0 & S_\beta &C_\beta\end{bmatrix}$

复合变换$R = R_xR_z = \begin{bmatrix}C_\alpha&-S_\alpha& 0\\ S_\alpha& C_\alpha & 0\\ 0& 0 & 1\end{bmatrix} \begin{bmatrix}1 & 0 & 0\\ 0 & C_\beta & -S_\beta \\0 & S_\beta &C_\beta\end{bmatrix}$

$R = \begin{bmatrix} C_\alpha & -S_\alpha C_\beta & S_\alpha S_\beta \\ S_\alpha & C_\alpha C_\beta & -C_\alpha S_\beta \\ 0 & S_\beta & C_\beta \end{bmatrix}$

由图解可知，$y$ 轴于平面$\Pi$法向量平行$w = \begin{bmatrix}A\\ B\\C\end{bmatrix} = \begin{bmatrix}-S_\alpha C_\beta\\ C_\alpha C_\beta\\S_\beta\end{bmatrix}$

设平面$\Pi$与$S$ 点距离为$d$

$d = \frac{|AS_x + BS_y + CS_z + D|}{\sqrt{A^2 + B^2 + C^2}} = \frac{A\lambda t + B\mu t + C \nu t}{\sqrt{A^2 + B^2 + C^2}} = A\lambda t + B\mu t + C \nu t = -S_\alpha C_\beta \lambda t + C_\alpha C_\beta\mu t + S_\beta\nu t$

$R$ 为正交矩阵$R^{-1}= R^T$

在以平面法向量方向为$y$ 轴方向，$x$轴平行于地面平面的坐标系下，设$U_{SP}$ 为逆旋转之后的向量坐标

$U_{SP} = R^{-1}V_{SP}$

$U_{SP} = \begin{bmatrix} C_\alpha & S_\alpha & 0 \\-S_\alpha C_\beta & C_\alpha C_\beta & S_\beta\\ S_\alpha S_\beta & -C_\alpha S_\beta & C_\beta \end{bmatrix} \begin{bmatrix} \lambda t \\ \mu t \\ \nu t\end{bmatrix}$

$U_{SP} = \begin{bmatrix}\lambda C_\alpha t + \mu S_\beta t\\-S_\alpha C_\beta \lambda t + C_\alpha C_\beta\mu t + S_\beta\nu t \\ S_\alpha S_\beta\lambda t -C_\alpha S_\beta \mu t + C_\beta \nu t\end{bmatrix}$

带入参数方程可得

$U_{SP} = \begin{bmatrix}\lambda C_\alpha t + \mu S_\beta t\\ d \\ S_\alpha S_\beta\lambda t -C_\alpha S_\beta \mu t + C_\beta \nu t\end{bmatrix}$

$U_{SP}$ 的$x,z$ 值就是屏幕上的坐标，以屏幕左下角为原点，$d$ 为平面到$S$点的距离