Consider an $(x, y, z)$ system where positive $x$ points to the right, positive $y$ points upwards, and positive $z$ points outside of the screen.
I create a new system $(x', y', z')$ by applying two rotations:
- Counter-clockwise rotation of $(x, y, z)$ around the $z$ axis, an angle $\theta$. This generates an intermediate system $(x_i, y_i, z_i)$, where $z_i=z$.
- Clockwise rotation of the $(x_i, y_i, z_i)$ system around the $x_i$ axis, an angle $\phi$. This generates the final $(x', y', z')$ system.
I know both $\theta, \phi$ angles. I also know the transformation equations between both systems $(x, y, z)$ and $(x', y', z')$:
$$
x' = x\,cos\theta + y\,sin\theta \\
y' = -x\,sin\theta\,cos\phi + y\,cos\theta\,cos\phi – z\,sin\phi \\
z' = -x\,sin\theta\,sin\phi + y\,cos\theta\,sin\phi + z\,cos\phi
$$
What I need is the equation for the plane defined by $(x', y')$, in the form:
$$ax+by+cz+d=0$$
How can I calculate this equation?
Best Answer
The first rotation around the $z$ axis transform the versor $\hat x=(1,0,0)^T$ of the $x$ axis to $\hat u=(u_1,u_2,u_3)^T=(\cos \theta, \sin \theta,0)^T$.
If I well understand you want the equation of the plane $x,y$ rotated of an angle $\phi$ around the axis oriented by such versor.
You can find such equation rotating in the same way the versor $\hat z=(0,0,1)^T$, orthogonal to the plane $x,y$, so to obtain the versor $\hat v=R( \hat z)$:
Given: $\hat u=(u_1,u_2,u_3)^T$ the rotation matrix is:
$$ R_{\hat u,\phi}=\left ( \begin{array}{cccc} \cos 2\psi + 2u_1^2 \sin^2 \psi &-u_3\sin 2\psi+2u_1u_2 \sin^2 \psi & u_2\sin 2\psi+2u_1u_3 \sin^2 \psi \\ u_3 \sin 2\psi +2u_1u_2 \sin^2 \psi& \cos 2\psi+ 2u_2^2 \sin^2 \psi& -u_1 \sin 2 \psi + 2u_2u_3 \sin^2 \psi\\ -u_2 \sin 2 \psi + 2u_1u_3 \sin^2 \psi & u_1 \sin 2 \psi + 2u_2u_3 \sin^2 \psi& \cos 2\psi+ 2u_3^2 \sin^2 \psi \end {array} \right) $$ where $2\psi=-\phi$ (I prefer this form of the matrix that is derived by representing rotation with quaternions, but it is the same as you can see here, and there is a minus sign because the angle $\phi$ is clockwise)
So, find $\hat v=R_{\hat u,\phi} \hat z$ and the equation of the rotated plane is: $$ \hat v \cdot \vec x=0 $$ with $\vec x=(x,y,z)^T$. Note that the plane passes thorough the origin that is the common fixed point of the two rotations.