Computing the transformation of a plane $\eta_1$ to another plane $\eta_2$

Question

Suppose we have two planes given by the parametric equations

$$\begin{align} &\eta_1~:~\vec{x}_1~=~\vec{o}_1+\vec{R}_{11}t_{11}+\vec{R}_{12}t_{12}\\ &\eta_2~:~\vec{x}_2~=~\vec{o}_2+\vec{R}_{21}t_{21}+\vec{R}_{22}t_{22} \end{align}$$

where all occuring vectors are elements of $\mathbb{R}^3$. We can also describe these planes by the following equations

$$\begin{align} &\eta_1~:~(\vec{x}_1-\vec{o}_1)\cdot(\vec{R}_{11}\times\vec{R}_{12})\\ &\eta_2~:~(\vec{x}_2-\vec{o}_2)\cdot(\vec{R}_{21}\times\vec{R}_{22}) \end{align}$$

The main target I attempt to fulfill is to transform $\eta_1$ to $\eta_2$. I want to this in two steps $(1)$ Making both planes parallel $(2)$ Shift $\eta_1$ to be equal to $\eta_2$.

I tried to approach by using the normal vectors of the both planes and so to solve the equation $\textbf{T}\vec{n}_1=\vec{n}_2$ for $\textbf{T}\in \mathbb{R}^{3\times3}$. To be honest I have to clue from hereon and I guess this is not even the right attempt. What I can say about the matrix-vector equation is that the eigenvalues, or atleast one of the eigenvalues, of the matrix $\textbf{T}$ has to $1$ and therefore $\vec{n}_2$ would be an eigenvector of $\textbf{T}$. I am familiar with the concept of rotation matrices around the different axis for an angle $\alpha$ which are given by

$$\begin{align*} \textbf{T}_x&=\begin{pmatrix}1&0&0\\0&\cos(\alpha)&-\sin(\alpha)\\0&\sin(\alpha)&\cos(\alpha)\end{pmatrix}\\ \textbf{T}_y&=\begin{pmatrix}\cos(\alpha)&0&-\sin(\alpha)\\0&1&0\\\sin(\alpha)&0&\cos(\alpha)\end{pmatrix}\\ \textbf{T}_z&=\begin{pmatrix}\cos(\alpha)&-\sin(\alpha)&0\\\sin(\alpha)&\cos(\alpha)&0\\0&0&1\end{pmatrix} \end{align*}$$

but since I do not know the angle $\alpha$ this does not help me at all. Another attempt would be to compute all angles alone and then jsut multiply the matrices $\textbf{T}_x$, $\textbf{T}_y$ and $\textbf{T}_z$ but hence this does not seem that efficient I am not sure about this.

Mainly I want to know I there is a way to construct $\textbf{T}$ out of $\vec{n}_1$ and $\vec{n}_2$ or atleast out of the given defintion of $\eta_1$ and $\eta_2$. If it is not possible could you please explain to me why this is so. Furthermore could maybe someone provide an example transform of a plane $\eta_1$ to plane $\eta_2$ by using a general algorithm if there exist such as thing.

Thanks in advance!

Answer 1

You might be getting stuck because there’s in fact an infinite number of rotations that will align the planes. Imagine “tilting” one normal until it’s aligned with the other. You can still freely rotate about the resulting axis while maintaining parallelism. Looking at it from another point of view, every line through the origin that lies on the angle bisector of a pair of vectors can serve as the axis of a rotation that will align the two vectors.

If you don’t care about preserving shapes and areas—i.e., you don’t need a rigid motion—it’s very easy to construct a transformation that will perform this mapping. Working in homogeneous coordinates since a translation is involved, the matrix $$M_1 = \begin{bmatrix}\vec R_{11} & \vec R_{12} & \vec R_{11}\times\vec R_{12} & \vec o_1 \\ 0&0&0&1 \end{bmatrix}$$ maps the $x$-$y$ plane to $\eta_1$. Similarly, $$M_2 = \begin{bmatrix}\vec R_{21} & \vec R_{22} & \vec R_{21}\times\vec R_{22} & \vec o_2 \\ 0&0&0&1 \end{bmatrix}$$ maps to $\eta_2$. (For the third column, you can use any vector that’s not parallel to $\eta_1$ or $\eta_2$, respectively. The cross products are an easy way to generate them.) The matrix $M_2M_1^{-1}$ then maps from $\eta_1$ to $\eta_2$.

If you need a rigid motion, a slight modification of the above will do the trick. The first and third columns of the two matrices are the same, but normalized. For the second columns, use $(\vec R_{i1}\times\vec R_{i2})\times\vec R_{i1}$, also normalized, so that the linear parts of the resulting transformation matrices are orthogonal with determinant $1$, in other words, rotations.

The above combines the rotation and translation into a single transformation matrix, but you can of course proceed in steps as you proposed. You can construct the appropriate pure rotation exactly the same way as above: generate an orthonormal set of vectors for each plane consisting of a unit normal for the plane and a pair of orthogonal unit vectors parallel to it and use them as the columns of two $3\times3$ matrices $M_1$ and $M_2$. $M_2M_1^{-1}$ is a rotation that aligns the two planes. Since $A^{-1}=A^T$ for an orthogonal matrix, you don’t even need to perform a matrix inversion. The three vectors that you generated for $\eta_1$ are the rows of $M_1^{-1}$. The order in which they appear in the matrices doesn’t matter, either, as long as it’s consistent.