How to map point from one csys (coordinate systems) to another csys

Question

I have two $3$-dimensional coordinate system csys1, csys2. I have a known point in csys1 called $p_1(x,y,z)$. I want to define the same point with respect to csys2.(i.e. find a $p_1(x,y,z)$ with respect to csys2) This csys2 has some offset and angled with respect to csys1.i need common solution approach for angled as well as offset. Kindly anyone helps me out.

Answer 1

I would suggest defining operations as $4\times4$ matrices on homogeneous coordinates:

$$\begin{pmatrix}x'\\y'\\z'\\1\end{pmatrix}= \begin{pmatrix}a&b&c&d\\e&f&g&h\\i&j&k&l\\0&0&0&1\end{pmatrix} \cdot\begin{pmatrix}x\\y\\z\\1\end{pmatrix}$$

This is the general shape of an affine transformation. If your transformation is a rigid motion, the matrix would satisfy additional requirements. But the nice thing about this representation is that you can combine elementary operations. So if you have a translation by $(s,t,u)$ followed by a rotation by angle $\varphi$ around the $x$ axis. You can write this as

$$ \begin{pmatrix}1&0&0&0\\0&\cos\varphi&-\sin\varphi&0\\ 0&\sin\varphi&\cos\varphi&0\\0&0&0&1\end{pmatrix}\cdot \begin{pmatrix}0&0&0&s\\0&0&0&t\\0&0&0&u\\0&0&0&1\end{pmatrix} $$

Remember the order: since you're multiplying the input from the right, the right of the two matrices gets applied first. If your overall transformation is defined in terms of a sequence of rotations and translations (and this is what I assume from reading your question), the above setup will help you combine all of these to form a single operation. It doesn't really matter whether you think about this in terms of transforming points to their images, or transforming coordinate systems instead while keeping the points fixed.