I have a lattice in 3D with $a$, $b$ and $c$ being its unit vectors. Say we have a simple cubic crystal then $a = (1, 0, 0)$, $b = (0, 1, 0)$ and $c = (0, 0, 1)$ are its building blocks. In principle, this is a periodic structure such that I have hundreds of these cubes where the distance between the cubes obviously is $a$ in $x$-direction ....
Now, I am supposed to project this 3D structure to a given plane and build a new 2D periodic structure with new unit vectors ($a'$ and $b'$) which should be derived from the old $a$, $b$ and $c$.
My problem is in finding the correct $a'$ and $b'$.
To make it clear I give you a simple example:
Suppose the plane is constructed based on two vectors $(1, 0, 0)$ and $(0, 1, 0)$ meaning that the plane normal vector is $(0, 0, 1)$ or in the other words we are in $xy$ plane. So, as we know the coordinates in the $xy$ plane then simply the $a' = a$ and $b' = b$.
The point is not always as simple as this example. So, let's say the two vectors forming the plane are $(0, 1, 1)$ and $(1, 1, 0)$ then the normal vector is $(1, -1, 1)$ and obviously, we cannot simply say we are in $xy$ or $xz$ or $yz$ plane. So, what is the best way to compute the new $a'$ and $b'$ in such situations?
I am not a mathematician so sorry if this is a trivial question. I have spent so much time on that and would greatly appreciate any help!
Dave