Suppose that we have an arbitrary (not necessarily cubic) unit cell with cell parameters $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ (lattice vectors) and $\alpha$, $\beta$, $\gamma$ (lattice angles).
We have a particle at point $\mathbf{r}_i$ and we want to calculate its interactions with the other particles inside a cutoff radius of $r$. (E.g. interactions can be neglected outside this radius).
How can I find scaled versions of $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ such that the $i$-th particle interacts only with one periodic image of the (lets say) $j$-th particle?
According to this paper an approach is the following (see p. 6). We define the perpendicular widths of the cell as: $$ W_a = \frac{|\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})|}{|\mathbf{b} \times \mathbf{c}|} $$ $$ W_b = \frac{|\mathbf{b} \cdot (\mathbf{c} \times \mathbf{a})|}{|\mathbf{c} \times \mathbf{a}|} $$ $$ W_c = \frac{|\mathbf{c} \cdot (\mathbf{a} \times \mathbf{b})|}{|\mathbf{a} \times \mathbf{b}|} $$
($W_a, W_b, W_c$ are merely the shortest distance between opposing faces of the unit cell.)
and the condition to satisfy minimum image convention is:
$$ r_c < \frac{1}{2} \min(W_a, W_b, W_c) $$
I am not able to follow the explanation for the validity proposed in Appendix (p. 7). Can someone provide a graphical illustration of the proof?
Special case - cubic unit cell
If the unit cell is cubic, then the above equations reduces to:
$$ r_c < \frac{1}{2} L $$ where $L$ is the length of each side of the unit cell. Suppose that the distance between $i$ and $j$ particles is within the cutoff, that is $r_{ij} \leq r_c$. How do we prove that the distance between $i$ and a periodic image of $j$, call it $j^*$, is not within the cutoff?
I am adding the following visualization so it is easier to state my problem.
