For 8 points set arbitrarily, for any directed line $\iota$ set arbitrarily on the plane, let the projections of $A_{1}, A_{2}, \cdots, A_{8}$ on this line be $P_{1}, P_{2}, \cdots, P_{8}$ respectively. If these 8 projections are pairwise distinct and are arranged in sequence according to the direction of line $l$ as $P_{i_{1}}, P_{i_{2}}, \cdots, P_{i_{8}}$, then we get a permutation $i_{1}, i_{2}, \cdots, i_{8}$ of $\int 1,2, \cdots, 8$ (in the figure, this permutation is $2,1,8, 3,7,4,6,5$). Let the number of different permutations obtained by the projections of these 8 points on all directed lines in the plane be $N_{8}=N\left(A_{1}, A_{2}, \cdots, A_{8}\right)$, try to find the maximum value of $N_{8}$.
(1) For two parallel and co-directional directed lines, the projection order of $A_{1}, A_{2}, \cdots, A_{8}$ must be the same, so we only need to discuss all directed lines passing through a fixed point $O$.
(2) If the chosen directed line is perpendicular to the line connecting two points, then the projections of these two points must coincide, so no corresponding permutation is generated; when the chosen directed line is not perpendicular to any line connecting two points, the projections of $A_{1}, A_{2}, \cdots, A_{8}$ must be distinct, thus corresponding to a permutation.
(3) Let the number of all lines passing through point $O$ and perpendicular to the line connecting any two points be $k$, then $k \leqslant \mathrm{C}_{8}^{2}=28$, which generates $2 k$ directed lines, arranged in a counterclockwise direction, denoted as $l_{1}, l_{2}, \cdots, l_{2 k}$. For any directed line $l$ (different from $l_{1}, \cdots, l_{2 k}$), there must exist two adjacent directed lines $l_{j}, l_{j+1}$ such that $l_{i}, l, l_{j+1}$ are arranged in a counterclockwise direction. Clearly, for a fixed $j$, the permutations generated by such $l$ must be the same.
(4) If for two directed lines $l$ and $l^{\prime}$ (different from $l_{1}, \cdots, l_{2 k}$), there does not exist a $j$ such that both $l_{j}, l, l_{j+1}$ and $l_{j}, l^{\prime}, l_{j+1}$ satisfy (3), then there must be a $j$ such that $l^{\prime}, l_{j}, l$ are arranged in a counterclockwise direction. Suppose $l_{j}$ is perpendicular to the line connecting $A_{j_{1}}$ and $A_{j_{2}}$, it is evident that the projections of points $A_{j_{1}}$ and $A_{j_{2}}$ on the directed lines $l$ and $l^{\prime}$ must have different orders, so the corresponding permutations must be different.
(5) From (3) and (4), the number of different permutations is $2 k$. Noting that $k=\mathrm{C}_{8}^{2}$ can be achieved, so $N_{8}=56$.