Eight, let $A_{1}, A_{2}, \cdots, A_{8}$ be any 8 points taken on a plane. For any directed line $l$ taken 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 the direction of line $l$ as $P_{1_{1}}, P_{2_{2}}, \cdots, P_{i_{8}}$, then we obtain a permutation $i_{1}, i_{2}, \cdots, i_{8}$ of $1,2, \cdots, 8$ (in Figure 1, this permutation is $2,1,8,3,7,4$, $6,5$). Let the number of different permutations obtained by projecting these 8 points onto all directed lines on the plane be $N_{8}=N\left(A_{1}, A_{2}, \cdots, A_{8}\right)$. Try to find the maximum value of $N_{8}$.
Proof 3: First, prove that $\triangle D E F$ is the triangle with the shortest perimeter among all inscribed triangles in the acute $\triangle A B C$.

Let point $D^{\prime}$ be any fixed point on side $B C$, as shown in Figure 4. Construct the reflection of point $D^{\prime}$ about $A B$ and $A C$.
Assume the number of all lines passing through point $O$ and perpendicular to the line segments connecting any two points is $k$. It is evident that $k \leqslant \mathrm{C}_{8}^{2} = 28$. This generates $2 k$ directed lines, arranged in a counterclockwise direction, denoted as $l_{1}, l_{2}, \cdots, l_{2 h}$, as shown in Figure 6.

For any directed line $l$ (different from $l_{1}, \cdots, l_{24}$), there must exist two adjacent directed lines $l_{j}, l_{j+1}$ such that $l_{j}, l, l_{j+1}$ are arranged in a counterclockwise direction. It is evident that for a fixed $j$, the corresponding arrangement generated by such an $l$ must be the same.

If for two directed lines $l, l^{\prime}$ (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 the requirement mentioned in the previous paragraph, then there must be a $j$ such that $l^{\prime}, l_{j}, l$ are arranged in a counterclockwise direction. Suppose $l$ is perpendicular to the line segment connecting $A_{j_{1}}$ and $A_{j_{2}}$. It is evident that the projections of points $A_{J_{1}}$ and $A$ on the directed lines $l, l^{\prime}$ must have different orders, resulting in different corresponding arrangements.

As described above, the number of different arrangements is $2 k$. Noting that $k=\mathrm{C}_{8}^{2}$ can be achieved, we have $N_{8}=56$.

Eight, for two parallel and co-directional directed lines, the projection order of $A_{1}, A_{2}, \cdots, A_{8}$ must be the same. Therefore, we only need to consider all directed lines passing through a fixed point $O$.

If the chosen directed line is perpendicular to the line segment connecting any two points, the projections of these two points must coincide, and thus no corresponding arrangement is generated. Otherwise, the projections of $A_{1}, A_{2}, \cdots, A_{8}$ must be distinct, and therefore, there is a corresponding arrangement.