We establish that the minimum number of arcs required to partition the Galois projective plane $\text{PG}(2,q)$ is $q+1$. Furthermore, we determine the exact value for a fractional variant of this problem. We extend our analysis to affine planes $\text{AG}(2,q)$, proving that they can be partitioned into $q$ arcs. In particular, we show that this partition is tight when $q$ is an odd prime power, and that a $(q-1)$-partition is attainable for $q=2^k$ with $k \in \{1,2,3\}$. For $q=2^k$ with $k \geq 4$, we provide bounds between two possible values. Finally, we apply these results to Euclidean grids, demonstrating that a partition into $(1+ε)n$ sets in general position exists for any $ε> 0$ and sufficiently large $n$. We also present exact minimal partitions for small Euclidean grids.

翻译：我们证明了划分 Galois 射影平面 $\text{PG}(2,q)$ 所需的最小弧数为 $q+1$。此外，我们确定了该问题一个分数变体的精确值。我们将分析推广至仿射平面 $\text{AG}(2,q)$，证明其可被划分为 $q$ 条弧。特别地，我们证明了当 $q$ 为奇素数幂时，该划分是紧的；并且当 $q=2^k$ 且 $k \in \{1,2,3\}$ 时，可以实现 $(q-1)$-划分。对于 $q=2^k$ 且 $k \geq 4$ 的情况，我们给出了介于两个可能值之间的界。最后，我们将这些结果应用于欧几里得网格，证明了对于任意 $ε> 0$ 和充分大的 $n$，存在一个划分为 $(1+ε)n$ 个处于一般位置的集合。我们还给出了小型欧几里得网格的精确最小划分。