The geometric design of fully-actuated and omnidirectional N-rotor aerial vehicles is conventionally formulated as a parametric optimization problem, seeking a single optimal set of N orientations within a fixed architectural family. This work departs from that paradigm to investigate the intrinsic topological structure of the optimization landscape itself. We formulate the design problem on the product manifold of Projective Lines \RP^2^N, fixing the rotor positions to the vertices of polyhedral chassis while varying their lines of action. By minimizing a coordinate-invariant Log-Volume isotropy metric, we reveal that the topology of the global optima is governed strictly by the symmetry of the chassis. For generic (irregular) vertex arrangements, the solutions appear as a discrete set of isolated points. However, as the chassis geometry approaches regularity, the solution space undergoes a critical phase transition, collapsing onto an N-dimensional Torus of the lines tangent at the vertexes to the circumscribing sphere of the chassis, and subsequently reducing to continuous 1-dimensional curves driven by Affine Phase Locking. We synthesize these observations into the N-5 Scaling Law: an empirical relationship holding for all examined regular planar polygons and Platonic solids (N <= 10), where the space of optimal configurations consists of K=N-5 disconnected 1D topological branches. We demonstrate that these locking patterns correspond to a sequence of admissible Star Polygons {N/q}, allowing for the exact prediction of optimal phases for arbitrary N. Crucially, this topology reveals a design redundancy that enables optimality-preserving morphing: the vehicle can continuously reconfigure along these branches while preserving optimal isotropic control authority.

翻译：全驱动全向N旋翼飞行器的几何设计通常被表述为参数优化问题，即在固定的架构族内寻找一组最优的N个旋翼朝向。本文突破这一范式，转而研究优化景观本身的内在拓扑结构。我们将设计问题表述在射影直线流形 \RP^2^N 的乘积空间上，将旋翼位置固定于多面体机架的顶点，同时改变其作用线方向。通过最小化坐标无关的对数体积各向同性度量，我们发现全局最优解的拓扑结构严格由机架的对称性决定。对于一般的（不规则）顶点排布，解表现为一组离散的孤立点。然而，随着机架几何形状趋近于规则，解空间经历一个临界相变：首先坍缩到一个N维环面，该环面上的作用线与机架外接球在顶点处相切；随后在仿射锁相驱动下进一步降维为连续的一维曲线。我们将这些观察综合为N-5标度律：一个对所有已考察的规则平面多边形和柏拉图立体（N <= 10）均成立的经验关系，即最优构型空间由 K=N-5 条互不连通的一维拓扑分支构成。我们证明这些锁相模式对应于一系列允许的星形多边形 {N/q}，从而能够精确预测任意N下的最优相位。至关重要的是，该拓扑结构揭示了一种设计冗余，使得保持最优性的形态变化成为可能：飞行器可以沿着这些分支连续重构，同时保持最优的各向同性控制能力。