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：全驱动与全向N旋翼飞行器的几何设计传统上被表述为参数优化问题，即在固定架构族中寻找N个朝向的单一最优集合。本研究突破该范式，探讨优化问题空间本身的内在拓扑结构。我们将设计问题表述在射影线\RP^2^N的乘积流形上，固定旋翼位置到多面体机架的顶点，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对称性决定。对于一般（不规则）顶点排列，解表现为离散的孤立点集。然而，当机架几何趋近正则性时，解空间经历关键相变，坍缩为N维环面（由顶点处与机架外接球相切的直线构成），随后进一步降维至由仿射相位锁定驱动的连续一维曲线。我们将这些观察综合为N-5缩放定律：对所有考察的正则平面多边形与柏拉图立体（N ≤ 10）成立的实证关系，其中最优构型空间由K=N-5个不连通的一维拓扑分支构成。我们证明这些锁定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的最优相位。关键在于，该拓扑揭示了一种保持最优性的设计冗余——飞行器可沿这些分支连续重构，同时保持最优各向同性控制能力。