Fluid antenna systems (FASs) are emerging as a reconfigurable-aperture technology that expands physical-layer design beyond fixed, rigid antenna geometries. While the \emph{fading diversity} of FASs -- which exploits spatial channel fluctuations for signal enhancement and interference avoidance -- has been widely studied, the \emph{geometry diversity} created by reconfigurable port placement remains far less understood, particularly for planar architectures under finite-aperture constraints. This paper develops a systematic analytical framework for finite-aperture planar fluid antenna arrays (FAAs). First, we derive a closed-form characterization of the minimum inter-port distance under uniform random placement over a rectangular aperture and show that it follows a Rayleigh law. Its mean scales as $\mathcal{O}(M^{-1})$, in sharp contrast to the $\mathcal{O}(M^{-2})$ behavior in the linear case in which $M$ represents the number of candidate ports, revealing a fundamentally more favorable packing geometry in two dimensions. Secondly, we establish a universal Cramér-Rao bound (CRB) for joint elevation-azimuth estimation, governed by a $2\times 2$ \emph{geometric inertia matrix} whose determinant and eigenstructure fully capture the role of port placement in estimation precision. We further prove that both the trace and determinant of this matrix are invariant to the azimuth look direction. Third, we uncover an intrinsic \emph{precision--ambiguity trade-off}: maximizing the geometric determinant to minimize the CRB drives ports toward the aperture boundary, but simultaneously increases sidelobe-induced spatial ambiguity.

翻译：流体天线系统（FASs）正作为一种可重构孔径技术兴起，将物理层设计扩展到固定、刚性天线几何结构之外。尽管FAS的衰落分集——利用空间信道波动实现信号增强和干扰规避——已被广泛研究，但由可重构端口布局产生的几何分集仍远未被充分理解，特别是在有限孔径约束下的平面架构中。本文为有限孔径平面流体天线阵列（FAAs）建立了系统分析框架。首先，我们推导了矩形孔径上均匀随机布局下最小端口间距的闭式表征，证明其服从瑞利分布。其均值按$\mathcal{O}(M^{-1})$量级缩放，与线性情况下的$\mathcal{O}(M^{-2})$行为形成鲜明对比（其中$M$表示候选端口数），揭示二维空间中本质上更有利的封装几何特性。其次，我们建立了联合俯仰-方位角估计的通用克拉美-罗界（CRB），其由$2\times 2$几何惯性矩阵支配，该矩阵的行列式和特征结构完全刻画了端口布局对估计精度的影响。我们进一步证明了该矩阵的迹和行列式对方位角观测方向具有不变性。第三，我们揭示了一个内在的精度-模糊度权衡：最大化几何行列式以最小化CRB会使端口趋向孔径边界，但同时会增加旁瓣引起的空间模糊度。