We develop a complete theory of projective cross-ratios in n-dimensional Plane-Based Geometric Algebra (PGA), R(n,0,1), covering geometric objects of every grade: finite and ideal points, hyperplanes, and intermediate flats. For each object type and configuration, we establish an explicit cross-ratio formula, prove that it recovers the appropriate classical invariant, and identify the canonical pairwise measurement operator. A systematic duality analysis further revealed that all eight configurations organize into four dual pairs under the Hodge dual, and that all measurement operators reduce to either the commutator or the commutator dual, depending solely on the geometric configuration rather than on object grade. In each case the formula recovers the appropriate classical invariant: signed distance ratios for parallel configurations and sine cross-ratios for secant ones. These results establish the cross-ratio as a grade-agnostic projective invariant within PGA, and provide a constructive foundation for defining n-dimensional homographies directly from prescribed invariants.

翻译：我们发展了基于平面的n维几何代数（PGA）R(n,0,1)中射影交叉比的完整理论，涵盖了每个阶数的几何对象：有限点和无穷远点、超平面以及中间平坦子空间。针对每种对象类型与配置，我们建立了明确的交叉比公式，证明其能够恢复相应的经典不变量，并标识了典范的成对测量算子。系统对偶分析进一步揭示：所有八种配置在霍奇对偶下组织为四个对偶对，且所有测量算子要么归结为换位子，要么归结为换位子对偶——这一判定仅取决于几何配置本身，而与对象的阶数无关。每种情形下的公式均能恢复相应的经典不变量：平行配置对应带符号距离比，割线配置对应正弦交叉比。这些结果确立了交叉比作为PGA中与阶数无关的射影不变量，并为从指定不变量直接定义n维单应变换提供了建设性基础。