在高维几何代数中统一定义了投影交叉比，实现跨阶几何对象的不变量计算。
Generalize cross-ratios in n-dimensional Plane-Based Geometric Algebra
- 基于平面几何代数构建n维投影交叉比理论，覆盖所有几何元素类型。
- 证明交叉比可还原经典不变量：平行时为有向距离比，相交时为正弦交叉比。
- 发现测量算子仅由几何构型决定，与对象阶无关，适用于任意维度同胚变换定义。
我们在n维平面基几何代数（PGA, R(n,0,1)）中建立了完整的投影交叉比理论，涵盖所有阶的几何对象：有限点、理想点、超平面及中间平坦结构。针对每种对象类型与配置，我们给出显式交叉比公式，证明其恢复相应的经典不变量，并识别出对应的规范成对测量算子。系统的对偶性分析揭示，全部八种配置在霍奇对偶下形成四组对偶对，且所有测量算子均可归约为交换子或其对偶形式，仅取决于几何构型而非对象阶。在每种情况下，公式均正确还原经典不变量：平行配置下为有向距离比，相交配置下为正弦交叉比。这些结果确立了交叉比作为PGA中阶无关的投影不变量，为直接从预设不变量构造n维射影变换提供了可构造的基础。
原文摘要 · Abstract (English)
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.
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