Abstract Plucker coordinates in both ray and axis order presented in 1865 have been used by Klein in his algebraic presentation and by Duffy in his argument of the instantaneous planar kinematics. This paper is to investigate the geometrical interpretation of each set of coordinates for a line vector by revealing the position vectors for ray-order coordinates and the plane normals for axis-order coordinates. It was found that the position vectors form a point-triangle and both primary and secondary parts of the Plucker coordinates in ray order are reflected in this triangle and represented by these two position vectors. In its dual, the plane normals form a plane-triangle and both primary and secondary parts of the Plucker coordinates in axis order are reflected in this plane-triangle and represented by these two plane normals. This reveals the geometry for the first time of both sets of the coordinates and identifies the duality in both geometrical and algebraic terms. The derivation and analysis relate ray-order coordinates to axis-order coordinates to present algebraically and geometrically the consistency in the geometrical presentation of the corresponding and reciprocal parts in each of the coordinate sets, and to obtain both cross-part correlated ratios and the correlation coefficient. This results in a derivation of a correlation between two sets of coordinates and naturally gives rise to the use of the correlation operator, leading to a conformability graph and a correlation table, both of that reveal the conformability of two sets of coordinates.

中文翻译：

---

线几何中普吕克坐标的射线和轴序相关特性的几何揭示

摘要 1865 年提出的 Plucker 坐标在射线和轴顺序中已被 Klein 在他的代数表示中和 Duffy 在他的瞬时平面运动学论证中使用。本文旨在通过揭示射线阶坐标的位置向量和轴阶坐标的平面法线来研究线向量的每组坐标的几何解释。发现位置向量形成一个点三角形，并且在光线顺序中 Plucker 坐标的主要和次要部分都反映在这个三角形中，并由这两个位置向量表示。在它的对偶中，平面法线形成一个平面三角形，并且按照轴顺序的 Plucker 坐标的主要和次要部分都反映在这个平面三角形中，并由这两个平面法线表示。这首次揭示了两组坐标的几何形状，并在几何和代数方面识别了对偶性。推导和分析将射线阶坐标与轴阶坐标联系起来，以代数和几何的方式表示每个坐标集中对应部分和互易部分的几何表示的一致性，并获得交叉部分的相关比和相关性系数。这导致了两组坐标之间相关性的推导，并且自然会产生相关算子的使用，导致一致性图和相关性表，这两者都揭示了两组坐标的一致性。