Geometrical revelation of correlated characteristics of the ray and axis order of the Plücker coordinates in line geometry
Jian S. Dai, Jie Sun
Abstract
Plücker 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 Plücker 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 Plücker 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.