Vector Products and Combinations of Points, Lines, and Planes

TitleVector Products and Combinations of Points, Lines, and Planes
Publication TypePresentation
Year of Publication2023
AuthorsBongardt B, Groh F
Conference NameConference on Geometry: Theory and Applications (CGTA)
Date Published06/2023

In the case that a robot is constructed as a system of rigid bodies, its kinematics can be described by a set of copies of the Euclidean space, briefly called a Euclidean system, in which each Euclidean space is attached to one of the rigid bodies. Since “a rigid body can be considered as an assemblage of points, or planes, or lines, or described as a combination of all three of them”, the relative constraints and displacements between the different bodies can be described with respect to points, planes, lines, and their combinations. Next to linear transforms for points, planes, and lines, several parameterizations for combinations of them - especially for a point on a line, a pointed line - have been developed. Within the context of efficient kinematics computations, it has been observed that next to the direction n and the moment axn of a line, also the scalar product of anchor and direction a*n as well as the expression 2a(a*n) - (a*a)n only depend in linear manner on a rotation about an axis in space. Due to this characteristic, these terms have been classified as Lee–Liang products. Geometrically, the action of the term 2a(a*n) - (a*a)n corresponds to a rotation of pi radians of the vector n about the vector a, which is also called a half-turn. For a unit a with |a| = 1, the term equals the negative Householder reflection and the sandwich product with the geometric product. In the planned conference presentation, geometric and algebraic properties of the pi-rotation shall be analyzed, in particular with regard to the potential for modeling constraints and displacements in rigid body systems in a coherent and favorable manner. For this purpose, the elemental triangle of points, lines, and planes in Euclidean space shall be used as a navigation map.