ABSTRACT
In Chapter 5, we considered subspaces, i.e., lines and planes passing through the origin, as well as the origin itself and the entire space. In this chapter, we consider points not necessarily at the origin and lines and planes not necessarily passing through the origin. We first show that points, lines, and planes in 3D can be regarded as subspaces in 4D by adding an extra dimension. This enables us to deal with them by the Grassmann algebra in that 4D space. There, a position in 3D and a direction in 3D are represented differently; the latter is identified with a “point at infinity.” There exist duality relations among points, lines, and planes, and exploiting the duality, we can describe the “join” (the line passing through two points and the plane passing through a point and a line or through three points) and the “meet” (the intersection of a line with a plane and of two or three planes) in a systematic manner.
