ABSTRACT

In the preceding chapter, we considered a 4D space obtained by adding the origin e0 to the 3D space. In this chapter, we consider a 5D space, called the “conformal space,” obtained by further adding the point at infinity e∞. Basic geometric elements of this space are spheres and circles: a point is regarded as a sphere of radius 0, a plane as a sphere of radius ∞ passing through e∞, and a line as a circle of radius ∞ passing through e∞. In this space, translation, being interpreted to be rotation around an axis placed infinitely far apart, is treated equivalently as rotation. The “conformal mappings” that map a sphere to a sphere are generated from translation, rotation, reflection, inversion, and dilation, which are described in terms of the geometric product of the Clifford algebra. The content of this chapter is the core of what is now known as “geometric algebra.”