Spherical Conformal Geometry with Geometric Algebra

The study of spheres dates back to the first century in the book Sphaerica of Menelaus. Spherical trigonometry was thoroughly developed in modern form by Euler in his 1782 paper [?]. Spherical geometry in n-dimensions was first studied by Schläfli in his 1852 treatise, which was published posthumously in 1901 [?]. The most important transformation in spherical geometry, Möbius transformation, was considered by Möbius in his 1855 paper [?]. The first person who studied spherical trigonometry with vectors was Hamilton [?]. In his 1987 book [?], Hestenes studied spherical trigonometry with Geometric Algebra, which laid down foundations for later study with Geometric Algebra. This chapter is a continuation of the previous chapter. In this chapter, we consider the homogeneous model of the spherical space, which is similar to that of the Euclidean space. We establish conformal geometry of spherical space in this model, and discuss several typical conformal transformations. Although it is well known that the conformal groups of n-dimensional Euclidean and spherical spaces are isometric to each other, and are all isometric to the group of isometries of hyperbolic (n+1)-space [?], [?], spherical conformal geometry on one hand has its unique conformal transformations, on the other hand can provide good understanding for hyperbolic conformal geometry. It is an indispensible part of the unification of all conformal geometries in the homogeneous model, which is going to be addressed in the next chapter.