  I apply the algebraic framework developed in arXiv:1101.4542 to study
geometry of elliptic spaces in 1, 2, and 3 dimensions. The background material
on projectivised Clifford algebras and their application to Cayley-Klein
geometries is described in arXiv:1307.2917. The use of Clifford algebra largely
obviates the need for spherical trigonometry as elementary geometric
transformations such as projections, rejections, reflections, and rotations can
be accomplished with geometric multiplication. Furthermore, the same
transformations can be used in 3-dimensional elliptic space where effective use
of spherical trigonometry is problematic. I give explicit construction of
Clifford parallels and discuss their properties in detail. Clifford
translations are represented in a uniform fashion by geometric multiplication
as well. The emphasis of the exposition is on geometric structures and
computation rather than proofs.
