  A metric introduced on a projective space yields a homogeneous metric space
known as a Cayley-Klein geometry. This construction is applicable not only to
Euclidean and non-Euclidean spaces but also to kinematic spaces (space-times).
A convenient algebraic framework for Cayley-Klein geometries called the
projective model is developed in [1, 2]. It is based on Grassmann and Clifford
algebras and provides a set of algebraic tools for modeling points, lines,
planes and their geometric transformations such as projections and isometries.
Isometry groups and their Lie algebras find a natural and intuitive expression
in the projective model. The aim of this paper is to translate the foundational
concepts of the projective model from the language of projective geometry to a
more familiar language of vector algebra and thereby facilitate its spread and
adoption among physicists and applied mathematicians. I apply the projective
model to Minkowski, de-Sitter, and anti-de-Sitter space-times in two
dimensions. In particular, I show how the action of the Poincare group can be
captured by the Clifford algebra in a uniform fashion with respect to rotations
(boosts) and translations.
