We carry out a sequence of
coordinate changes for the planar three-body problem, which successively eliminate
the translation and rotation symmetries, regularize all three double collision
singularities and blow-up the triple collision. Parametrizing the configurations
by the three relative position vectors maintains the symmetry among the
masses and simplifies the regularization of binary collisions. Using size and
shape coordinates facilitates the reduction by rotations and the blow-up of
triple collision while emphasizing the role of the shape sphere. By using
homogeneous coordinates to describe Hamiltonian systems whose configurations
spaces are spheres or projective spaces, we are able to take a modern, global
approach to these familiar problems. We also show how to obtain the reduced
and regularized differential equations in several convenient local coordinates
systems.
