Reflection subgroups of finite and affine Weyl groups

We discuss the classification of reflection subgroups of finite and affine Weyl groups from the point of view of their root systems. A short case free proof is given of the well known classification of the isomorphism classes of reflection subgroups using completed Dynkin diagrams, for which there seems to be no convenient source in the literature. This is used as a basis for treating the affine case, where finer classifications of reflection subgroups are given, and combinatorial aspects of root systems are shown to appear. Various parameter sets for certain types of subsets of roots are interpreted in terms of alcove geometry and the Tits cone, and combinatorial identities are derived.