Painting Self-Consistent Beam Distributions in Rings

We define self-consistent beam distributions to have the following properties: 1) time-independence or periodicity, 2) linear space charge forces, and 3) maintainance of their defining shape and density under all linear transformations. The periodic condition guarantees zero space-charge-induced halo growth and beam loss during injection. Some self-consistent distributions can be manipulated into flat, or possibly even point-like, beams, which makes them interesting to colliders and to heavy-ion fusion. This paper discusses methods for painting 2D and 3D self-consistent distributions and for their manipulation to produce flat and point-like beams.