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The transformation blows up the single point in phase space where the collision occurs into a collision manifold, the phase space point is cut out and in its place a smooth manifold is pasted. This allows the phase space singularity to be studied in detail.
What McGehee found was a distorted sphere with four horns pulled out to infinity and the points at their tips deleted. McGehee then went on to study the flow on the collision manifold.
- Celestial Encounters,The Origins of Chaos and Stability, Diacu/Holmes, ISBN 0-691-00545-1, Princeton Science Library