**Lie Transformations**

Any element of the group O(3,2) of orthogonal transformations of **R**3,2 maps any null one dimensional subspaces of **R**3,2 to another such subspace. Hence the group O(3,2) acts on the Lie quadric. These transformations of cycles are called "Lie transformations". They preserve the incidence relation between cycles. The action is transitive and so all cycles are Lie equivalent. In particular, points are not preserved by general Lie transformations. The subgroup of Lie transformations preserving the point cycles is essentially the subgroup of orthogonal transformations which preserve the chosen timelike direction. This subgroup is isomorphic to the group O(3,1) of Möbius transformations of the sphere. It can also be characterized as the centralizer of the involution *ρ*, which is itself a Lie transformation.

Lie transformations can often be used to simplify a geometrical problem, by transforming circles into lines or points.

Read more about this topic: Lie Sphere Geometry, Lie Sphere Geometry in The Plane

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