Any element of the group O(3,2) of orthogonal transformations of R3,2 maps any null one dimensional subspaces of R3,2 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.
Other articles related to "lie, lie transformations":
... Lie sphere geometry in n-dimensions is obtained by replacing R3,2 (corresponding to the Lie quadric in n = 2 dimensions) by Rn + 1, 2 ... is Rn + 3 equipped with the symmetric bilinear form The Lie quadric Qn is again defined as the set of ∈ RPn+2 = P(Rn+1,2) with x · x = 0 ... The group of Lie transformations is now O(n + 1, 2) and the Lie transformations preserve incidence of Lie cycles ...
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“Bring me an axe and spade,
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—William Blake (17571827)