Lie sphere geometry

This provides a natural realisation of the osculating circle to a curve, and the curvature spheres of a surface.

It also allows for a natural treatment of Dupin cyclides and a conceptual solution of the problem of Apollonius.

Lie sphere geometry can be defined in any dimension, but the case of the plane and 3-dimensional space are the most important.

spheres) and their tangential contact have a more natural formulation in a more general context in which circles, lines and points (resp.

Finally, for technical reasons, the circles (or spheres), including the lines (or planes) are given orientations.

These groups also have a direct physical interpretation: As pointed out by Harry Bateman, the Lie sphere transformations are identical with the spherical wave transformations that leave the form of Maxwell's equations invariant.

The planar Lie quadric Q consists of the points [x] in projective space represented by vectors x with x · x = 0.

The orthogonal space to (1,0,0,0,0), intersected with the Lie quadric, is the two dimensional celestial sphere S in Minkowski space-time.

Thus the isometric reflection map x → x + 2 (x · (1,0,0,0,0)) (1,0,0,0,0) induces an involution ρ of the Lie quadric which reverses the orientation of circles and lines, and fixes the points of the plane (including infinity).

It therefore remains to consider the case that neither [x] nor [y] are in S. Without loss of generality, we can then take x= (1,0,0,0,0) + v and y = (1,0,0,0,0) + w, where v and w are spacelike unit vectors in (1,0,0,0,0)⊥.

The incidence of cycles in Lie sphere geometry provides a simple solution to the problem of Apollonius.

By the incidence of cycles, a solution to the Apollonian problem compatible with the chosen orientations is given by a point [q] ∈ Q such that q is orthogonal to x, y and z.

If these three vectors are linearly dependent, then the corresponding points [x], [y], [z] lie on a line in projective space.

In the case that the subspace has signature (1,0), the unique solution q lies in the span of x, y and z.

This difficulty can be mitigated by the observation that there is a Lie invariant notion of contact element.

The set of all lines on the Lie quadric is a 3-dimensional manifold called the space of contact elements Z 3.

In more familiar terms, if λ is the contact lift of a curve γ in the plane, then the preferred cycle at each point is the osculating circle.

In other words, after taking contact lifts, much of the basic theory of curves in the plane is Lie invariant.

This is Rn + 3 equipped with the symmetric bilinear form The Lie quadric Qn is again defined as the set of [x] ∈ RPn+2 = P(Rn+1,2) with x · x = 0.

The quadric parameterizes oriented (n – 1)-spheres in n-dimensional space, including hyperplanes and point spheres as limiting cases.

The incidence relation carries over without change: the spheres corresponding to points [x], [y] ∈ Qn have oriented first order contact if and only if x · y = 0.

Lie noticed a remarkable similarity with the Klein correspondence for lines in 3-dimensional space (more precisely in RP3).

These are characterized as the common envelope of two one parameter families of spheres S(s) and T(t), where S and T are maps from intervals into the Lie quadric.

In order for a common envelope to exist, S(s) and T(t) must be incident for all s and t, i.e., their representative vectors must span a null 2-dimensional subspace of R4,2.

This map is Legendrian if and only if the derivatives of S (or T) are orthogonal to T (or S), i.e., if and only if there is an orthogonal decomposition of R4,2 into a direct sum of 3-dimensional subspaces σ and τ of signature (2,1), such that S takes values in σ and T takes values in τ. Conversely such a decomposition uniquely determines a contact lift of a surface which envelops two one parameter families of spheres; the image of this contact lift is given by the null 2-dimensional subspaces which intersect σ and τ in a pair of null lines.

Such a decomposition is equivalently given, up to a sign choice, by a symmetric endomorphism of R4,2 whose square is the identity and whose ±1 eigenspaces are σ and τ.

To summarize, Dupin cyclides are determined by quadratic forms on R4,2 such that the associated symmetric endomorphism has square equal to the identity and eigenspaces of signature (2,1).

A cyclide consists of the points [0,x1,x2,x3,x4,x5] ∈ Q3 which satisfy an additional quadratic relation for some symmetric 5 × 5 matrix A = (aij).