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Recent advances in twistor theory are applied to geometric optics in R 3. The general formulae for reflection of a wavefront in a surface are derived and in three special cases explicit descriptions are provided: when the reflecting surface is a plane, when the incoming wave is a plane and when the incoming wave is spherical. In each case particular… (More)

We introduce a local coordinate description for the correspondence between the space of oriented affine lines in Euclidean R 3 and the tangent bundle to the 2-sphere. These can be utilised to give canonical coordinates on surfaces in R 3 , as we illustrate with a number of explicit examples. The correspondence between oriented affine lines in R 3 and the… (More)

A number of results for C 2-smooth surfaces of constant width in Euclidean 3-space E 3 are obtained. In particular, an integral inequality for constant width surfaces is established. This is used to prove that the ratio of volume to cubed width of a constant width surface is reduced by shrinking it along its normal lines. We also give a characterization of… (More)

We study the neutral Kähler metric on the space of time-like lines in Lorentzian E 3 1 , which we identify with the total space of the tangent bundle to the hyperbolic plane. We find all of the infinitesimal isometries of this metric, as well as the geodesics, and interpret them in terms of the Lorentzian metric on E 3 1. In addition, we give a new… (More)

We study surfaces in TN that are area-stationary with respect to a neutral Kähler metric constructed on TN from a riemannian metric g on N. We show that holomorphic curves in TN are area-stationary, while lagrangian surfaces that are area-stationary are also holomorphic and hence totally null. However, in general, area stationary surfaces are not… (More)

The space L of oriented lines, or rays, in R 3 is a 4-dimensional space with an abundance of natural geometric structure. In particular, it boasts a neutral Kähler metric which is closely related to the Euclidean metric on R 3. In this paper we explore the relationship between the focal set of a line congruence (or 2-parameter family of oriented lines in R… (More)

We study the geodesic flow on the global holomorphic sections of the bundle π : TS 2 → S 2 induced by the neutral Kähler metric on the space of oriented lines of R 3 , which we identify with TS 2. This flow is shown to be completely integrable when the sections are symplectic and the behaviour of the geodesics is described.

We present a new proof for the existence of a simple closed geo-desic on a convex surface M. This result is due originally to Poincaré. The proof uses the 2k-dimensional Riemannian manifold k ΛM = (briefly) Λ of piecewise geodesic closed curves on M with a fixed number k of corners, k chosen sufficiently large. In Λ we consider a submanifold ≈ Λ 0 formed by… (More)