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- (with W. Klingenberg) Reflection in a translation invariant surface, Math. Phys. Anal. Geom. 9 (2006) 225-231. Preprint
We prove that the focal set generated by the reflection of a point source off a translation invariant surface consists of two sets: a
curve and a surface. The focal curve lies in the plane orthogonal to the symmetry direction containing the source, while the focal surface
is translation invariant. In addition, we show that the focal curve is not physically visible.
- (with W. Klingenberg), On Hamilton's characteristic functions for reflection, Irish Math. Soc. Bulletin 57 (2006) 29-40. Preprint
We review the complex differential geometry of the space of oriented affine lines in R³ and give a description of Hamilton's characteristic
functions for reflection in an oriented C¹ surface in terms of this geometry.
- (with W. Klingenberg) Reflection of a wave off a surface, J. Geom. 84 (2006) 55-72. Preprint
Recent investigations of the space of oriented lines in R³ are applied to geometric optics. 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 examples are computed exactly and the results
plotted to illustrate the outgoing wavefront.
- (with W. Klingenberg and S. Sen) The Casimir effect between non-parallel plates by geometric optics, Reviews in Math. Phys. 17 (2005) 859-880. Preprint
Recent work by Jaffe and Scardicchio has expressed the optical approximation to the Casimir effect as a sum over geometric quantities.
The first two authors have developed a technique which uses the complex geometry of the space of oriented affine lines in R³ to describe
reflection of rays off a surface. This allows the quantities in the optical approximation to the Casimir effect to be calculated.
To illustrate this we determine explicitly and in closed form the geometric optics approximation of the Casimir force between two non-parallel
plates. By making one of the plates finite we regularise the divergence that is caused by the intersection of the planes. In the parallel plate
limit we prove that our expression reduces to Casimir's original result.
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