Reflection of a wave off a surface

Guilfoyle, Brendan; Klingenberg, Wilhelm

doi:10.1007/s00022-005-0022-0

Cited by 3 publications

(11 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We turn now to the reflection of an oriented line in a surface in R 3 . This is equivalent to the action of a certain group on the space of oriented lines, as described by [4]:…”

Section: Definitionmentioning

confidence: 99%

Reflection in a Translation Invariant Surface

Guilfoyle

Klingenberg²

2007

Math Phys Anal Geom

Self Cite

View full text Add to dashboard Cite

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. This is done by constructing explicitly the focal set of the reflected line congruence (2-parameter family of oriented lines in R 3 ) with the aid of the natural complex structure on the space of all oriented affine lines.Date: September 26, 2005. 1991 Mathematics Subject Classification. 53A25 78A05 53C80.

show abstract

“…We turn now to the reflection of an oriented line in a surface in R 3 . This is equivalent to the action of a certain group on the space of oriented lines, as described by [4]:…”

Section: Definitionmentioning

confidence: 99%

Reflection in a Translation Invariant Surface

Guilfoyle

Klingenberg²

2007

Math Phys Anal Geom

Self Cite

View full text Add to dashboard Cite

show abstract

“…In a recent series of papers [3][4][5], the first two authors have presented a study of the local and global geometry of the space of oriented affine lines in R 3 . This has proved useful in geometric optics, for example, in the modeling of reflection.…”

Section: Introductionmentioning

confidence: 99%

“…Section 2 outlines our approach to the geometry of reflection in terms of the complex geometry on T S 2 -further details can be found in [5].…”

Section: Introductionmentioning

confidence: 99%

“…In this paper we present a complex geometry scheme [4] which, in principle, can be used to evaluate the Casimir effect for a range of boundaries in the approximation scheme of Jaffe et al The basic ingredient of our method is a description of reflection in R 3 via the space of oriented affine lines, which we identify with TP 1 . In this representation, reflection consists of the combined actions of PSL(2, C) and fibre mappings on TP 1 .…”

Section: Introductionmentioning

confidence: 99%

“…By virtue of the general reflection formulae derived in [4] and summarised in Proposition 3, we compute the sequence of rays reflected k-times off the pair of planes. The results (Proposition 7) are then used to determine the initial direction required for the ray to return to a given point in the wedge after k-bounces (Propositions 8 and 10).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Casimir Effect Between Non-Parallel Plates by Geometric Optics

2005

Self Cite

View full text Add to dashboard Cite

The first two authors have developed a technique which uses the complex geometry of the space of oriented affine lines in R 3 to describe the reflection of rays off a surface. This can be viewed as a parametric approach to geometric optics which has many possible applications. Recently, Jaffe and Scardicchio have developed a geometric optics approximation to the Casimir effect and the main purpose of this paper is to show that the quantities involved can be easily computed by this complex formalism. 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 regularize 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.

show abstract