Reflection of a wave off a surface (Q2490671)

According to Huygens' principle the propagation of light can be described by either tracing the rays or the wavefronts. The authors use properties of the geometry of oriented lines in \(\mathbb R^3\) to switch back and forth between these two descriptions [see the authors, Arch. Math. 82, No. 1, 81--84 (2004; Zbl 1060.51015)]. They give an explicit description of a wavefront reflected off a surface. For point sources this has been answered earlier by \textit{J. B. Keller} and \textit{H. B. Keller} [J. Opt. Soc. Am. 40, 48--52 (1950)] using a completely different approach. The construction presented in the paper under review involves the long known correspondence between the space of oriented affine lines in \(\mathbb R^3\) with the tangent bundle of the 2-sphere. In addition to the general solution explicit formulae are given in the cases when the reflecting surface or the incoming wave is a plane or when the incoming wave is spherical. The paper closes with many applications including the construction of caustics and the Casimir force.