Title:

Singularities of systems of chords in affine space

In this thesis I continue a series of studies of singular varieties associated to a pair of submanifolds in an affine space in a natural way. The origin of the study of similar objects called Wigner caustics can be traced back to the work of M. Berry [5] in the 1970s. He was motivated by its appearance in the semiclassical limit of theWigner function of a pure quantum state whose classical limit corresponds to the given smooth curve in R2 with canonical symplectic structure. Other similar objects were analysed by Janeczko [22] in which he generalised the concept of centre of symmetry for a convex body in the plane by considering the bifurcation set of a certain family of ratios called the centre symmetry set. Then in [12] P. Giblin and P. Holtom described an analogous method that uses envelopes. First find all the parallel tangent pairs (pairs of points where the tangent lines are parallel) join them with a chord (infinite straight line) and find the envelope of these chords. The envelope of these chords is also the Centre Symmetry Set (CSS). In that paper nonconvex bodies were considered and the envelopes were found to contain additional components and singularities that resemble the boundary singularities of V. Arnold. A series of papers by V. Zakalyukin, P. Giblin and J. Warder [15], [16], [14] then generalised the idea to smooth hypersurfaces in ndimensions. In these papers the authors took the CSS to be the envelope of chords between points at which the tangent hyperplanes are parallel. The generic singularities of the CSS were shown to be special singularities of wave fronts and caustics in the context of the theory of Lagrange and Legendre mappings as developed by Arnold and others [4]. In [31] motivation for the CSS in given, citing applications in computer vision. In [10] W. Domitrz and M. Rios study the Global Centre Symmetry Set (GCS) which generalises the concept of the CSS enabling them to consider the global properties of mdimensional submanifolds of the affine space Rn for n ≤ 2m. The paper also contains some motivation for the study citing applications in quantum mechanics. In [13] Giblin and Janeczko gave a new approach to studying the centre symmetry sets via a family of maps obtained by reflection in the midpoints of chords of a submanifold in affine space. In that paper 2dimensional surfaces in both R3 and R4 are discussed. In particular the conditions for the caustic of the CSS for 2 surface pieces in R4 in terms of their geometry is given. When considering a more general case for two submanifolds Mk1 and Nk2 of some affine space RK, we define the chords to be the infinite straight lines which join pairs of points from M with points from N which share a common normal. It seems, at least at first, as though it does not make sense here to talk in terms of symmetry if the two submanifolds are of different dimensions. Therefore, following the work of Stunzhas [32], we call the envelope of the family of chords the Minkowski set of the pair M and N. In this thesis two main cases are investigated. The first case is the Minkowski set for a space curve and a surface in three space (chapter 3) and the second case is the Minkowski set for two surfaces in four space (chapter 4). In both cases the generic singularities of the Minkowski set are classified and where possible some geometric interpretation is given. The construction of the Minkowski set generalises that of the family of normals of a surface in Euclidean space and also the family of affine normals of a surface in affine space. The concept of offsets (sometimes called parallels or equidistants) in Euclidean geometry can also be generalised in the same way. We define the wave fronts as the set of points of the chords which divide the chord segments between the base points with a fixed ratio λ, also called the affine time. Note that the wave fronts are sometimes called affine equidistants [14]. When λ varies these wave fronts sweep out the singularities of the Minkowski set (see figure 2). In [14] the bifurcations of the family of affine equidistants for two curves in the plane and two surfaces in three space were studied. The half way equidsitant or Wigner Caustic, that is when the ratio λ = 1 2 , is often cited as being of particular interest, see for example [5], [13], [10] and [31]. In chapter 3 the wave fronts for a curve and surface in three space are studied and in chapter 4 the wave fronts for two surfaces in four space are considered. The wave fronts in the case of a curve and surface in R3 are of particular interest. With a few changes the present problem can describe that of the wave propagations from an initial space curve with indicatrix described by a surface (see [30], [37]). In fact, the list of generic singularity types from the problem studied in [30] coincides with the list for the case of a curve and a surface considered in chapter 3.
