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• W759419585 abstract "In this thesis I continue a series of studies of singular varieties associated to a pair ofsubmanifolds in an affine space in a natural way. The origin of the study of similar objectscalled Wigner caustics can be traced back to the work of M. Berry [5] in the 1970s. He wasmotivated by its appearance in the semi-classical limit of theWigner function of a pure quantumstate whose classical limit corresponds to the given smooth curve in R2 with canonical symplecticstructure.Other similar objects were analysed by Janeczko [22] in which he generalised the conceptof centre of symmetry for a convex body in the plane by considering the bifurcation set of acertain 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) jointhem with a chord (infinite straight line) and find the envelope of these chords. The envelopeof these chords is also the Centre Symmetry Set (CSS). In that paper non-convex bodies wereconsidered and the envelopes were found to contain additional components and singularitiesthat resemble the boundary singularities of V. Arnold.A series of papers by V. Zakalyukin, P. Giblin and J. Warder [15], [16], [14] then generalisedthe idea to smooth hyper-surfaces in n-dimensions. In these papers the authors took the CSSto be the envelope of chords between points at which the tangent hyperplanes are parallel. Thegeneric singularities of the CSS were shown to be special singularities of wave fronts and causticsin the context of the theory of Lagrange and Legendre mappings as developed by Arnold andothers [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 generalisesthe concept of the CSS enabling them to consider the global properties of m-dimensionalsubmanifolds of the affine space Rn for n ≤ 2m. The paper also contains some motivation forthe study citing applications in quantum mechanics.In [13] Giblin and Janeczko gave a new approach to studying the centre symmetry sets viaa family of maps obtained by reflection in the mid-points of chords of a submanifold in affine space. In that paper 2-dimensional surfaces in both R3 and R4 are discussed. In particular theconditions for the caustic of the CSS for 2 surface pieces in R4 in terms of their geometry isgiven.When considering a more general case for two submanifolds Mk1 and Nk2 of some affinespace RK, we define the chords to be the infinite straight lines which join pairs of points fromM with points from N which share a common normal. It seems, at least at first, as though itdoes not make sense here to talk in terms of symmetry if the two submanifolds are of differentdimensions. Therefore, following the work of Stunzhas [32], we call the envelope of the familyof 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 spacecurve and a surface in three space (chapter 3) and the second case is the Minkowski set for twosurfaces in four space (chapter 4). In both cases the generic singularities of the Minkowski setare classified and where possible some geometric interpretation is given.The construction of the Minkowski set generalises that of the family of normals of a surfacein 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 geometrycan also be generalised in the same way. We define the wave fronts as the set of points of thechords which divide the chord segments between the base points with a fixed ratio λ, also calledthe affine time. Note that the wave fronts are sometimes called affine equidistants [14]. Whenλ 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 andtwo surfaces in three space were studied. The half way equidsitant or Wigner Caustic, that iswhen the ratio λ = 12 , 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 chapter4 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. Witha few changes the present problem can describe that of the wave propagations from an initialspace curve with indicatrix described by a surface (see [30], [37]). In fact, the list of genericsingularity types from the problem studied in [30] coincides with the list for the case of a curveand a surface considered in chapter 3." @default.
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• W759419585 date "2012-01-01" @default.
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• W759419585 title "Singularities of systems of chords in affine space" @default.
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