Abstract
Let M be a smooth surface in real affine 3-space. Consider the pairs of points of this surface at which the tangent planes are parallel and in particular the chords (infinite lines) joining these pairs. We study in detail and classify the singularities of the envelope of these chords, that is a (singular) surface tangent to all of them. This is called the Centre Symmetry Set (CSS) of M. The study is local in character and is based upon a more general investigation by the authors of the n-dimensional case. The construction of the CSS is affinely invariant and generalises the focal set of a surface in euclidean space and the affine focal set of a surface in affine space. Many standard and some unusual singularities occur in a natural way as the singularities of the CSS. There are illustrations of the various cases by means of concrete examples.
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