Pure Maths Colloquium: Centre Symmetry Sets of Curves and Surfaces
13 March 2006 16:00 in CM221
"The Centre Symmetry Set (CSS) of a curve in the plane (or a surface in 3-space) can be defined as the envelope of chords joining pairs of points where the tangent lines (or planes) are parallel. For a centrally symmetric curve or surface the CSS is a single point, the centre of symmetry. For a generic convex plane curve the CSS is a curve with an odd number of cusps. For a non-convex curve or a surface the CSS exhibits many standard singularities and some not-so-standard ones. One interest of the CSS is therefore that it is exhibits singularities of various kinds in a natural geometric context. The construction is affinely invariant, and in fact generalises various classical concepts such as euclidean and affine focal sets.
In this talk, intended to be comprehensible to postgraduates and a general mathematical audience, I shall describe the construction of the CSS, giving on the way many examples and also making general observations about envelopes of lines in the plane and in space. The work is joint with V.Zakalyukin."
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