Geometry of curves and surfaces through the contact map

We introduce a new approach to the study of affine equidistants and centre symmetry sets via a family of maps obtained by reflexion in the midpoints of chords of a submanifold of affine space. We apply this to surfaces in R3, previously studied by Giblin and Zakalyukin, and then apply the same ideas to surfaces in R4, elucidating some of the connexions between their geometry and the family of reflexion maps. We also point out some connexions with symplectic topology.