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 R 3 , previously studied by Giblin and Zakalyukin, and then apply the same ideas to surfaces in R 4 , elucidating some of the connexions between their geometry and the family of reflexion maps. We also point out some connexions with symplectic topology.