On Geometry of the Midlocus Associated to a Smooth Curve in Plane and Space

Abstract

The singularities of the midlpoint map associated to a smooth plane curve, which
is a map from the plane to the plane, are classied. The midlocus associated to a
regular space curve is introduced. The geometric conditions for the midlocus of a
space curve to have a crosscap or an S
±
1 singularities are investigated. A more general
map, the -point map, associated to a space curve is introduced and many known
surface singularities are realized as a special cases of this construction.