Abstract
The singularities of the midpoint map associated to a smooth plane curve, which is a map from the plane to the plane, are classified. 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 lambda-point map, associated to a space curve is introduced and many known surface singularities are realized as a special cases of this construction.