On the connection of geometric properties of curves and properties of their motion groups

Connections between geometric properties of curves and properties of their motion groups are established. Motions of two types, namely, positive and negative motions, are defined for curves. A necessary and sufficient condition for a smooth curve to possess a motion of one of these types is given. The notion of a motion group of a curve is defined in different ways, and classes of curves are specified for which these notions coincide. Closed curves are investigated in terms of their motion groups, and a necessary and sufficient condition for a smooth curve to be closed is presented.