ABSTRACT
Among basic concepts of the theory of curves in intrinsic geometry, we have considered by now only the length of a curve and the angle between curves. We now pass to two other basic concepts, the direction of a curve at a given point and the swerve of the direction of a curve, which is a generalization of the concept of the curvature of a curve. Since shortest arcs can emanate from a given point on a convex surface not in an arbitrary direction, the intrinsic definition of the direction of a curve via a shortest arc that touches this curve seems inappropriate. For example, there are no shortest arc on a right circular cone which touches its base circle and also there are no shortest arcs on a doubly convex lens which touch its edge. Meanwhile, both the base circle of the right circular cone and the edge of the doubly convex lens are circles both in the space and in the sense of intrinsic geometry, so that it would be rather extravagant to avoid considering them as smooth curves. Also, we can give examples of curves that are not edges of a convex surface and are smooth in the spatial sense but which have no shortest arcs tangent to them at some points.1
