Total curvature

The total curvature of a regular curve isIt equals the length traced by the unit tangent on the unit sphere, so lifting a planar circle into a helix can reduce total curvature even when both curves make one revolution around the same axis.

Every closed regular space curve has total curvature at least $2\pi$. Equality holds exactly for convex plane curves traversed once. Here curvature is unsigned and integration is with respect to arc length.