As an application of Fenchel's theorem, we show that (a) a closed geodesic on a regular surface whose Gauss image is of length $< 2\pi$ must somewhere have an asymptotic and somewhere a principal direction, and (b) the polar reciprocal of any closed space curve whose central projection on a unit sphere is a regular curve without inflections for which the integral of the geodesic curvature is of absolute value $< 2\pi$ must have singular points.