We determine in \mathbb{R}^{n} the form of curves C corresponding to strictly monotone functions as well as the components of affine connections \Delta for which any image of C under a compact-free group Ω of affinities containing the translation group is a geodesic with respect to \Delta . Special attention is paid to the case that Ω contains many dilatations or that C is a curve in \mathbb{R}^{3}. If C is a curve in \mathbb{R}^{3} and Ω is the translation group then we calculate not only the components of the curvature and the Weyl tensor but we also decide when \Delta yields a flat or metrizable space and compute the corresponding metric tensor., Josef Mikeš, Karl Strambach., and Obsahuje seznam literatury