There is extensive experimental evidence linking instantaneous velocity to curvature in drawing and hand-writing movements. The empirical relationship between these characteristics of motion and path is well described by a power law in which the velocity varies in proportion to the one-third power of the radius of curvature. It was recently shown that a similar relationship can be observed during locomotion along curved elliptical paths raising the possibility that these very different motor activities might, at some level, share the same planning strategies. It has, however, been noted that the ellipse is a special case with respect to the one-third power law and therefore these previous results might not provide strong evidence that the one-third power law is a general feature of locomotion around curved paths. For this reason the experimental study of locomotion and its comparison with hand writing is extended here to non-elliptical paths. Subjects walked along predefined curved paths consisting of two complex shapes drawn on the ground: the cloverleaf and the limacon. It was found that the data always supported a close relationship between instantaneous velocity and curvature. For these more complex paths, however, the relationship is shape-dependent--although velocity and curvature can still be linked by a power law, the exponent depends on the geometrical form of the path. The results demonstrate the existence of a close relationship between instantaneous velocity and curvature in locomotion that is more general than the one-third power law. The origins of this relationship and its possible explanation in the mechanical balance of forces and in central planning are discussed.