Numerous studies have shown that the power of 1/3 is important in relating Euclidean velocity to radius of curvature (R) in the generation and perception of planar movement. Although the relation between velocity and curvature is clear and very intuitive, no valid explanation for the specific 1/3 value has yet been found. We show that if instead of computing the Euclidean velocity we compute the affine one, a velocity which is invariant to affine transformations, then we obtain that the unique function of R which will give (constant) affine invariant velocity is precisely R1/3. This means that the 1/3 power law, experimentally found in the studies of hand-drawing and planar motion perception, implies motion at constant affine velocity. Since drawing/perceiving at constant affine velocity implies that curves of equal affine length will be drawn in equal time, we performed an experiment to further support this result. Results showed agreement between the 1/3 power law and drawing at constant affine velocity. Possible reasons for the appearance of affine transformations in the generation and perception of planar movement are discussed.