One long-established simplifying principle behind the large repertoire and high versatility of human hand movements is the two-thirds power law - an empirical law stating a relationship between local geometry and kinematics of human hand trajectories during planar curved movements. It was further generalized to various types of human movements, as well as to motion perception and prediction, but was unsuccessful in explaining unconstrained three-dimensional movements. Recently, movement obeying the power-law was proved to be equivalent to moving with constant planar equi-affine speed. Generalizing such motion to three-dimensional space, i.e. to movement at constant spatial equi-affine speed, predicts the emergence of a new power-law, whose utility for describing spatial scribbling movements we have previously demonstrated. In this empirical investigation of the new power-law, subjects repetitively traced six different three-dimensional geometrical shapes with their hand. We show that the three-dimensional power-law explains the data consistently better than both the two-thirds power-law and an additional power-law that was previously suggested for spatial hand movements. We also found small yet systematic modifications of the power-law's exponents across the various shapes, which further scrutiny suggested to be correlated with global geometric factors of the traced shape. Nevertheless, averaging over all subjects and shapes, the power-law exponents are generally in accordance with constant spatial equi-affine speed. Taken together, our findings provide evidence for the potential role of non-Euclidean geometry in motion planning and control. Moreover, these results seem to imply a relationship between geometry and kinematics that is more complex than the simple local one stipulated by the two-thirds power law and similar models.