Vertical sine-wave gratings of varying spatial frequency were stepped instantaneously to the right or to the left at differing phase angles (theta). Separate paradigms measured the contrast threshold for the detection of such a step and for the discrimination of the direction of the same step. By considering the grating before and after its displacement as a rotating phasor, we made the following predictions: (1) Contrast sensitivity for the detection of a displacement should rise as sin(theta). (2) Contrast sensitivity for the discrimination of the direction of the displacement should rise as sin(theta/2). Both predictions were confirmed using a range of spatial frequencies and phase angles. From the results of additional experiments, by measuring the discrimination of the direction thresholds as a function of contrast, we derived a nonlinear contrast response function for the motion system. This function appears to saturate fully at fairly low levels, in the neighborhood of 2 to 3% under the conditions examined. Our results suggest a direct connection among the contrast sensitivity, the contrast response function, and motion-hyperacuity thresholds.