Pursuing an object with smooth eye movements requires an accurate estimate of its two-dimensional (2D) trajectory. This 2D motion computation requires that different local motion measurements are extracted and combined to recover the global object-motion direction and speed. Several combination rules have been proposed such as vector averaging (VA), intersection of constraints (IOC), or 2D feature tracking (2DFT). To examine this computation, we investigated the time course of smooth pursuit eye movements driven by simple objects of different shapes. For type II diamond (where the direction of true object motion is dramatically different from the vector average of the 1-dimensional edge motions, i.e., VA ≠ IOC = 2DFT), the ocular tracking is initiated in the vector average direction. Over a period of less than 300 ms, the eye-tracking direction converges on the true object motion. The reduction of the tracking error starts before the closing of the oculomotor loop. For type I diamonds (where the direction of true object motion is identical to the vector average direction, i.e., VA = IOC = 2DFT), there is no such bias. We quantified this effect by calculating the direction error between responses to types I and II and measuring its maximum value and time constant. At low contrast and high speeds, the initial bias in tracking direction is larger and takes longer to converge onto the actual object-motion direction. This effect is attenuated with the introduction of more 2D information to the extent that it was totally obliterated with a texture-filled type II diamond. These results suggest a flexible 2D computation for motion integration, which combines all available one-dimensional (edge) and 2D (feature) motion information to refine the estimate of object-motion direction over time.