A vector-sum process produces curved aiming paths under rotated visual-motor mappings

Under a 90 degrees rotation of motor space relative to visual space, human two-dimensional aiming movements frequently take the form of smooth arcs such as spirals and semi-circles. A time-independent differential equation explains this tendency in terms of a rotation-induced vector field made up, at each point in the two-dimensional space, of two input vectors. One vector represents a visual error signal and the other represents a motor error signal. A trajectory's instantaneous direction of movement at each point can be described as the resultant of the two vectors. This mathematical formulation incorporates plausible visual-motor mechanisms and, when expressed in polar coordinates, leads to a new method for analyzing the spatial properties of movements (i.e., movement paths). Plots of the angle between the resultant and the target vector (phi) against distance from the target (r, in the polar representation) summarize the arc-shaped movement paths as a simple relation that can be analyzed statistically with respect to properties such as monotonicity. The polar representation is a plausible representation of visually-guided movements, with the visual error vector functioning as an objective function relative to which behavior is optimized. We extend the model and the r, phi movement path analysis to non-90 degrees rotations, and we find that the model predicts an observed qualitative shift in behavior for rotations greater than 90 degrees. It also predicts qualitatively different path shapes observed under visual-motor reflections.