Predicting the shape of distance functions in curve tracing: evidence for a zoom lens operator

Jolicoeur, Ullman, and MacKay (1986) showed that the time to confirm that two dots are on the same curve increases monotonically, but nonlinearly, as the curve distance between the two dots increases. These displays contained two curves and two dots. On same trials, the two dots were on the same curve (target curve), while the other curve served as a foil (distractor curve). The monotonically increasing effects of curve distance on response times for same trials suggested that the intervening curve segment was traced. In the present investigation of the source of the nonlinearity in these distance functions, it was hypothesized that differences in the distractor curves may have allowed a curve tracing operator with zoom lens properties to widen its receptive field while tracing parts of certain target curves. The wider receptive field may have allowed faster tracing over certain segments, owing to a reduced number of shifts required by the operator to scan the curve. The consequence of training certain segments of the curve more quickly than other segments of the curve would be a nonlinear effect of distance. A new set of stimuli was created for testing this hypothesis directly. Fairly linear distance effects were found for stimuli that contained a distractor curve that constrained the breadth of the postulated curve tracing operator, whereas stimuli that contained a distractor curve that could allow for a larger receptive field yielded nonlinear distance functions. The results are compared with the predictions of three quantitative models: pixel-by-pixel tracing; Jolicoeur, Ullman, and MacKay's (1991) bipartite operator; and a new zoom lens model, analogous to the zoom lens model of visual attention. The results were fit best by the latter model, in which tracing is accomplished by tracking the curve with a variably sized local operator.