When comparing two target elements placed on the same convoluted curve, response times are dependent on the distance between the targets along the curve, despite being separated by a constant Euclidean distance. The present study assessed whether such line tracing is obligatory across the whole of the line even when the task demands do not require it, or whether it is an optional strategy that can be disregarded when the circumstances favor a different method of attentional deployment. Three experiments were conducted to assess whether attention can select only a portion of a curve to trace when it is strategically sensible to do so. The results suggest that attention can indeed jump over portions of a line that are irrelevant to task performance before tracing has begun. However, the final experiment suggests that line tracing may continue beyond the task-relevant portion of the line. We conclude that line tracing is a strategy whose initial deployment can be influenced by top-down factors, rather than an obligatory response triggered by the stimuli—although, once engaged, line tracing may be hard to stop.