The formation of temporal geodesics (i.e., fast routes) in a velocity field seems to offer a means of detecting curves by parallel hardware in a wide range of images with little prior design information. This hypothesis was tested by devising and constructing a dotted-curve detector based on it, and applying the detector to several synthetic dotted images covering a wide range of dotted curves and dotted partially coherent noise. The results are encouraging. The dotted-curve detector searches for fast routes in a velocity field determined locally by the approximate collinearity or cocircularity of clusters of dots, and globally by the tortuousness of the route and the uniformity of the spacing of dots along the route. The routes are found by a parallel adaptive classification of the dots into curve dots and noise dots, followed by a parallel merging and splitting of graphs spanning temporal close dots or subgraphs at successively higher levels of abstraction.>