Objects in our visual environment are perceived as integral wholes even when their retinal images are incomplete. We ask whether the perceptual precision of subjective interpolation between isolated image parts depends on the overall proportion of visible image information or rather on its geometrical arrangement. We used Varin-type subjective shapes that provide less physical stimulus information than Kanizsa-type figures because partially occluded solid inducers are replaced by partially occluded concentric arcs. We tested whether perceptual precision varies as a function of contour support, or alternatively, depends on the number of, and the distance between, line endings within the inducers. We measured performance in a probe localization task, where a small target is presented at different distances around a subjective boundary. Sensitivity, captured by the just noticeable position difference between in- and outside probes, crucially depended on the geometric arrangement of line ends in the Varin figures. This is objective evidence that the apparent subjective contour strength does not primarily depend on contour support but is determined by the number and the separation between inducers' line endings. The results suggest that neuronal mechanisms sensitive to highly localized 2D features are crucial for determining the perceived shape of visual objects.