- The phenomenon of visual curve completion, where the visual system completes the missing part (e.g., due to occlusion) between two contour fragments, is a major problem in perceptual organization research. Previous computational approaches for the shape of the completed curve typically follow formal descriptions of desired, image-based perceptual properties (e.g, minimum total curvature, roundedness, etc.). Unfortunately, however, it is difficult to determine such desired properties psychophysically and indeed there is no consensus in the literature for what they should be. Instead, in this paper we suggest to exploit the fact that curve completion occurs in early vision in order to formalize the problem in a space that explicitly abstracts the primary visual cortex. We first argue that a suitable abstraction is the unit tangent bundle R 2 × S 1 and then we show that a basic principle of “minimum energy consumption” in this space, namely a minimum length completion, entails desired perceptual properties for the completion in the image plane. We present formal theoretical analysis and numerical solution methods, we show results on natural images and their advantage over existing popular approaches, and we discuss how our theory explains recent findings from the perceptual literature using basic principles only.