We propose a theory for cortical representation and computation of visually completed curves that are generated by the visual system to fill in missing visual information (e.g., due to occlusions). Recent computational theories and physiological evidence suggest that although such curves do not correspond to explicit image evidence along their length, their construction emerges from corresponding activation patterns of orientation-selective cells in the primary visual cortex. Previous theoretical work modeled these patterns as least energetic 3D curves in the mathematical continuous space , which abstracts the mammalian striate cortex. Here we discuss the biological plausibility of this theory and present a neural architecture that implements it with locally connected parallel networks. Part of this contribution is also a first attempt to bridge the physiological literature on curve completion with the shape problem and a shape theory. We present completion simulations of our model in natural and synthetic scenes and discuss various observations and predictions that emerge from this theory in the context of curve completion.