Visual curve completion is typically handled in an axiomatic fashion where the shape of the sought-after completed curve follows formal descriptions of desired, image-based perceptual properties (e.g, minimum curvature, roundedness, etc...). Unfortunately, however, these desired properties are still a matter of debate in the perceptual literature. Instead of the image plane, here we study the problem in the mathematical space × that abstracts the cortical areas where curve completion occurs. In this space one can apply basic principles from which perceptual properties in the image plane are derived rather than imposed. In particular, we show how a "least action" principle in × entails many perceptual properties which have support in the perceptual curve completion literature. We formalize this principle in a variational framework for general parametric curves, we derive its differential properties, we present numerical solutions, and we show results on a variety of images.