- The analysis of texture patterns, and texture segregation in particular, are at the heart of perceptual organization. In this paper we question the widely accepted view that the detection (both perceptual and computational) of salient perceptual singularities between perceptually coherent texture regions is tightly dependent upon feature gradients. Specifically, we study smooth orientation-defined textures (ODTs) and show that they exhibit striking perceptual singularities even without any outstanding gradients in their defining feature, namely orientation. We further show how these generic singularities are not only unpredictable from the orientation gradient, but that they also defy popular segmentation algorithms and neural models. We then examine smooth ODTs from a (differential) geometric point of view and develop a theory that fully predicts their perceptual singularities from two ODT curvatures. The computational results exhibit striking correspondence to segregation performed by human subjects and provide a conclusive evidence for the role of curvature in texture segregation. Extensions and implications of our results are developed for various aspects of visual processing.