- Locally parallel dense patterns - or texture flows - define a perceptually coherent structure important for image segmentation, edge classification, shading analysis, and shape interpretation. The classical psychophysics of texture flows is dominated by straight, exactly parallel patterns, in which orientation change is interpreted as a region boundary. We show how the geometry of texture flows extends this limited model by introducing two curvatures, one in the "tangential" direction and the other in the "normal" direction, and how these curvatures are naturally related to long-range horizontal interactions in primate V1 neurophysiology. The resulting model leads to psychophysical stimuli that demonstrate (i) amodal completion of texture flows and (ii) the coherence of flows with orientation changes and singularities. Thus not all orientation changes, and flow discontinuities, should be interpreted as boundaries. Implications for shading analysis are also developed.