While it is widely assumed that the long-range horizontal connections in V1 are present to support contour integration, there has been only limited consideration of other possible relationships between anatomy and physiology (the horizontal connections) and visual function beyond contour integration. We introduce the possibility of other relationships directly from the perspective of computation and differential geometry by identifying orientation columns in visual physiology with the (unit) tangent bundle in differential geometry. This suggests abstracting early vision in a space that incorporates both position and orientation, from which we show that the physiology is capable of supporting a number of functional computations beyond contour integration, including texture-flow and shading-flow integration, as well as certain relationships between them. The geometric abstraction emphasizes the role of curvature, which necessitates a coupled investigation into how it might be estimated. The result is an elaboration of layer-to-layer interactions within an orientation column, with non-linearities possibly implemented by shunting inhibition. Finally, we show how the same computational framework naturally lends itself to solving stereo correspondence, with binocular tangents abstracting curves in space.