Human vision is adept at inferring the shape of a surface from the image of curves lying across the surface. The strongest impression of 3-D shape derives from parallel (but not necessarily equally spaced) contours. In [Stevens 1981a] the computational problem of inferring 3-D shape from image configurations is examined, and a theory is given for how the visual system constrains the problem by certain assumptions. The assumptions are three: that neither the viewpoint not the placement of the physical curves on the surface is misleading, and that the physical curves are lines of curvature across the surface. These assumptions imply that parallel image contours correspond to parallel curves lying across an approximately cylindrical surface. Moreover, lines of curvature on a cylinder are geodesic and planar. These properties provide strong constraint on the local surface orientation. We describe a computational method embodying these geometric constraints that is able to determine the surface orientation even in places where locally it is very weakly constrained, by extrapolating from places where it is strongly constrained. This computation has been implemented, and predicts local surface orientation that closely matches the apparent orientation. Experiments with the implementation support the theory that our visual interpretation of surface shape from contour assumes the image contours correspond to lines of curvature.