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The mammalian cerebral cortex is organized into a variety of two-dimensional areas whose anatomical and physiological organization is obscured by the folding of the cortical mantle in three dimensions. This organization can be revealed by unfolding the cortex so as to produce a two-dimensional representation or map of its surface. To produce such mappings, we have developed algorithms for computational cartography which maximally preserve intrinsic geometry while unfolding the cortical surface. Our computational algorithms were used to produce the first computational maps of the entire primary visual cortex of the macaque monkey (Carman and Van Essen, 1985), and the first completely noninvasive mapping of in vivo human visual cortex (Carman and Mora, 1989). In order to measure the geometry of the region of cortex to be mapped, a reconstruction of a surface or layer of cortex must be obtained from a typically sparse sample of contours of section. We obtain a solution to this reconstruction problem by computing a flow which fuses pairs of images containing successive contours of the surface. These flows are governed by a pair of complex harmonic potentials which represent translations, rotations, and scalings which combine to produce a conformal mapping of the two images onto a third fused and interpolated image. Since these potentials are harmonic, their values over a region of the images can be computed from samples taken only along the boundary of that region by solution of the Dirchlet problem. Thus, a coarse to fine series of samples on concentric annuli, similar to the sampling of the primate retina, can be used to compute such flows at a continuum of spatial scales. A number of visual problems arising in the analysis of motion, stereo, and shape information are formally equivalent to this reconstruction problem and can therefore also be solved by computing such flows. Remarkably, the equations which determine these flows can reproduce many aspects of the topography of the first stages of the primate visual pathway, suggesting that such flows may also be computed by the mappings of the cerebral cortex.