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Abstract. We present a novel approach to creating flat maps of the brain. It is impossible to flatten a curved surface in 3D space without metric and areal distortion; however, the Riemann Mapping Theorem implies that it is theoretically possible to preserve conformal (angular) information under flattening. Our approach attempts to preserve the conformal structure between the original cortical surface in 3-space and the flattened surface. We demonstrate this with data from the human cerebellum and we produce maps in the conventional Euclidean plane, as well as in the hyperbolic plane and on a sphere. Conformal mappings are uniquely determined once certain normalizations have been chosen, and this allows one to impose a coordinate system on the surface when flattening in the hyperbolic or spherical setting. Unlike existing methods, our approach does not require that cuts be introduced in the original surface. In addition, hyperbolic and spherical maps allow the map focus to be transformed interactively to correspond to any anatomical landmark.