Recent atlases of the cortical surface are based on a modelization of the cerebral cortex as a topological sphere. This captures effectively its organization as a regular bidimensional sheet of layers parallel to the surface and with perpendicular cortical columns. Yet, while in the vertical direction cortices are almost the same throughout phylia, in the sense of its surface the cerebral cortex is one of the most variable and distinctive parts of the nervous system. Indeed, gyri and sulci appear to have a crucial organizing role in an architectonic, connectional, and functional sense. This organization is not explicitly captured by the surface model of the cortex. We propose a geometric model of the cortical anatomy based on flat representations of principal sulci obtained from surface reconstructions of MRI data, and on neuroanatomical and theoretical considerations concerning the folding patterns of the cortex. The cortex is modeled by a sphere where primary sulci are included as axes. The arrangement of the axes is a simplification of the arrangement of principal sulci observed in flat stereographic representations of the whole cortical surface. The position of secondary and tertiary sulci is then defined by a field of orientations parallel and orthogonal to the axes. We consider the use of the geometric model as a synthetic reference cortex for addressing reconstructions of cortical surfaces. We present a method which establishes a bijection between the geometric model and a cortical surface reconstruction by using the axes of the model as boundary conditions for a set of partial differential equations solved over both surfaces. Using the geometric model as atlas provides a natural parameterization of the cortical surface that, unlike angular coordinates, allows for a localization based on the surface distance to its main organizing landmarks and folding patterns.