Angle-Preserving (Conformal) Flat Maps of the Human Brain
The cortical surface of the brain is very convoluted, with many folds and fissures. The cortical surface is topologically equivalent to a sheet, so it is possible to "unfold" it and create a cortical flat map of the brain, which may reveal functional and anatomical spatial relationships that were not previously apparent. It is impossible to flatten a surface with intrinsic curvature (such as the brain) without introducing linear and areal distortion, but a 150-year old mathematics theorem (the Riemann Mapping Theorem) proves that it is possible to preserve angular (conformal) information under flattening. I will describe a novel computational method which uses the mathematical theory of circle packings to create quasi-conformal flat maps. I will present flat maps of the cortical surface obtained using this approach. These maps exhibit conformal behavior in that angular distortion is controlled and can be created in the Euclidean and hyperbolic planes and on a sphere.