When axonal fibres approach or leave the cortex, their trajectories tend to closely follow the cortical convolutions. To quantify this tendency, we propose a three-dimensional coordinate system based on the gyral geometry. For every voxel in the brain, we define a “radial” axis orthogonal to nearby surfaces, a “sulcal” axis along the sulcal depth gradient that preferentially points from deep white matter to the gyral crown, and a “gyral” axis aligned with the long axis of the gyrus. When compared with high-resolution, in-vivo diffusion MRI data from the Human Connectome Project, we find that in superficial white matter the apparent diffusion coefficient (at b = 1000) along the sulcal axis is on average 16% larger than along the gyral axis and twice as large as along the radial axis. This is reflected in the vast majority of observed fibre orientations lying close to the tangential plane (median angular offset < 7°), with the dominant fibre orientation typically aligning with the sulcal axis. In cortical grey matter, fibre orientations transition to a predominantly radial orientation. We quantify the width and location of this transition and find strong reproducibility in test-retest data, but also a clear dependence on the resolution of the diffusion data. The ratio of radial to tangential diffusion is fairly constant throughout most of the cortex, except for a decrease of the diffusivitiy ratio in the sulcal fundi and the primary somatosensory cortex (Brodmann area 3) and an increase in the primary motor cortex (Brodmann area 4). Although only constrained by cortical folds, the proposed gyral coordinate system provides a simple and intuitive representation of white and grey matter fibre orientations near the cortex, and may be useful for future studies of white matter development and organisation.