Illustration of isotropic sampling schemes derived from the midpoints of the edges of the icosahedron (top left) and the dodecahedron (top right). Both the icosahedron and the dodecahedron have 30 edges. The midpoints of these edges coincide, but the edges from the two sets are orthogonal. The midpoints are further antipodal, so only 15 directions are defined by these points (middle). The 15 blue coordinate systems/b-tensor, and the 15 red ones are complimentary, and together defines a set of 30 coordinate systems.

It can be shown that the proposed icosahedral and dodecahedral midpoint sampling schemes have the same isotropic distribution of the directions in each of the three axes independently and simultaneously. The figures show that the distributions for the three eigenvectors $$$e_1$$$ (yellow), $$$e_2$$$ (magenta), and $$$e_3$$$ (cyan) are equal, ensuring minimal directional biases. (The eigenvectors have been drawn as headless arrows since if $$$e_1$$$ is an eigenvector, so is $$$-e_1$$$.)