On spherical harmonics possessing octahedral symmetry

In this paper, we present the implicit equations for one special class of real-valued spherical harmonics with octahedral symmetry. Based on this representation, we construct the rotationally invariant measure of deviation from the specified symmetry. The spherical harmonics we consider have some applications in the area of directional fields design due to their ability to represent mutually orthogonal axes in 3D space, not relative to their order and orientation.