† Joint first authors (a) (b) (c) (d) Figure 1: Closed space-filling curves optimized under (a) anisotropy, (b) boundary alignment (c) isotropy inside the E and boundary alignment inside the G. Isotropy and anisotropy are optimized in terms of distribution of angles, while alignment is specified in a vector field (here parallel to the boundary). For the three curves, the number of transitions between the two letters of the logo is minimized. (d) Multi-material 3D printed cube, with isotropic layers optimized by our approach.

We present a method for direction field design on surface and volumetric meshes supporting non-orthogonality. Our approach is a generalization of the representation of 3D cross fields in spherical harmonic basis. As such it induces a geometrically meaningful measure of smoothness, allows orthogonality control by a simple parameter and enables orientation constraints of a single direction. To the best of our knowledge this is the first work to propose non-orthogonal 3D frame field design. We demonstrate the applicability of our method to generate anisotropic quadrangular and hexahedral meshes which are particularly useful for remeshing CAD models.