Figure 1 : Snapshots of the optimization procedure to construct a boundary aligned 3D cross-frame field. The top row shows the internal streamlines. The next row contains another visualization with cubes spread by a parameterization along the current cross-frame field and rotated by the current local frame R(Φ). The corresponding number of iteration is shown at the bottom. Abstract In this paper, we present a method for constructing a 3D crossframe field, a 3D extension of the 2D cross-frame

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... ld as applied to surfaces in applications such as quadrangulation and texture synthesis. In contrast to the surface cross-frame field (equivalent to a 4-Way Rotational-Symmetry vector field), symmetry for 3D crossframe fields cannot be formulated by simple one-parameter 2D rotations in the tangent planes. To address this critical issue, we represent the 3D frames by spherical harmonics, in a manner invariant to combinations of rotations around any axis by multiples of π/2. With such a representation, we can formulate an efficient smoothness measure of the cross-frame field. Through minimization of this measure under certain boundary conditions, we can construct a smooth 3D cross-frame field that is aligned with the surface normal at the boundary. We visualize the resulting cross-frame field through restrictions to the boundary surface, streamline tracing in the volume, and singularities. We also demonstrate the application of the 3D cross-frame field to producing hexahedron-dominant meshes for given volumes, and discuss its potential in high-quality hexahedralization, much as its 2D counterpart has shown in quadrangulation.