Abstract: In this paper,we present a method for constructing a 3D cross-frame field,a 3D extension of the 2D cross-frame field 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 cross-frame 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.