Exact conversion from Bézier tetrahedra to Bézier hexahedra

Modeling and computing of trivariate parametric volumes is an important research topic in the field of three-dimensional isogeometric analysis. In this paper, we propose two kinds of exact conversion approaches from Bézier tetrahedra to Bézier hexahedra with the same degree by reparametrization technique. In the first method, a Bézier tetrahedron is converted into a degenerate Bézier hexahedron, and in the second approach, a non-degenerate Bézier tetrahedron is converted into four non-degenerate Bézier hexahedra. For the proposed methods, explicit formulas are given to compute the control points of the resulting tensor–product Bézier hexahedra. Furthermore, in the second method, we prove that tetrahedral spline solids with C^k-continuity can be converted into a set of tensor–product Bézier volumes with G^k-continuity. The proposed methods can be used for the volumetric data exchange problems between different trivariate spline representations in CAD/CAE. Several experimental results are presented to show the effectiveness of the proposed methods.