Parameterizations, which map parametric domains into certain domains, are widely used in computer aided design, computer aided geometric design, computer graphics, isogeometric analysis, and related fields. The parameterizations of curves, surfaces, and volumes are injective means that they do not have self-intersections. A 3D toric volume is defined via a set of 3D control points with weights that correspond to a set of finite 3D lattice points. Rational tensor product or tetrahedral Bézier volumes are special cases of toric volumes. In this paper, we proved that a toric volume is injective for any positive weights if and only if the lattice points set and control points set are compatible. An algorithm is also presented for checking the compatibility of the two sets by the mixed product of three vectors. Some examples illustrate the effectiveness of the proposed method. Moreover, we improve the algorithm based on the properties and results of clean and empty tetrahedrons in combinatorics.

将参数化域映射到某些域的参数化被广泛用于计算机辅助设计，计算机辅助几何设计，计算机图形，等几何分析以及相关领域。曲线，曲面和体积的参数化是可注射的，这意味着它们没有自相交。通过一组3D控制点定义3D复曲面体积，其权重对应于一组有限的3D晶格点。有理张量积或四面体贝塞尔体积是复曲面体积的特例。在本文中，我们证明了当且仅当格点集和控制点集兼容时，复曲面体积对于任何正权都是可注射的。还提出了一种算法，用于通过三个向量的混合乘积来检验两组的兼容性。一些例子说明了该方法的有效性。此外，我们根据组合语中干净和空的四面体的性质和结果对算法进行了改进。