We provide effective methods to construct and manipulate trilinear birational maps $\phi:(\mathbb{P}^1)^3\dashrightarrow \mathbb{P}^3$ by establishing a novel connection between birationality and tensor rank. These yield four families of nonlinear birational transformations between 3D spaces that can be operated with enough flexibility for applications in computer-aided geometric design. More precisely, we describe the geometric constraints on the defining control points of the map that are necessary for birationality, and present constructions for such configurations. For adequately constrained control points, we prove that birationality is achieved if and only if a certain $2\times 2\times 2$ tensor has rank one. As a corollary, we prove that the locus of weights that ensure birationality is $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$. Additionally, we provide formulas for the inverse $\phi^{-1}$ as well as the explicit defining equations of the irreducible components of the base loci. Finally, we introduce a notion of "distance to birationality" for trilinear rational maps, and explain how to continuously deform birational maps.

翻译：我们通过建立双有理性与张量秩之间的新联系，提出了构造和操作双有理三线性映射 $\phi:(\mathbb{P}^1)^3\dashrightarrow \mathbb{P}^3$ 的有效方法。由此产生了四族三维空间之间的非线性双有理变换，这些变换具有足够的操作灵活性，可应用于计算机辅助几何设计。更精确地说，我们描述了映射定义控制点所需满足的几何约束条件以确保双有理性，并给出了此类配置的构造方法。对于充分约束的控制点，我们证明了双有理性当且仅当某个 $2\times 2\times 2$ 张量的秩为一时得以实现。作为推论，我们证明了确保双有理性的权值轨迹为 $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$。此外，我们给出了逆映射 $\phi^{-1}$ 的公式，以及基轨迹不可约分量的显式定义方程。最后，我们为三线性有理映射引入了"双有理性距离"的概念，并解释了如何连续形变双有理映射。