  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.
