Construction of birational trilinear volumes via tensor rank criteria

TL;DR

The paper connects birationality of trilinear maps φ:(\mathbb{P}^1)^3 \dashrightarrow \mathbb{P}^3 to a rank-one test on a 2×2×2 tensor $W=(\frac{w_{ijk}}{\Delta_{ijk}})$, enabling four explicit families—hexahedral, pyramidal, scaffold, and tripod—to be constructed with geometric control points. Each class yields an inverse map, a detailed base-locus description, and a tensor-rank criterion (rank-one flattenings) that exactly characterizes birationality; weights satisfy $w_{ijk}=\alpha_i\beta_j\gamma_k\Delta_{ijk}$. The authors also introduce a practical distance to birationality and CPD-based deformation tools, allowing continuous, birational-preserving morphing of trilinear volumes for CAGD applications. Together, the framework provides exact, linear-algebraic checks for birationality, explicit inverses, and robust manipulation workflows that can be used to design and optimize birational trilinear volumes in computer-aided geometric design. The results enable efficient preimage computation, principled weight selection, and controlled deformation within the birational components of the parameter space, with potential impact on 3D modeling and mesh generation.

Abstract

We provide effective methods to construct and manipulate trilinear birational maps $φ:(\mathbb{P}^1)^3\dashrightarrow \mathbb{P}^3$ 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 $φ^{-1}$ 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.

Figures (5)

• Figure 1: A deformation of a model, enclosed inside the unit cube, by means of a trilinear rational map. Leftmost: the boundary lines $s_{jk}$ (red), $t_{ik}$ (green), and $u_{ij}$ (blue), for each $0\leq i,j,k\leq 1$. From left to right: the boundary surfaces $\Sigma_0,\Sigma_1$ (red), $T_0,T_1$ (green), and $Y_0,Y_1$ (blue).
• Figure 1: Left: the unit cube enclosing a model. Middle: deformation of the unit cube and the model using the trilinear rational map constructed in \ref{['example: quadcube']}, with uniform weights $w_{ijk} = 1$ for every $0\leq i,j,k \leq 1$. Right: the lines $s$ (red), $t$ (green), and $u$ (blue) used in the construction. These three lines intersect at a common point $\mathbf{A}$. Additionally, all the boundary $s$-lines (red), $t$-lines (green), and $u$-lines respectively intersect $s,t$, and $u$.
• Figure 2: The distinct geometric constraints on the control points that are necessary for birationality. From left to right: hexahedral, pyramidal, scaffold, and tripod.
• Figure 2: The two pyramidal maps presented in \ref{['example: birational approx']}. In the left image, we use uniform weights, specifically $w_{ijk} = 1$ for every $0\leq i,j,k \leq 1$. The right image showcases the effect of utilizing the computed birational weights. These weight adjustments subtly influence the deformation, while simultaneously ensuring the existence of an inverse transformation.
• Figure 3: A birational deformation of the hexahedral birational map of \ref{['example: deformation']}. The boundary plane $\Sigma_1$, previously defined by the vector $\boldsymbol{\sigma}_1 = ( 1.25, \, -0.63, \, -0.32, \, -0.63 )$, is updated to the plane defined by $\boldsymbol{\sigma}_1 = ( 2.31, \, -0.84, \, -0.2, \, -0.32 )$, yielding new control points. Additionally, the weights are also updated to preserve birationality.

Theorems & Definitions (53)

• Definition 1.1
• Definition 2.1
• Definition 2.2
• Definition 2.5
• Lemma 2.6
• Proof 1
• Remark 2.7
• Definition 3.1
• Proof 2
• Theorem 3.5