### 6.2.1. The contractions of a tripod birational map.

**NOTATION 6.9.** Let $\phi$ be tripod with weights as in **COROLLARY 6.8**. In particular, $\phi$ is birational of type $(2, 2, 2)$, and by **PROPOSITION 6.2** we have the following:

1. The rank-two quadric $\Pi \cup \Pi_1$ (resp. $\Pi \cup \Pi_2$ and $\Pi \cup \Pi_3$) belongs to the pencil of s-surfaces (resp. t- and u-surfaces). Namely, we find $(\lambda_{00} : \lambda_{10})$ (resp. $(\mu_{00} : \mu_{10})$ and $(\nu_{00} : \nu_{10})$) in $\mathbb{P}_{\mathbb{R}}^1$ such that $\Pi \cup \Pi_1$ (resp. $\Pi \cup \Pi_2$ and $\Pi \cup \Pi_3$) is defined by

$$ \lambda_{00} \sigma_0 + \lambda_{10} \sigma_1 = 0 \quad (\text{resp. } \mu_{00} \tau_0 + \mu_{10} \tau_1 = 0 \quad \text{and} \quad \nu_{00} v_0 + \nu_{10} v_1 = 0) $$

2. The cone $Q$ of apex $A$ through the plane conic $C$ belongs to the pencils of s-, t-, and u-surfaces. In particular, $Q$ is defined by a quadratic form $q = q(x)$, and we find points $(\lambda_{01} : \lambda_{11})$, $(\mu_{01} : \mu_{11})$, and $(\nu_{01} : \nu_{11})$ in $\mathbb{P}_{\mathbb{R}}^1$ such that

$$ q = \lambda_{01} \sigma_0 + \lambda_{11} \sigma_1 = \mu_{01} \tau_0 + \mu_{11} \tau_1 = \nu_{01} v_0 + \nu_{11} v_1 $$

As for the previous classes of birational maps, **THEOREM 6.7** readily implies the following corollary.

**COROLLARY 6.10.** Let $\phi$ be tripod with weights as in **COROLLARY 6.8**, and let

$$ (6.8) \qquad a_i(s_0, s_1) = \begin{vmatrix} s_0 & s_1 \\ -\alpha_{1i} & \alpha_{0i} \end{vmatrix}, \quad b_j(t_0, t_1) = \begin{vmatrix} t_0 & t_1 \\ -\beta_{1j} & \beta_{0j} \end{vmatrix}, \quad c_k(u_0, u_1) = \begin{vmatrix} u_0 & u_1 \\ -\gamma_{1k} & \gamma_{0k} \end{vmatrix} $$

for each $0 \le i, j, k \le 1$, as well as

$$ h(s_0, s_1, t_0, t_1, u_0, u_1) = \omega_1 a_0 b_1 c_1 + \omega_2 a_1 b_0 c_1 + \omega_3 a_1 b_1 c_0 $$

for some $\omega_1, \omega_2, \omega_3$ in $\mathbb{R}$ such that $q = \omega_1 \pi_2 \pi_3 + \omega_2 \pi_1 \pi_3 + \omega_3 \pi_1 \pi_2$. Then, we have the following:

1. $\phi$ contracts the surface $a_1 = 0$ to the line $s$

2. $\phi$ contracts the surface $b_1 = 0$ to the line $t$

3. $\phi$ contracts the surface $c_1 = 0$ to the line $u$

4. $\phi$ contracts the surface $h = 0$ to the plane conic $C$

5. The image of the surface $a_0 = 0$ is dense in the plane $\Pi_1$

6. The image of the surface $b_0 = 0$ is dense in the plane $\Pi_2$

7. The image of the surface $c_0 = 0$ is dense in the plane $\Pi_3$

**Proof.** Follows from the proof of **THEOREM 6.7**. In particular, the cone $Q$ of apex $A = s \cap t \cap u$ through the plane conic $C$ is defined by $q = \omega_1 \pi_2 \pi_3 + \omega_2 \pi_1 \pi_3 + \omega_3 \pi_1 \pi_2$. Since $\phi$ contracts the surfaces $a_1 = 0, b_1 = 0$, and $c_1 = 0$ respectively to the lines $s, t$, and $u$, we find the factorization

$$ q(f_0, f_1, f_2, f_3) = a_1 b_1 c_1 (\omega_1 a_1 b_0 c_0 + \omega_2 a_0 b_1 c_0 + \omega_3 a_0 b_0 c_1) = a_1 b_1 c_1 h. $$

By the proof of 3 implies 1 in **THEOREM 6.7**, the image of $h = 0$ lies on the plane $\Pi$. Hence, $\phi$ contracts $h = 0$ to $C = \Pi \cap Q$. $\square$

### 6.3. Inverse and base locus.
As in § 3.3, § 4.3, and § 5.3, we close the section giving explicit formulas for the inverse of a tripod birational map.

**THEOREM 6.11.** Let $\phi$ be tripod with weights as in **COROLLARY 6.8**. Then, $\phi^{-1}$ is given by

$$ (x_0 : x_1 : x_2 : x_3) \mapsto (\alpha_{1i} : \lambda_{1i} : \sigma_1 : \alpha_{0i} : \lambda_{0i} : \sigma_0) \times (\beta_{1j} : \mu_{1j} : \tau_1 : \beta_{0j} : \mu_{0j} : \tau_0) \times (\gamma_{1k} : \nu_{1k} : v_1 : \gamma_{0k} : \nu_{0k} : v_0), $$

where any $0 \le i, j, k \le 1$ are valid. Additionally, with the notation of **COROLLARY 6.10**, we have the following:

1. The base locus of $\phi$ is defined by the ideal

$$ (6.9) \qquad I : \mathfrak{N}^{\infty} = (a_0, b_0, c_0) \cap (a_1^2, b_1^2, c_1^2, a_1 b_1, a_1 c_1, b_1 c_1, \omega_1 a_1 b_0 c_0 + \omega_2 a_0 b_1 c_0 + \omega_3 a_0 b_0 c_1) $$

2. The base locus of $\phi^{-1}$ is $s \cup t \cup u \cup C$
### 6.2.1. The contractions of a tripod birational map.

**NOTATION 6.9.** Let $\phi$ be tripod with weights as in **COROLLARY 6.8**. In particular, $\phi$ is birational of type $(2, 2, 2)$, and by **PROPOSITION 6.2** we have the following:

1. The rank-two quadric $\Pi \cup \Pi_1$ (resp. $\Pi \cup \Pi_2$ and $\Pi \cup \Pi_3$) belongs to the pencil of s-surfaces (resp. t- and u-surfaces). Namely, we find $(\lambda_{00} : \lambda_{10})$ (resp. $(\mu_{00} : \mu_{10})$ and $(\nu_{00} : \nu_{10})$) in $\mathbb{P}_{\mathbb{R}}^1$ such that $\Pi \cup \Pi_1$ (resp. $\Pi \cup \Pi_2$ and $\Pi \cup \Pi_3$) is defined by

$$ \lambda_{00} \sigma_0 + \lambda_{10} \sigma_1 = 0 \quad (\text{resp. } \mu_{00} \tau_0 + \mu_{10} \tau_1 = 0 \quad \text{and} \quad \nu_{00} v_0 + \nu_{10} v_1 = 0) $$

2. The cone $Q$ of apex $A$ through the plane conic $C$ belongs to the pencils of s-, t-, and u-surfaces. In particular, $Q$ is defined by a quadratic form $q = q(x)$, and we find points $(\lambda_{01} : \lambda_{11})$, $(\mu_{01} : \mu_{11})$, and $(\nu_{01} : \nu_{11})$ in $\mathbb{P}_{\mathbb{R}}^1$ such that

$$ q = \lambda_{01} \sigma_0 + \lambda_{11} \sigma_1 = \mu_{01} \tau_0 + \mu_{11} \tau_1 = \nu_{01} v_0 + \nu_{11} v_1 $$

As for the previous classes of birational maps, **THEOREM 6.7** readily implies the following corollary.

**COROLLARY 6.10.** Let $\phi$ be tripod with weights as in **COROLLARY 6.8**, and let

$$ (6.8) \qquad a_i(s_0, s_1) = \begin{vmatrix} s_0 & s_1 \\ -\alpha_{1i} & \alpha_{0i} \end{vmatrix}, \quad b_j(t_0, t_1) = \begin{vmatrix} t_0 & t_1 \\ -\beta_{1j} & \beta_{0j} \end{vmatrix}, \quad c_k(u_0, u_1) = \begin{vmatrix} u_0 & u_1 \\ -\gamma_{1k} & \gamma_{0k} \end{vmatrix} $$

for each $0 \le i, j, k \le 1$, as well as

$$ h(s_0, s_1, t_0, t_1, u_0, u_1) = \omega_1 a_0 b_1 c_1 + \omega_2 a_1 b_0 c_1 + \omega_3 a_1 b_1 c_0 $$

for some $\omega_1, \omega_2, \omega_3$ in $\mathbb{R}$ such that $q = \omega_1 \pi_2 \pi_3 + \omega_2 \pi_1 \pi_3 + \omega_3 \pi_1 \pi_2$. Then, we have the following:

1. $\phi$ contracts the surface $a_1 = 0$ to the line $s$

2. $\phi$ contracts the surface $b_1 = 0$ to the line $t$

3. $\phi$ contracts the surface $c_1 = 0$ to the line $u$

4. $\phi$ contracts the surface $h = 0$ to the plane conic $C$

5. The image of the surface $a_0 = 0$ is dense in the plane $\Pi_1$

6. The image of the surface $b_0 = 0$ is dense in the plane $\Pi_2$

7. The image of the surface $c_0 = 0$ is dense in the plane $\Pi_3$

**Proof.** Follows from the proof of **THEOREM 6.7**. In particular, the cone $Q$ of apex $A = s \cap t \cap u$ through the plane conic $C$ is defined by $q = \omega_1 \pi_2 \pi_3 + \omega_2 \pi_1 \pi_3 + \omega_3 \pi_1 \pi_2$. Since $\phi$ contracts the surfaces $a_1 = 0, b_1 = 0$, and $c_1 = 0$ respectively to the lines $s, t$, and $u$, we find the factorization

$$ q(f_0, f_1, f_2, f_3) = a_1 b_1 c_1 (\omega_1 a_1 b_0 c_0 + \omega_2 a_0 b_1 c_0 + \omega_3 a_0 b_0 c_1) = a_1 b_1 c_1 h. $$

By the proof of 3 implies 1 in **THEOREM 6.7**, the image of $h = 0$ lies on the plane $\Pi$. Hence, $\phi$ contracts $h = 0$ to $C = \Pi \cap Q$. $\square$

### 6.3. Inverse and base locus.
As in § 3.3, § 4.3, and § 5.3, we close the section giving explicit formulas for the inverse of a tripod birational map.

**THEOREM 6.11.** Let $\phi$ be tripod with weights as in **COROLLARY 6.8**. Then, $\phi^{-1}$ is given by

$$ (x_0 : x_1 : x_2 : x_3) \mapsto (\alpha_{1i} : \lambda_{1i} : \sigma_1 : \alpha_{0i} : \lambda_{0i} : \sigma_0) \times (\beta_{1j} : \mu_{1j} : \tau_1 : \beta_{0j} : \mu_{0j} : \tau_0) \times (\gamma_{1k} : \nu_{1k} : v_1 : \gamma_{0k} : \nu_{0k} : v_0), $$

where any $0 \le i, j, k \le 1$ are valid. Additionally, with the notation of **COROLLARY 6.10**, we have the following:

1. The base locus of $\phi$ is defined by the ideal

$$ (6.9) \qquad I : \mathfrak{N}^{\infty} = (a_0, b_0, c_0) \cap (a_1^2, b_1^2, c_1^2, a_1 b_1, a_1 c_1, b_1 c_1, \omega_1 a_1 b_0 c_0 + \omega_2 a_0 b_1 c_0 + \omega_3 a_0 b_0 c_1) $$

2. The base locus of $\phi^{-1}$ is $s \cup t \cup u \cup C$
3. $\phi$ blows up the base point $(a_0, b_0, c_0)$ to the plane $\Pi$

4. $\phi$ blows up the nonreduced base point $(a_1, b_1, c_1)$ to the cone $Q$

**Proof.** By COROLLARY 6.8, $\phi$ is birational of type $(2, 2, 2)$. As explained in the proofs of THEOREM 4.10 and THEOREM 5.10, the inverses on each factor of $(\mathbb{P}^1)^3$ can be regarded as line isomorphisms $\sigma, \tau, v : \mathbb{P}^1 \to (\mathbb{P}^9)^\vee$, to the space $(\mathbb{P}^9)^\vee$ of quadrics in $\mathbb{P}^3$, satisfying

$$\sigma : (1:0) \mapsto \sigma_0, \quad (0:1) \mapsto \sigma_1, \quad \tau : (1:0) \mapsto \tau_0, \quad (0:1) \mapsto \tau_1, \quad v : (1:0) \mapsto v_0, \quad (0:1) \mapsto v_1.$$

By COROLLARY 6.10, $\phi$ contracts the surface $a_1 = 0$ (resp. $b_1 = 0$ and $c_1 = 0$) to the line $s \subset Q$ (resp. $t \subset Q$ and $u \subset Q$). On the other hand, $\phi$ also maps the surface $a_0 = 0$ (resp. $b_0 = 0$ and $c_0 = 0$) to the plane $\Pi_1$ (resp. $\Pi_2$ and $\Pi_3$). These two observations readily imply

$$\sigma : (-\alpha_{10} : \alpha_{00}) \mapsto \pi_1\pi, \quad (-\alpha_{11} : \alpha_{01}) \mapsto q$$

since $\Pi_1 \cup \Pi$ and $Q$ are the quadrics in the pencil of $s$-surfaces respectively containing the plane $\Pi_1$ and the line $s$. Hence, $\sigma : \mathbb{P}^1 \to (\mathbb{P}^9)^\vee$ is explicitly given by

$$(s_0 : s_1) \mapsto \begin{vmatrix} s_0 & s_1 \\ \alpha_{1i} & \lambda_{1i} \end{vmatrix}$$

where both $i = 0, 1$ are admissible. Similarly, we find

$$\tau : (t_0 : t_1) \mapsto \begin{vmatrix} t_0 & t_1 \\ \beta_{1j} \mu_{1j} \tau_1 & \beta_{0j} \mu_{0j} \tau_0 \end{vmatrix}, \quad v : (u_0 : u_1) \mapsto \begin{vmatrix} s_0 & s_1 \\ \gamma_{1k} \nu_{1k} \gamma_1 & \gamma_{0k} \nu_{0k} \gamma_0 \end{vmatrix}$$

for any $0 \le j, k \le 1$, and $\phi^{-1}$ is as in the statement.

The base locus of $\phi^{-1}$ follows immediately from PROPOSITION 6.2. On the other hand, we have the identity

$$I'_{(1,1,1)} = \mathbb{R}\langle a_0 b_1 c_1, a_1 b_0 c_1, a_1 b_1 c_0, h \rangle = \mathbb{R}\langle\langle \pi_1, f \rangle\rangle, \langle\langle \pi_2, f \rangle\rangle, \langle\langle \pi_3, f \rangle\rangle, \langle\langle \pi, f \rangle\rangle = \mathbb{R}\langle f_0, f_1, f_2, f_3 \rangle$$

where $I'$ is the ideal on the right-hand side of (6.9), which is saturated. Hence, $I'$ defines the base locus of $\phi$. The blow-ups follow from the proof of PROPOSITION 6.2. $\square$

**7. Effective manipulation of birational trilinear volumes.** We conclude the paper introducing several effective methods for approximating and manipulating birational trilinear volumes. They rely on the results of the previous sections, and exploit the connection between birationality and tensor rank. We illustrate these methods through several examples, that emphasize how they are meaningful for applications.

**7.1. Distance to birationality.** First we address QUESTION 1.4, namely the problem of measuring how “far” is a trilinear rational map from being birational. The intuition behind this question is clear, but the question itself lacks precision. To formalize it, the first step is to introduce a notion of distance that provides a precise definition of the term “far”.

Let $\mathrm{Rat}_{(1,1,1)}$ denote the space of trilinear rational maps, and let $\mathrm{Bir}_{(1,1,1)} \subset \mathrm{Rat}_{(1,1,1)}$ be the locus of birational maps. In full generality, measuring the distance from $\phi$ to $\mathrm{Bir}_{(1,1,1)}$ presents a delicate challenge. The first complication arises from the high dimensionality, since a trilinear rational map is determined by 31 independent parameters (in the monomial basis: 8 × 4 coefficients up to scalar; in the Bernstein basis: 3 coordinates × 8 control points + 8 weights up to scalar) and hence $\mathrm{Rat}_{(1,1,1)} \cong \mathbb{P}^{31}$. Secondly, by [6, Theorem 4.1] $\mathrm{Bir}_{(1,1,1)}$ has eight irreducible components of various dimensions, and the distance from $\phi$ to each of the components is in general different.

Interestingly, THEOREM 3.5, THEOREM 4.6, THEOREM 5.6, and THEOREM 6.7 provide a way to quantify this distance. Specifically, since birationality is characterized by means of a rank-one condition on a $2 \times 2 \times 2$ tensor, we rely on the Frobenius norm for tensors [33] to define our notion of “distance to birationality”. Namely, it is the relative distance from the tensor $W$ that governs birationality to the locus of rank-one tensors, i.e. the affine cone over the Segre embedding of $\mathbb{P}_{\mathbb{R}}^1 \times \mathbb{P}_{\mathbb{R}}^1 \times \mathbb{P}_{\mathbb{R}}^1$.
3. $\phi$ blows up the base point $(a_0, b_0, c_0)$ to the plane $\Pi$

4. $\phi$ blows up the nonreduced base point $(a_1, b_1, c_1)$ to the cone $Q$

**Proof.** By COROLLARY 6.8, $\phi$ is birational of type $(2, 2, 2)$. As explained in the proofs of THEOREM 4.10 and THEOREM 5.10, the inverses on each factor of $(\mathbb{P}^1)^3$ can be regarded as line isomorphisms $\sigma, \tau, v : \mathbb{P}^1 \to (\mathbb{P}^9)^\vee$, to the space $(\mathbb{P}^9)^\vee$ of quadrics in $\mathbb{P}^3$, satisfying

$$ \sigma : (1:0) \mapsto \sigma_0, \quad (0:1) \mapsto \sigma_1, \quad \tau : (1:0) \mapsto \tau_0, \quad (0:1) \mapsto \tau_1, \quad v : (1:0) \mapsto v_0, \quad (0:1) \mapsto v_1. $$

By COROLLARY 6.10, $\phi$ contracts the surface $a_1 = 0$ (resp. $b_1 = 0$ and $c_1 = 0$) to the line $s \subset Q$ (resp. $t \subset Q$ and $u \subset Q$). On the other hand, $\phi$ also maps the surface $a_0 = 0$ (resp. $b_0 = 0$ and $c_0 = 0$) to the plane $\Pi_1$ (resp. $\Pi_2$ and $\Pi_3$). These two observations readily imply

$$ \sigma : (-\alpha_{10} : \alpha_{00}) \mapsto \pi_1\pi, \quad (-\alpha_{11} : \alpha_{01}) \mapsto q $$

since $\Pi_1 \cup \Pi$ and $Q$ are the quadrics in the pencil of $s$-surfaces respectively containing the plane $\Pi_1$ and the line $s$. Hence, $\sigma : \mathbb{P}^1 \to (\mathbb{P}^9)^\vee$ is explicitly given by

$$ (s_0 : s_1) \mapsto \begin{vmatrix} s_0 & s_1 \\ \alpha_{1i} & \lambda_{1i} \\ \alpha_{0i} & \lambda_{0i} \end{vmatrix} $$

where both $i = 0, 1$ are admissible. Similarly, we find

$$ \tau : (t_0 : t_1) \mapsto \begin{vmatrix} t_0 & t_1 \\ \beta_{1j} \mu_{1j} \tau_1 & \beta_{0j} \mu_{0j} \tau_0 \end{vmatrix}, \quad v : (u_0 : u_1) \mapsto \begin{vmatrix} s_0 & s_1 \\ \gamma_{1k} \nu_{1k} \gamma_1 & \gamma_{0k} \nu_{0k} \gamma_0 \end{vmatrix} $$

for any $0 \le j, k \le 1$, and $\phi^{-1}$ is as in the statement.

The base locus of $\phi^{-1}$ follows immediately from PROPOSITION 6.2. On the other hand, we have the identity

$$ I'_{(1,1,1)} = \mathbb{R}\langle a_0 b_1 c_1, a_1 b_0 c_1, a_1 b_1 c_0, h \rangle = \mathbb{R}\langle\langle \pi_1, f \rangle\rangle, \langle\langle \pi_2, f \rangle\rangle, \langle\langle \pi_3, f \rangle\rangle, \langle\langle \pi, f \rangle\rangle = \mathbb{R}\langle f_0, f_1, f_2, f_3 \rangle $$

where $I'$ is the ideal on the right-hand side of (6.9), which is saturated. Hence, $I'$ defines the base locus of $\phi$. The blow-ups follow from the proof of PROPOSITION 6.2. $\square$

7. **Effective manipulation of birational trilinear volumes.** We conclude the paper introducing several effective methods for approximating and manipulating birational trilinear volumes. They rely on the results of the previous sections, and exploit the connection between birationality and tensor rank. We illustrate these methods through several examples, that emphasize how they are meaningful for applications.

**7.1. Distance to birationality.** First we address QUESTION 1.4, namely the problem of measuring how “far” is a trilinear rational map from being birational. The intuition behind this question is clear, but the question itself lacks precision. To formalize it, the first step is to introduce a notion of distance that provides a precise definition of the term “far”.

Let $\mathrm{Rat}_{(1,1,1)}$ denote the space of trilinear rational maps, and let $\mathrm{Bir}_{(1,1,1)} \subset \mathrm{Rat}_{(1,1,1)}$ be the locus of birational maps. In full generality, measuring the distance from $\phi$ to $\mathrm{Bir}_{(1,1,1)}$ presents a delicate challenge. The first complication arises from the high dimensionality, since a trilinear rational map is determined by 31 independent parameters (in the monomial basis: $8 \times 4$ coefficients up to scalar; in the Bernstein basis: 3 coordinates $\times 8$ control points + 8 weights up to scalar) and hence $\mathrm{Rat}_{(1,1,1)} \cong \mathbb{P}^{31}$. Secondly, by [6, Theorem 4.1] $\mathrm{Bir}_{(1,1,1)}$ has eight irreducible components of various dimensions, and the distance from $\phi$ to each of the components is in general different.

Interestingly, THEOREM 3.5, THEOREM 4.6, THEOREM 5.6, and THEOREM 6.7 provide a way to quantify this distance. Specifically, since birationality is characterized by means of a rank-one condition on a $2 \times 2 \times 2$ tensor, we rely on the Frobenius norm for tensors [33] to define our notion of “distance to birationality”. Namely, it is the relative distance from the tensor $W$ that governs birationality to the locus of rank-one tensors, i.e. the affine cone over the Segre embedding of $\mathbb{P}_{\mathbb{R}}^1 \times \mathbb{P}_{\mathbb{R}}^1 \times \mathbb{P}_{\mathbb{R}}^1$.
**DEFINITION 7.1.** Let $\phi$ be either hexahedral, pyramidal, scaffold, or tripod. Let $W$ be any of the tensors characterizing birationality (in the statements of THEOREM 3.5, THEOREM 4.6, THEOREM 5.6, and THEOREM 6.7) for the corresponding class. We define the distance to birationality of $\phi$, denoted by $dist_{bir}(\phi)$, as

$$ (7.1) \qquad dist_{bir}(\phi) := \min_{P \in V} \frac{\|W - P\|}{\|W\|}, $$

where $V$ is the affine cone over the Segre embedding of $\mathbb{P}_{\mathbb{R}}^1 \times \mathbb{P}_{\mathbb{R}}^1 \times \mathbb{P}_{\mathbb{R}}^1$.

**REMARK 7.2.** The following observations are important:

1. For scaffold and tripod maps, several choices for $W$ are possible. However, THEOREM 5.6 and THEOREM 6.7 assert that $dist_{bir}(\phi) = 0$ represents the same condition for any choice.

2. $dist_{bir}(\phi)$ is well defined. More explicitly, although the vectors of $w_{ijk}$'s and $\Delta_{ijk}$'s are defined up to a nonzero scalar, they yield proportional tensors. Hence, the relative distances coincide.

Remarkably, this distance depends solely on the weights of $\phi$ (8 parameters up to scalar). This is a design-oriented approach, since a designer will typically move the control points of a rational map but will rarely modify the weights. Within this formulation, a “closest birational map” refers to a point in $V$ that minimizes the distance to $W$. In particular, it is an orthogonal projection of $\phi$ onto $V$. For a general $\phi$, there are exactly 6 such projections, where at least two of them are real. Equivalently, the Euclidean distance degree of $V$ is 6 [22, Example 8.2].

**7.2. Computation of a birational approximation.** We now deal with **QUESTION 1.5**. In order to compute a birational approximation of $\phi$, we can compute a rank-one tensor $W_{bir}$ such that $dist_{bir}(\phi) = \|W\|^{-1}\|W - W_{bir}\|$. Equivalently, we can solve the optimization problem

$$ (7.2) \qquad \min_{W_{bir}} \frac{\|W - W_{bir}\|}{\|W\|} \quad \text{where } W_{bir} = \alpha \otimes \beta \otimes \gamma \text{ for some } \alpha, \beta, \gamma \text{ in } \mathbb{R}^2 $$

There is an extensive body of work related to tensor rank and fixed-rank approximations to solve this problem. An important tool is the Canonical Polyadic Decomposition (CPD) (e.g. [16, 32, 30, 33]). We rely on it to efficiently compute rank-one approximations. The following example illustrates how our results can be applied in practice to compute birational approximations of trilinear rational maps.

**EXAMPLE 7.3** (Birational approximation of a pyramidal rational map). Consider the trilinear rational map $\phi$ defined by the control points

$$ \begin{align*} \mathbf{P}_{000} &= (1, -\frac{8}{5}, 0, 1), \quad \mathbf{P}_{100} = (1, 0, -\frac{9}{5}, \frac{1}{2}), \quad \mathbf{P}_{010} = (1, 0, \frac{27}{20}, \frac{1}{2}), \quad \mathbf{P}_{110} = (1, \frac{4}{5}, 0, 1), \\ \mathbf{P}_{001} &= (1, -\frac{11}{10}, 0, \frac{9}{4}), \quad \mathbf{P}_{101} = (1, 0, -\frac{4}{5}, 3), \quad \mathbf{P}_{011} = (1, 0, \frac{3}{5}, 3), \quad \mathbf{P}_{111} = (1, \frac{11}{20}, 0, \frac{9}{4}), \end{align*} $$

and $w_{ijk} = 1$ for every $0 \le i, j, k \le 1$. Since the four boundary u-lines, i.e. $u_{ij} = \overline{\mathbf{P}_{ij0}\mathbf{P}_{ij1}}$ for each $0 \le i, j, k \le 1$, meet at the point $\mathbf{A} = (1, 0, 0, 5)$ it follows that $\phi$ is pyramidal. In particular, we compute (recall **NOTATION 4.4**)

$$ \Delta_{000} = \frac{5}{6}, \quad \Delta_{100} = \frac{20}{21}, \quad \Delta_{010} = \frac{80}{63}, \quad \Delta_{110} = \frac{5}{3}, \quad \Delta_{001} = \frac{40}{33}, \quad \Delta_{101} = \frac{15}{7}, \quad \Delta_{011} = \frac{20}{7}, \quad \Delta_{111} = \frac{80}{33}. $$

Thus, the tensor

$$ W = \left( \frac{w_{ijk}}{\Delta_{ijk}} \right)_{0 \le i,j,k \le 1} = \left( \frac{1}{\Delta_{ijk}} \right)_{0 \le i,j,k \le 1} $$

does not have rank one, and by THEOREM 4.6 $\phi$ is not birational. However, we can render $\phi$ birational by computing new weights using the formulas in COROLLARY 4.7 for any choice of $(\alpha_0 : \alpha_1) \times (\beta_0 : \beta_1) \times (\gamma_0 : \gamma_1) \in (\mathbb{P}_{\mathbb{R}}^1)^3$. Nevertheless, the new birational map $\phi$ might differ significantly from the original $\phi$ if we make a “bad” choice. A smarter idea is to compute a rank-one CP decomposition of $W$. Using the Python library TensorFox [19], we compute the rational rank-one approximation of $W$ given by

$$ W_{bir} = (\alpha_0, \alpha_1) \otimes (\beta_0, \beta_1) \otimes (\gamma_0, \gamma_1) = (0.95, 0.91) \otimes (1.06, 0.78) \otimes (1.08, 0.75). $$
**DEFINITION 7.1.** Let $\phi$ be either hexahedral, pyramidal, scaffold, or tripod. Let $W$ be any of the tensors characterizing birationality (in the statements of THEOREM 3.5, THEOREM 4.6, THEOREM 5.6, and THEOREM 6.7) for the corresponding class. We define the distance to birationality of $\phi$, denoted by $dist_{bir}(\phi)$, as

$$ (7.1) \qquad dist_{bir}(\phi) := \min_{P \in V} \frac{\|W - P\|}{\|W\|}, $$

where $V$ is the affine cone over the Segre embedding of $\mathbb{P}_{\mathbb{R}}^1 \times \mathbb{P}_{\mathbb{R}}^1 \times \mathbb{P}_{\mathbb{R}}^1$.

**REMARK 7.2.** The following observations are important:

1. For scaffold and tripod maps, several choices for $W$ are possible. However, THEOREM 5.6 and THEOREM 6.7 assert that $dist_{bir}(\phi) = 0$ represents the same condition for any choice.

2. $dist_{bir}(\phi)$ is well defined. More explicitly, although the vectors of $w_{ijk}$'s and $\Delta_{ijk}$'s are defined up to a nonzero scalar, they yield proportional tensors. Hence, the relative distances coincide.

Remarkably, this distance depends solely on the weights of $\phi$ (8 parameters up to scalar). This is a design-oriented approach, since a designer will typically move the control points of a rational map but will rarely modify the weights. Within this formulation, a “closest birational map” refers to a point in $V$ that minimizes the distance to $W$. In particular, it is an orthogonal projection of $\phi$ onto $V$. For a general $\phi$, there are exactly 6 such projections, where at least two of them are real. Equivalently, the Euclidean distance degree of $V$ is 6 [22, Example 8.2].

**7.2. Computation of a birational approximation.** We now deal with **QUESTION 1.5**. In order to compute a birational approximation of $\phi$, we can compute a rank-one tensor $W_{bir}$ such that $dist_{bir}(\phi) = \|W\|^{-1}\|W - W_{bir}\|$. Equivalently, we can solve the optimization problem

$$ (7.2) \qquad \min_{W_{bir}} \frac{\|W - W_{bir}\|}{\|W\|} \quad \text{where } W_{bir} = \alpha \otimes \beta \otimes \gamma \text{ for some } \alpha, \beta, \gamma \text{ in } \mathbb{R}^2 $$

There is an extensive body of work related to tensor rank and fixed-rank approximations to solve this problem. An important tool is the Canonical Polyadic Decomposition (CPD) (e.g. [16, 32, 30, 33]). We rely on it to efficiently compute rank-one approximations. The following example illustrates how our results can be applied in practice to compute birational approximations of trilinear rational maps.

**EXAMPLE 7.3** (Birational approximation of a pyramidal rational map). Consider the trilinear rational map $\phi$ defined by the control points

$$ \begin{aligned} \mathbf{P}_{000} &= (1, -\frac{8}{5}, 0, 1), \quad \mathbf{P}_{100} = (1, 0, -\frac{9}{5}, \frac{1}{2}), \quad \mathbf{P}_{010} = (1, 0, \frac{27}{20}, \frac{1}{2}), \quad \mathbf{P}_{110} = (1, \frac{4}{5}, 0, 1), \\ \mathbf{P}_{001} &= (1, -\frac{11}{10}, 0, \frac{9}{4}), \quad \mathbf{P}_{101} = (1, 0, -\frac{4}{5}, 3), \quad \mathbf{P}_{011} = (1, 0, \frac{3}{5}, 3), \quad \mathbf{P}_{111} = (1, \frac{11}{20}, 0, \frac{9}{4}), \end{aligned} $$

and $w_{ijk} = 1$ for every $0 \le i, j, k \le 1$. Since the four boundary u-lines, i.e. $u_{ij} = \overline{\mathbf{P}_{ij0}\mathbf{P}_{ij1}}$ for each $0 \le i, j, k \le 1$, meet at the point $\mathbf{A} = (1, 0, 0, 5)$ it follows that $\phi$ is pyramidal. In particular, we compute (recall **NOTATION 4.4**)

$$ \Delta_{000} = \frac{5}{6}, \quad \Delta_{100} = \frac{20}{21}, \quad \Delta_{010} = \frac{80}{63}, \quad \Delta_{110} = \frac{5}{3}, \quad \Delta_{001} = \frac{40}{33}, \quad \Delta_{101} = \frac{15}{7}, \quad \Delta_{011} = \frac{20}{7}, \quad \Delta_{111} = \frac{80}{33}. $$

Thus, the tensor

$$ W = \left( \frac{w_{ijk}}{\Delta_{ijk}} \right)_{0 \le i,j,k \le 1} = \left( \frac{1}{\Delta_{ijk}} \right)_{0 \le i,j,k \le 1} $$

does not have rank one, and by THEOREM 4.6 $\phi$ is not birational. However, we can render $\phi$ birational by computing new weights using the formulas in COROLLARY 4.7 for any choice of $(\alpha_0 : \alpha_1) \times (\beta_0 : \beta_1) \times (\gamma_0 : \gamma_1) \in (\mathbb{P}_{\mathbb{R}}^1)^3$. Nevertheless, the new birational map $\phi$ might differ significantly from the original $\phi$ if we make a “bad” choice. A smarter idea is to compute a rank-one CP decomposition of $W$. Using the Python library TensorFox [19], we compute the rational rank-one approximation of $W$ given by

$$ W_{bir} = (\alpha_0, \alpha_1) \otimes (\beta_0, \beta_1) \otimes (\gamma_0, \gamma_1) = (0.95, 0.91) \otimes (1.06, 0.78) \otimes (1.08, 0.75). $$
<figure><img src="image_5.png" /><figcaption>Fig. 4: The two pyramidal maps presented in EXAMPLE 7.3. In the left image, we use uniform weights, specifically $w_{ijk} = 1$ for every $0 \le i, j, k \le 1$. 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.</figcaption></figure>

Specifically, this leads to the exact rational weights $w_{ijk} = \alpha_i \beta_j \gamma_k \Delta_{ijk}$ for each $0 \le i, j, k \le 1$. Additionally, we compute

$$ dist_{bir}(\phi) = \frac{\|W - W_{bir}\|}{\|W\|} \sim 0.2594. $$

In FIGURE 4, we present a comparison between the deformations resulting from the original rational map, with uniform weights $w_{ijk} = 1$ for every $0 \le i, j, k \le 1$, and its birational approximation, utilizing the new weights.

**7.3. Exact computation of the inverse.** THEOREM 3.7, THEOREM 4.10, THEOREM 5.10, and THEOREM 6.11 provide explicit formulas for the inverse of a trilinear birational map, which are directly applicable in practice. The following example shows the explicit computation of the inverse of a pyramidal birational map.

**EXAMPLE 7.4.** We continue with EXAMPLE 7.3, using the computed birational weights. In this case, the vectors and matrices defining the boundary surfaces are

$$ \sigma_0 = (5, -\frac{5}{2}, \frac{10}{3}, 1) , \quad \sigma_1 = (4, -4, 2, -\frac{4}{5}) , $$

$$ \tau_0 = (5, \frac{5}{2}, \frac{5}{2}, -1) , \quad \tau_1 = (-3, 3, 2, \frac{3}{5}) , $$

$$ v_0 = \begin{pmatrix} 390 & -30 & -\frac{45}{2} & -294 \\ -30 & 75 & 0 & 60 \\ -\frac{45}{2} & 0 & -50 & \frac{45}{2} \\ -294 & 60 & \frac{45}{2} & 102 \end{pmatrix} , \quad v_1 = \begin{pmatrix} 1530 & -\frac{495}{2} & -45 & -438 \\ -\frac{495}{2} & -225 & 0 & \frac{165}{2} \\ -45 & 0 & 150 & 20 \\ -438 & \frac{165}{2} & 20 & 114 \end{pmatrix} . $$

Furthermore, using the formulas in NOTATION 4.4 we compute

$$ (\lambda_0 : \lambda_1) \times (\mu_0 : \mu_1) \times (\nu_0 : \nu_1) = (4 : -5) \times (3 : -5) \times (11 : -18). $$

Thus, by THEOREM 4.10 $\phi^{-1} : \mathbb{P}_R^3 \rightarrow \mathbb{P}_R^1 \times \mathbb{P}_R^1 \times \mathbb{P}_R^1$ is given by

$$ (x_0 : x_1 : x_2 : x_3) \mapsto (\alpha_1 \lambda_1 \sigma_1 : \alpha_0 \lambda_0 \sigma_0) \times (\beta_1 \mu_1 \tau_1 : \beta_0 \mu_0 \tau_0) \times (\gamma_1 \nu_1 \nu_2 : \gamma_0 \nu_0 \nu_3). $$

**7.4. Deformation of birational maps.** An important task in the manipulation of birational transformations is the modification of the defining parameters, namely control points and weights, while preserving birationality. More specifically, we are interested in modifying these parameters continuously. This process is known as
<figure><img src="image_5.png" /><figcaption>Fig. 4: The two pyramidal maps presented in EXAMPLE 7.3. In the left image, we use uniform weights, specifically $w_{ijk} = 1$ for every $0 \le i, j, k \le 1$. 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.</figcaption></figure>

Specifically, this leads to the exact rational weights $w_{ijk} = \alpha_i \beta_j \gamma_k \Delta_{ijk}$ for each $0 \le i, j, k \le 1$. Additionally, we compute

$$ dist_{bir}(\phi) = \frac{\|W - W_{bir}\|}{\|W\|} \sim 0.2594. $$

In FIGURE 4, we present a comparison between the deformations resulting from the original rational map, with uniform weights $w_{ijk} = 1$ for every $0 \le i, j, k \le 1$, and its birational approximation, utilizing the new weights.

**7.3. Exact computation of the inverse.** THEOREM 3.7, THEOREM 4.10, THEOREM 5.10, and THEOREM 6.11 provide explicit formulas for the inverse of a trilinear birational map, which are directly applicable in practice. The following example shows the explicit computation of the inverse of a pyramidal birational map.

**EXAMPLE 7.4.** We continue with EXAMPLE 7.3, using the computed birational weights. In this case, the vectors and matrices defining the boundary surfaces are

$$ \sigma_0 = (5, -\frac{5}{2}, \frac{10}{3}, 1), \quad \sigma_1 = (4, -4, 2, -\frac{4}{5}), $$

$$ \tau_0 = (5, \frac{5}{2}, \frac{5}{2}, -1), \quad \tau_1 = (-3, 3, 2, \frac{3}{5}), $$

$$ v_0 = \begin{pmatrix} 390 & -30 & -\frac{45}{2} & -294 \\ -30 & 75 & 0 & 60 \\ -\frac{45}{2} & 0 & -50 & \frac{45}{2} \\ -294 & 60 & \frac{45}{2} & 102 \end{pmatrix}, \quad v_1 = \begin{pmatrix} 1530 & -\frac{495}{2} & -45 & -438 \\ -\frac{495}{2} & -225 & 0 & \frac{165}{2} \\ -45 & 0 & 150 & 20 \\ -438 & \frac{165}{2} & 20 & 114 \end{pmatrix}. $$

Furthermore, using the formulas in NOTATION 4.4 we compute

$$ (\lambda_0 : \lambda_1) \times (\mu_0 : \mu_1) \times (\nu_0 : \nu_1) = (4 : -5) \times (3 : -5) \times (11 : -18). $$

Thus, by THEOREM 4.10 $\phi^{-1} : \mathbb{P}_R^3 \rightarrow \mathbb{P}_R^1 \times \mathbb{P}_R^1 \times \mathbb{P}_R^1$ is given by

$$ (x_0 : x_1 : x_2 : x_3) \mapsto (\alpha_1 \lambda_1 \sigma_1 : \alpha_0 \lambda_0 \sigma_0) \times (\beta_1 \mu_1 \tau_1 : \beta_0 \mu_0 \tau_0) \times (\gamma_1 \nu_1 \nu_2 : \gamma_0 \nu_0 \nu_3). $$

**7.4. Deformation of birational maps.** An important task in the manipulation of birational transformations is the modification of the defining parameters, namely control points and weights, while preserving birationality. More specifically, we are interested in modifying these parameters continuously. This process is known as
<figure><img src="image_6.png" /><figcaption>Fig. 5: A birational deformation of the hexahedral birational map of EXAMPLE 7.5. The boundary plane $\Sigma_1$, previously defined by the vector $\sigma_1 = (1.25, -0.63, -0.32, -0.63)$, is updated to the plane defined by $\sigma_1 = (2.31, -0.84, -0.2, -0.32)$, yielding new control points. Additionally, the weights are also updated to preserve birationality.</figcaption></figure>

deformation, and it is instrumental in practice. Specifically, applications typically require the manipulation of the control points until the designer achieves the desired shape (e.g. [44, 43, 47, 48]).

Let $1 \le d_1, d_2, d_3 \le 2$, and let $\phi_0, \phi_1$ be trilinear birational maps of type $(d_1, d_2, d_3)$. Formally, a deformation of $\phi_0$ to $\phi_1$ is a continuous map $\delta: [0, 1] \to W \times P_{(d_1, d_2, d_3)}$ (in the usual topology) such that $\delta(0) = \phi_0$ and $\delta(1) = \phi_1$, where $W$ and $P_{(d_1, d_2, d_3)}$ are respectively the spaces of possible weights and control points for birational maps of type $(d_1, d_2, d_3)$.

By FACT 3.2, PROPOSITION 4.2, PROPOSITION 5.2, and PROPOSITION 6.2 we can take $P_{(d_1, d_2, d_3)}$ as the set of configurations of control points $\{\mathbf{P}_{ijk}\}_{0 \le i,j,k \le 1}$ that are either hexahedral, pyramidal, scaffold, or tripod. A small caveat is that pyramidal (resp. scaffold) birational maps can be of type either (1, 1, 2), (1, 2, 1), and (2, 1, 1) (resp. (1, 2, 2), (2, 1, 2), and (2, 2, 1)), so the distinguished parameter should be specified. On the other hand, CONSTRUCTION 3.3, CONSTRUCTION 4.3, CONSTRUCTION 5.3, and CONSTRUCTION 6.3 provide effective methods for continuously manipulating the control points, while preserving the necessary constraints for birationality. Thus, they can be conveniently exploited for deformation. More explicitly, we can use the dominant rational parametrization of the irreducible component labelled by $(d_1, d_2, d_3)$ (see [6, Theorem 4.1]) given by

$$ (7.3) \qquad \begin{aligned} & \Phi_{(d_1, d_2, d_3)} : (\mathbb{P}_{\mathbb{R}}^1)^3 \times P_{(d_1, d_2, d_3)} \dashrightarrow W \times P_{(d_1, d_2, d_3)} \\ & (\alpha_0 : \alpha_1) \times (\beta_0 : \beta_1) \times (\gamma_0 : \gamma_1) \times (\dots, \mathbf{P}_{ijk}, \dots) \mapsto (\dots : w_{ijk} : \dots) \times (\dots, \mathbf{P}_{ijk}, \dots) = \\ & \qquad (\dots : \Delta_{ijk} \alpha_i \beta_j \gamma_k : \dots) \times (\dots, \mathbf{P}_{ijk}, \dots) \end{aligned} $$

In particular, by COROLLARY 3.6, COROLLARY 4.7, COROLLARY 5.7, and COROLLARY 6.8 for a fixed choice of control points, $\Phi_{(d_1, d_2, d_3)}$ yields all the weights that render $\phi$ birational of type $(d_1, d_2, d_3)$.

The previous parametrization explains how deformation can be performed. Specifically, a user can continuously decide new control points, and update the $\Delta_{ijk}$'s accordingly (which are continuous functions in the coordinates of the points) while keeping the same tensor decomposition $W = (\alpha_0, \alpha_1) \otimes (\beta_0, \beta_1) \otimes (\gamma_0, \gamma_1)$ for each instant $t \in [0, 1]$ of the deformation, by applying the formulas $w_{ijk} = \alpha_i \beta_j \gamma_k \Delta_{ijk}$ for each $0 \le i, j, k \le 1$.

The following is an example of the deformation of hexahedral birational maps. This case is particularly important in practice, since hexahedral meshes appear frequently in geometric modeling. Furthermore, the generation of the control points proposed in CONSTRUCTION 3.3 is simple and intuitive, making it appealing for applications.
<figure><img src="image_6.png" /><figcaption>Fig. 5: A birational deformation of the hexahedral birational map of EXAMPLE 7.5. The boundary plane $Σ_1$, previously defined by the vector $σ_1 = (1.25, -0.63, -0.32, -0.63)$, is updated to the plane defined by $σ_1 = (2.31, -0.84, -0.2, -0.32)$, yielding new control points. Additionally, the weights are also updated to preserve birationality.</figcaption></figure>

deformation, and it is instrumental in practice. Specifically, applications typically require the manipulation of the control points until the designer achieves the desired shape (e.g. [44, 43, 47, 48]).

Let $1 ≤ d_1, d_2, d_3 ≤ 2$, and let $φ_0, φ_1$ be trilinear birational maps of type $(d_1, d_2, d_3)$. Formally, a deformation of $φ_0$ to $φ_1$ is a continuous map $δ: [0, 1] → W × P(d_1,d_2,d_3)$ (in the usual topology) such that $δ(0) = φ_0$ and $δ(1) = φ_1$, where W and $P(d_1,d_2,d_3)$ are respectively the spaces of possible weights and control points for birational maps of type $(d_1, d_2, d_3)$.

By FACT 3.2, PROPOSITION 4.2, PROPOSITION 5.2, and PROPOSITION 6.2 we can take $P(d_1,d_2,d_3)$ as the set of configurations of control points $\{P_{ijk}\}_{0≤i,j,k≤1}$ that are either hexahedral, pyramidal, scaffold, or tripod. A small caveat is that pyramidal (resp. scaffold) birational maps can be of type either (1, 1, 2), (1, 2, 1), and (2, 1, 1) (resp. (1, 2, 2), (2, 1, 2), and (2, 2, 1)), so the distinguished parameter should be specified. On the other hand, CONSTRUCTION 3.3, CONSTRUCTION 4.3, CONSTRUCTION 5.3, and CONSTRUCTION 6.3 provide effective methods for continuously manipulating the control points, while preserving the necessary constraints for birationality. Thus, they can be conveniently exploited for deformation. More explicitly, we can use the dominant rational parametrization of the irreducible component labelled by $(d_1, d_2, d_3)$ (see [6, Theorem 4.1]) given by

$$ (7.3) \qquad \Phi_{(d_1,d_2,d_3)} : (\mathbb{P}_{\mathbb{R}}^1)^3 \times \mathcal{P}_{(d_1,d_2,d_3)} \dashrightarrow W \times \mathcal{P}_{(d_1,d_2,d_3)} \\ (\alpha_0 : \alpha_1) \times (\beta_0 : \beta_1) \times (\gamma_0 : \gamma_1) \times (\dots, \mathbf{P}_{ijk}, \dots) \mapsto (\dots : w_{ijk} : \dots) \times (\dots, \mathbf{P}_{ijk}, \dots) = \\ (\dots : \Delta_{ijk} \alpha_i \beta_j \gamma_k : \dots) \times (\dots, \mathbf{P}_{ijk}, \dots) $$

In particular, by COROLLARY 3.6, COROLLARY 4.7, COROLLARY 5.7, and COROLLARY 6.8 for a fixed choice of control points, $Φ_{(d_1,d_2,d_3)}$ yields all the weights that render $φ$ birational of type $(d_1, d_2, d_3)$.

The previous parametrization explains how deformation can be performed. Specifically, a user can continuously decide new control points, and update the $Δ_{ijk}$'s accordingly (which are continuous functions in the coordinates of the points) while keeping the same tensor decomposition $W = (α_0, α_1) ⊗ (β_0, β_1) ⊗ (γ_0, γ_1)$ for each instant $t ∈ [0, 1]$ of the deformation, by applying the formulas $w_{ijk} = α_i β_j γ_k Δ_{ijk}$ for each $0 ≤ i, j, k ≤ 1$.

The following is an example of the deformation of hexahedral birational maps. This case is particularly important in practice, since hexahedral meshes appear frequently in geometric modeling. Furthermore, the generation of the control points proposed in CONSTRUCTION 3.3 is simple and intuitive, making it appealing for applications.
**EXAMPLE 7.5 (Deformation of a hexahedral birational map).** Consider the boundary planes defined by

$$
\begin{align*}
\boldsymbol{\sigma}_0 &= (0.16, -0.45, -0.07, -0.14) , \quad \boldsymbol{\sigma}_1 = (1.25, -0.63, -0.32, -0.63) , \\
\boldsymbol{\tau}_0 &= (0, 0, 0, 1) , \quad \boldsymbol{\tau}_1 = (-1.18, 0.18, 0.51, 1) , \\
\boldsymbol{v}_0 &= (0, 0, 1, 0) , \quad \boldsymbol{v}_1 = (-1.17, 0.1, 0.8, 0.54) .
\end{align*}
$$

For each $0 \le i, j, k \le 1$, we can use the formula (3.1) and express $\mathbf{P}_{ijk} = \Sigma_i \cap T_j \cap Y_k$ as the exterior product

$$
(7.4) \qquad \mathbf{P}_{ijk} = \Delta_{ijk}^{-1} \boldsymbol{\sigma}_i \wedge \boldsymbol{\tau}_j \wedge \mathbf{v}_k,
$$

where each $\Delta_{ijk}$ can be computed using (3.2). If we initialize the weights as $w_{ijk} = 1$ for every $0 \le i, j, k \le 1$, a rank-one approximation of the tensor $W$ in THEOREM 3.5 can be computed with the same strategy of EXAMPLE 7.3,

$$
W = \left( \frac{w_{ijk}}{\Delta_{ijk}} \right)_{0 \le i,j,k \le 1} = (\alpha_0, \alpha_1) \otimes (\beta_0, \beta_1) \otimes (\gamma_0, \gamma_1) = (1.56, 1.24) \otimes (1.12, 1.65) \otimes (1.02, 1.71).
$$

By COROLLARY 3.6, setting $w_{ijk} = \alpha_i \beta_j \gamma_k \Delta_{ijk}$ renders $\phi$ birational. The left image in FIGURE 5 shows the deformation of a model enclosed in the unit cube $[0, 1]^3$ by $\phi$.

Now, suppose that we want to update the control points, as a user frequently does during the design process.
However, we want to do so preserving birationality, so the property of being hexahedral must be preserved (recall
FACT 3.2). Using CONSTRUCTION 3.3, we decide to update the boundary plane $\boldsymbol{\sigma}_1$ linearly in time, namely

$$
[0, 1] \ni t \mapsto \boldsymbol{\sigma}_1(t) = (1-t) (1.25\ -0.63\ -0.32\ -0.63) + t (2.31\ -0.84\ -0.2\ -0.32) .
$$

In particular, for each $0 \le j, k \le 1$ this induces a linear function $\Delta_{1jk}(t)$. Therefore, setting

$$
w_{1jk}(t) = \alpha_1 \beta_j \gamma_k \Delta_{1jk}(t)
$$

ensures birationality for each instant $t \in [0, 1]$ of the deformation.

REFERENCES

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[4] N. BOTBOL, L. BUSÉ, M. CHARDIN, S. H. HASSANZADEH, A. SIMIS, AND Q. H. TRAN, *Effective criteria for bigraded birational maps*, J. Symbolic Comput., 81 (2017), pp. 69–87, https://doi.org/10.1016/j.jsc.2016.12.001, https://doi.org/10.1016/j.jsc.2016.12.001.

[5] B. BURLEY AND W. D. A. STUDIOS, *Physically-based shading at disney*, in Acm Siggraph, vol. 2012, vol. 2012, 2012, pp. 1-7.

[6] L. BUSÉ, P. GONZÁLEZ-MAZÓN, AND J. SCHICHO, *Tri-linear birational maps in dimension three*, Math. Comp., 92 (2023), pp. 1837–1866, https://doi.org/10.1090/mcom/3804, https://doi.org/10.1090/mcom/3804.

[7] E. CASAS-ALVERO, *Analytic projective geometry*, EMS Textbooks in Mathematics, European Mathematical Society (EMS), Zürich, 2014, https://doi.org/10.4171/138, https://doi.org/10.4171/138.

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[12] L. CREMONA, *Sulle trasformazioni geometriche delle figure piane*, Tipi Gamberini e Parmeggiani, 1863.

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**EXAMPLE 7.5 (Deformation of a hexahedral birational map).** Consider the boundary planes defined by

$$ 
\begin{aligned}
\sigma_0 &= (0.16, -0.45, -0.07, -0.14) , \quad \sigma_1 = (1.25, -0.63, -0.32, -0.63) , \\
\tau_0 &= (0, 0, 0, 1) , \quad \tau_1 = (-1.18, 0.18, 0.51, 1) , \\
\nu_0 &= (0, 0, 1, 0) , \quad \nu_1 = (-1.17, 0.1, 0.8, 0.54) .
\end{aligned}
$$

For each $0 \le i, j, k \le 1$, we can use the formula (3.1) and express $\mathbf{P}_{ijk} = \Sigma_i \cap T_j \cap Y_k$ as the exterior product

$$ (7.4) \qquad \mathbf{P}_{ijk} = \Delta_{ijk}^{-1} \sigma_i \wedge \tau_j \wedge \nu_k, $$

where each $\Delta_{ijk}$ can be computed using (3.2). If we initialize the weights as $w_{ijk} = 1$ for every $0 \le i, j, k \le 1$, a rank-one approximation of the tensor $W$ in THEOREM 3.5 can be computed with the same strategy of EXAMPLE 7.3,

$$ W = \left( \frac{w_{ijk}}{\Delta_{ijk}} \right)_{0 \le i,j,k \le 1} = (\alpha_0, \alpha_1) \otimes (\beta_0, \beta_1) \otimes (\gamma_0, \gamma_1) = (1.56, 1.24) \otimes (1.12, 1.65) \otimes (1.02, 1.71). $$

By COROLLARY 3.6, setting $w_{ijk} = \alpha_i \beta_j \gamma_k \Delta_{ijk}$ renders $\phi$ birational. The left image in FIGURE 5 shows the deformation of a model enclosed in the unit cube $[0, 1]^3$ by $\phi$.

Now, suppose that we want to update the control points, as a user frequently does during the design process. However, we want to do so preserving birationality, so the property of being hexahedral must be preserved (recall FACT 3.2). Using CONSTRUCTION 3.3, we decide to update the boundary plane $\sigma_1$ linearly in time, namely

$$ [0, 1] \ni t \mapsto \sigma_1(t) = (1-t) (1.25 - 0.63 - 0.32 - 0.63) + t (2.31 - 0.84 - 0.2 - 0.32). $$

In particular, for each $0 \le j, k \le 1$ this induces a linear function $\Delta_{1jk}(t)$. Therefore, setting

$$ w_{1jk}(t) = \alpha_1 \beta_j \gamma_k \Delta_{1jk}(t) $$

ensures birationality for each instant $t \in [0, 1]$ of the deformation.

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[37] T. McREYNOLDS, D. BLYTHE, B. GRANTHAM, AND S. NELSON, *Advanced graphics programming techniques using opengl*, Computer Graphics, (1998), pp. 95–145.

[38] M. MICHAŁ EK AND B. STURMFELS, *Invitation to nonlinear algebra*, vol. 211 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, [2021] ©2021, https://doi.org/10.1090/gsm/211, https://doi.org/10.1090/gsm/211.

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[42] D. RUECKERT, L. I. SONODA, C. HAYES, D. L. HILL, M. O. LEACH, AND D. J. HAWKES, *Nonrigid registration using free-form deformations: application to breast mr images*, IEEE transactions on medical imaging, 18 (1999), pp. 712–721.

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[44] T. W. SEDERBERG AND J. ZHENG, *Birational quadrilateral maps*, Comput. Aided Geom. Design, 32 (2015), pp. 1–4, https://doi.org/10.1016/j.cagd.2014.11.001, https://doi.org/10.1016/j.cagd.2014.11.001.

[45] O. SORKINE, D. COHEN-OR, Y. LIPMAN, M. ALEXA, C. RÖSSL, AND H.-P. SEIDEL, *Laplacian surface editing*, in Proceedings of the 2004 Eurographics/ACM SIGGRAPH symposium on Geometry processing, 2004, pp. 175–184.

[46] A. SOTIRAS, C. DAVATZIKOS, AND N. PARAGIOS, *Deformable medical image registration: A survey*, IEEE transactions on medical imaging, 32 (2013), pp. 1153–1190.

[47] X. WANG, Y. HAN, Q. NI, R. LI, AND R. GOLDMAN, *Birational quadratic planar maps with generalized complex rational representations*, Mathematics, 11 (2023), p. 3609.

[48] X. WANG, M. WU, Y. LIU, AND Q. NI, *Constructing quadratic birational maps via their complex rational representation*, Comput. Aided Geom. Design, 85 (2021), pp. Paper No. 101969, 11, https://doi.org/10.1016/j.cagd.2021.101969, https://doi.org/10.1016/j.cagd.2021.101969.