An Atlas for the Pinhole Camera

Abstract

We introduce an atlas of algebro-geometric objects associated with image formation in pinhole cameras. The nodes of the atlas are algebraic varieties or their vanishing ideals related to each other by projection or elimination and restriction or specialization, respectively. This atlas offers a unifying framework for the study of problems in 3D computer vision. We initiate the study of the atlas by completely characterizing a part of the atlas stemming from the triangulation problem. We conclude with several open problems and generalizations of the atlas.

Appendices

Appendix A. Dimension Counts

Proposition A.1

Below, ${\bar{\mathbf {A}}} \in (\mathbb {P}^{11})^m, {\bar{\mathbf {q}}} \in (\mathbb {P}^3)^n, {\bar{\mathbf {p}}} \in \Gamma ^{m,n}_{\mathbf {p}}$ are generic whenever they appear.

$$\begin{aligned} \dim \Gamma ^{m,n}_{\mathbf {A},\mathbf {q},\mathbf {p}}&= 3n + 11m \end{aligned}$$

(A.1)

$$\begin{aligned} \dim \Gamma ^{m,n}_{\mathbf {A},\mathbf {p}}&= \min (2mn + 11m, 3n+11m) \end{aligned}$$

(A.2)

$$\begin{aligned} \dim \Gamma ^{m,n}_{\mathbf {q}, \mathbf {p}}&= \min (3n+2mn, 3n + 11m) \end{aligned}$$

(A.3)

$$\begin{aligned} \dim \Gamma ^{m,n}_{\mathbf {p}}&= \min (2mn, 11m + \max (3n - 15, 0)) \end{aligned}$$

(A.4)

$$\begin{aligned} \dim \Gamma ^{m,n}_{{\bar{\mathbf {A}}}, \mathbf {q}, \mathbf {p}}&= \dim \Gamma ^{m,n}_{\mathbf {A}, \mathbf {q}, \mathbf {p}} - \dim \Gamma ^{m,n}_\mathbf {A}= 3n \end{aligned}$$

(A.5)

$$\begin{aligned} \dim \Gamma ^{m,n}_{\mathbf {A}, \mathbf {q}, {\bar{\mathbf {p}}}}&= \dim \Gamma ^{m,n}_{\mathbf {A}, \mathbf {q}, \mathbf {p}} - \dim \Gamma ^{m,n}_{\mathbf {p}} \end{aligned}$$

(A.6)

$$\begin{aligned} \dim \Gamma ^{m,n}_{\mathbf {A}, {\bar{\mathbf {q}}}, \mathbf {p}}&= \dim \Gamma ^{m,n}_{\mathbf {A}, \mathbf {q}, \mathbf {p}} - \dim \Gamma ^{m,n}_{\mathbf {q}} = 11m \end{aligned}$$

(A.7)

$$\begin{aligned} \dim \Gamma ^{m,n}_{{\bar{\mathbf {A}}}, \mathbf {p}}&= \dim \Gamma ^{m,n}_{\mathbf {A}, \mathbf {p}} - \dim \Gamma ^{m,n}_{\mathbf {A}} = \min (2mn, 3n) \end{aligned}$$

(A.8)

$$\begin{aligned} \dim \Gamma ^{m,n}_{\mathbf {A}, {\bar{\mathbf {p}}}}&= \dim \Gamma ^{m,n}_{A, \mathbf {p}} - \dim \Gamma ^{m,n}_{\mathbf {p}} \end{aligned}$$

(A.9)

$$\begin{aligned} \dim \Gamma ^{m,n}_{\mathbf {q}, {\bar{\mathbf {p}}}}&= \dim \Gamma ^{m,n}_{\mathbf {q}, \mathbf {p}} - \dim \Gamma ^{m,n}_{\mathbf {p}} \end{aligned}$$

(A.10)

$$\begin{aligned} \dim \Gamma ^{m,n}_{{\bar{\mathbf {q}}} , \mathbf {p}}&= \dim \Gamma ^{m,n}_{\mathbf {q}, \mathbf {p}} - \dim \Gamma ^{m,n}_{\mathbf {q}} \end{aligned}$$

(A.11)

Remark A.2

For m and n sufficiently large, the formulas above involving $\min$ and $\max$ expressions can be simplified as follows:

$$\begin{aligned} \dim \Gamma ^{m,n}_{\mathbf {A},\mathbf {p}} = \dim \Gamma ^{m,n}_{\mathbf {q}, \mathbf {p}}&= 3n+11m \end{aligned}$$

(A.12)

$$\begin{aligned} \dim \Gamma ^{m,n}_{\mathbf {p}}&= 3n + 11m - 15 \end{aligned}$$

(A.13)

$$\begin{aligned} \dim \Gamma ^{m,n}_{{\bar{\mathbf {A}}}, \mathbf {q}, \mathbf {p}} = \dim \Gamma ^{m,n}_{{\bar{\mathbf {A}}}, \mathbf {p}}&= 3n\end{aligned}$$

(A.14)

$$\begin{aligned} \dim \Gamma ^{m,n}_{\mathbf {A}, \mathbf {q}, {\bar{\mathbf {p}}}} = \dim \Gamma ^{m,n}_{\mathbf {A}, {\bar{\mathbf {p}}}} = \dim \Gamma ^{m,n}_{\mathbf {q}, {\bar{\mathbf {p}}}}&= 15 \end{aligned}$$

(A.15)

$$\begin{aligned} \dim \Gamma ^{m,n}_{\mathbf {A}, {\bar{\mathbf {q}}}, \mathbf {p}} = \dim \Gamma ^{m,n}_{{\bar{\mathbf {q}}} , \mathbf {p}}&= 11m \end{aligned}$$

(A.16)

Proof

(A.1): This follows at once from the birational equivalence $\square$

$$\begin{aligned} \pi _\mathbf {p}: \Gamma _{\mathbf {A}, \mathbf {q}, \mathbf {p}}^{m,n}&\rightarrow (\mathbb {P}^{11})^m \times (\mathbb {P}^3)^n \\ ({\bar{\mathbf {A}}}, {\bar{\mathbf {q}}}, {\bar{\mathbf {p}}})&\mapsto ({\bar{\mathbf {A}}}, {\bar{\mathbf {q}}}) . \end{aligned}$$

(A.2): Consider the projection

$$\begin{aligned} \pi _\mathbf {q}: \Gamma _{\mathbf {A}, \mathbf {q}, \mathbf {p}}^{m,n}&\rightarrow (\mathbb {P}^{11})^m \times (\mathbb {P}^2)^{mn}\\ ({\bar{\mathbf {A}}}, {\bar{\mathbf {q}}}, {\bar{\mathbf {p}}})&\mapsto ({\bar{\mathbf {A}}}, {\bar{\mathbf {p}}}). \end{aligned}$$

A fiber $\pi _\mathbf {q}^{-1} ({\bar{\mathbf {A}}}, {\bar{\mathbf {p}}})$ can be identified with the projective linear space of all q satisfying $A_i q_j \sim p_{i j}$. Equivalently, each of the 2mn matrices $\left( \begin{matrix}A_i q_j&p_{i j}\end{matrix}\right)$ is of rank one. For generic $({\bar{\mathbf {A}}}, {\bar{\mathbf {p}}}) \in \Gamma ^{m,n}_{\mathbf {A},\mathbf {p}},$ the $2\times 2$ minors impose 2 linear conditions on ${\bar{\mathbf {q}}},$ so that $\pi _\mathbf {q}^{-1} ({\bar{\mathbf {A}}}, {\bar{\mathbf {p}}})$ is a projective linear space of dimension $\max (3n-2mn, 0).$ Hence, using (A.1) and the fiber-dimension theorem,

$$\begin{aligned} \dim \Gamma ^{m,n}_{\mathbf {A},\mathbf {p}}&= \dim \Gamma ^{m,n}_{\mathbf {A},\mathbf {q}, \mathbf {p}} - \max (3n-2mn, 0)\\&= \min (2mn + 11m, 3n + 11m). \end{aligned}$$

(A.3): Similar to (A.2), consider

$$\begin{aligned} \pi _\mathbf {A}: \Gamma ^{m,n}_{\mathbf {A},\mathbf {q}. \mathbf {p}}&\rightarrow (\mathbb {P}^{11})^m \times (\mathbb {P}^2)^{mn} \\ ({\bar{\mathbf {A}}}, {\bar{\mathbf {q}}}, {\bar{\mathbf {p}}})&\mapsto ({\bar{\mathbf {q}}}, {\bar{\mathbf {p}}}). \end{aligned}$$

For generic $({\bar{\mathbf {q}}}, {\bar{\mathbf {p}}}) \in \Gamma ^{m,n}_{\mathbf {q}, \mathbf {p}},$ the fibers $\pi _\mathbf {A}^{-1} ({\bar{\mathbf {q}}}, {\bar{\mathbf {p}}})$ are projective linear spaces of dimension $\max (11m - 2mn, 0)$ which are defined by the $\mathbf {A}$-linear $2\times 2$ minors of $\left( \begin{matrix}A_i q_j&p_{i j}\end{matrix}\right) .$ Hence,

$$\begin{aligned} \dim \Gamma ^{m,n}_{\mathbf {q},\mathbf {p}}&= \dim \Gamma ^{m,n}_{\mathbf {A},\mathbf {q}, \mathbf {p}} - \max (11m-2mn, 0)\\&= \min (3n + 2mn, 3n + 11m). \end{aligned}$$

(A.4): For generic $q_1, \ldots , q_5 \in \mathbb {P}^3,$ define $\sigma = \{ \sigma _1, \ldots , \sigma _4 \} \subset [5],$

$$\begin{aligned}^{-1} := \det \left( \begin{matrix}q_{\sigma _1}&q_{\sigma _2}&q_{\sigma _3}&q_{\sigma _4}\end{matrix}\right) ^{-1}, \end{aligned}$$

and consider the projective change of basis matrix

$$\begin{aligned} \begin{aligned} C_{q_1, \ldots , q_5} =&\, {{\,\mathrm{diag}\,}}\left( [5 2 3 4]^{-1}, \, [1 5 3 4]^{-1}, \, [1 2 5 4]^{-1}, \, [1 2 3 5]^{-1} \right) \cdot \left( \begin{matrix}q_1&q_2&q_3&q_4 \end{matrix}\right) ^{-1} \end{aligned} \end{aligned}$$

(A.17)

where $\bullet ^{-1}$ denotes matrix inversion. When defined, the projective transformation defined by $C_{q_1, \ldots , q_5}$ maps $q_1 \ldots q_5$ onto the standard projective basis:

$$\begin{aligned} C_{q_1, \ldots , q_5} \cdot q_1&\sim e_1, \\ C_{q_1, \ldots , q_5} \cdot q_2&\sim e_2, \\ C_{q_1, \ldots , q_5} \cdot q_3&\sim e_3, \\ C_{q_1, \ldots , q_5} \cdot q_4&\sim e_4, \\ C_{q_1, \ldots , q_5} \cdot q_5&\sim e_1 + e_2 + e_3 + e_4. \end{aligned}$$

Consider the projection

$$\begin{aligned} \pi _{\mathbf {A}, \mathbf {q}} : \Gamma _{\mathbf {A}, \mathbf {q}, \mathbf {p}}^{m,n}&\rightarrow \Gamma ^{m,n}_\mathbf {p}\\ ({\bar{\mathbf {A}}}, {\bar{\mathbf {q}}}, {\bar{\mathbf {p}}})&\mapsto {\bar{\mathbf {p}}} . \end{aligned}$$

We observe a version of projective ambiguity [23, p 265], stating that the fibers of $\pi _{\mathbf {A}, \mathbf {q}}$ are invariant under the action of ${{\,\mathrm{\mathrm PGL}\,}}_4$ described in Sect. 2.2. Suppose first that $n<6$. We need to show $\dim \Gamma ^{m,n}_\mathbf {p}= 2mn.$ Let ${\bar{\mathbf {p}}} \in \Gamma ^{m,n}_\mathbf {p}$ be generic and suppose $({\bar{\mathbf {A}}}, {\bar{\mathbf {q}}}, {\bar{\mathbf {p}}}) \in \pi _{\mathbf {A}, \mathbf {q}}^{-1} ({\bar{\mathbf {p}}}).$ Then for generic ${\bar{\mathbf {q}}} ' \in (\mathbb {P}^3)^n,$ we may find $({\bar{\mathbf {A}}}', {\bar{\mathbf {q}}} ' , {\bar{\mathbf {p}}} ) \in \pi _{\mathbf {A}, \mathbf {q}}^{-1} ({\bar{\mathbf {p}}})$ by projective change of basis $H = C_{{\tilde{\mathbf {q}}} '}^{-1} C_{{\tilde{\mathbf {q}}}},$ where ${\tilde{\mathbf {q}}}$ and ${\tilde{\mathbf {q}}}'$ extend ${\bar{\mathbf {q}}}$ and ${\bar{\mathbf {q}}} '$ to projective bases when $n<5.$ In other words, the generic fiber of $\Gamma ^{m,n}_{\mathbf {q}, \mathbf {p}} \rightarrow \Gamma ^{m,n}_\mathbf {p}$ has dimension 3n. Applying (A.1),

$$\begin{aligned} \dim \Gamma ^{m,n}_\mathbf {p}&= \dim \Gamma ^{m,n}_{\mathbf {q}, \mathbf {p}} - 3n = (3n + 2mn) - 3n = 2mn. \end{aligned}$$

Now suppose $n\ge 6$. For $m=1$ camera, (A.4) asserts that $\dim \Gamma ^{m,n}_\mathbf {p}= 2 n ,$ which follows since there are no constraints on image points. Otherwise, observe that the quantity

$$\begin{aligned} 2mn - (11m + 3n - 15)&= (2m-3) n - 11m + 15 = (2n - 11) m + 15 - 3n \end{aligned}$$

is increasing in n for fixed $m\ge 2$ and increasing in m for fixed $n\ge 6.$ Moreover, this quantity equals zero precisely in the minimal cases $(m,n)=(2,7), \, (3, 6).$ Thus, (A.4) asserts that $\dim \Gamma ^{m,n}_{\mathbf {p}} = 11m + 3n - 15$ whenever either $m\ge 2$ and $n\ge 7$ or $m\ge 3$ and $n\ge 6.$ This leaves one exceptional case for $n\ge 6,$ which is $(m,n) = (2,6)$; here, to show that $\dim \Gamma ^{m,n}_{\mathbf {p}} = 2mn = 24$, it suffices to verify that the Jacobian of $\pi _{\mathbf {A}, \mathbf {q}}$ evaluated at some point in local coordinates has rank 24. The same Jacobian check gives us $\dim \Gamma ^{m,n}_{\mathbf {p}} = 11m + 3n - 15$ for the two minimal cases; equivalently, $\dim \pi _{\mathbf {A}, \mathbf {q}}^{-1} ({\bar{\mathbf {p}}})= 15$ for generic ${\bar{\mathbf {p}}} \in \Gamma ^{m,n}_\mathbf {p}.$ Finally, if either $m\ge 2$ and $n\ge 8$ or $m\ge 3$ and $n\ge 7,$ note that the fiber $\pi _{\mathbf {A},\mathbf {q}}^{-1} ({\bar{\mathbf {p}}})$ for generic ${\bar{\mathbf {p}}} \in \Gamma ^{m,n}_\mathbf {p}$ is nonempty, and thus has dimension at least 15 by projective ambiguity. Since $\pi _{\mathbf {A},\mathbf {q}}^{-1} ({\bar{\mathbf {p}}})$ projects onto a fiber for one of the minimal cases, we also have $\dim \pi _{\mathbf {A}, \mathbf {q}}^{-1} ({\bar{\mathbf {p}}}) \le 15.$ Thus,

$$\begin{aligned} \dim \Gamma ^{m,n}_\mathbf {p}&= \dim \Gamma ^{m,n}_{\mathbf {A}, \mathbf {q}, \mathbf {p}} - \dim \dim \pi _{\mathbf {A}, \mathbf {q}}^{-1} ({\bar{\mathbf {p}}}) = 11m + 3n - 15. \end{aligned}$$

(A.5)–(A.11) In all cases, $\Gamma ^{m,n}_{{\bar{X}}, Y}$ is the generic fiber of $\Gamma ^{m,n}_{X ,Y} \rightarrow \Gamma ^{m,n}_{X},$ so these formulas follow from the fiber dimension theorem and (A.1)–(A.4). $\square$

Appendix B. Miscellaneous Proofs

1.1 B.1. Generic Cameras

Proposition B.1

If a camera arrangement ${\bar{\mathbf {A}}} = ({\bar{A}}_1,\ldots {\bar{A}}_m)$ is ultra minor generic then it is minor generic, and if ${\bar{\mathbf {A}}}$ is minor generic, then it has pairwise distinct centers.

Proof

If ${\bar{\mathbf {A}}}$ is ultra minor generic, then all $k \times k$ minors of $\left( \begin{array}{c|c|c}{{\bar{A}}_1}^\top&\cdots&{{\bar{A}}_m}^{\top {}}\end{array}\right)$ are nonzero for any $k \in [4]$. In particular, all $4 \times 4$ minors are nonzero and ${\bar{\mathbf {A}}}$ is minor generic. If ${\bar{\mathbf {A}}}$ is minor generic, then for any $1 \le i < j \le m$, the $4\times 6$ matrix $\left( \begin{array}{c|c}{{\bar{A}}_i}^\top&{{\bar{A}}_j}^\top \end{array} \right)$ has rank 4. This implies that ${\bar{A}}_i$ and ${\bar{A}}_j$ have district centers. $\square$

Theorem B.2

• (1) A camera arrangement ${\bar{\mathbf {A}}}$ has pairwise distinct centers if and only if it is equivalent to a minor generic camera arrangement under the group action (2.1).

• (2) A camera arrangement ${\bar{\mathbf {A}}}$ is minor generic if and only if it is equivalent to an ultra minor generic arrangement under the group action (2.2).

• (3) A camera arrangement ${\bar{\mathbf {A}}}$ has pairwise distinct centers if and only if it is equivalent to an ultra minor generic camera arrangement under the group action (2.3).

Proof

• (1) This statement was proved in [2, Lemma 3.6].

• (2) We already saw in Proposition B.1 that ultra minor genericity implies minor genericity. For the other direction, fix a minor generic arrangement $({\bar{A}}_1,\ldots {\bar{A}}_m)$. Let $\sigma \in \left( {\begin{array}{c}[4]\\ k\end{array}}\right)$ and $\tau \in \left( {\begin{array}{c}[3m]\\ k\end{array}}\right)$ be subsets indexing the rows and columns of some $k\times k$ minor of $\left( \begin{array}{c|c|c}{{\bar{A}}_1}^\top&\cdots&{{\bar{A}}_m}^{\top {}}\end{array}\right)$. Using the Cauchy-Binet theorem,

  $$\begin{aligned} \begin{aligned}&\det \left( \left( \begin{array}{c|c|c}{({\bar{A}}_1 H)}^\top&\cdots&{({\bar{A}}_m H)}^{\top {}}\end{array}\right) [\sigma , \tau ] \right) \\&\quad = \displaystyle \sum _{\upsilon \in \left( {\begin{array}{c}[4]\\ k\end{array}}\right) } \det \left( H^\top [\sigma , \upsilon ] \right) \cdot \det \left( \left( \begin{array}{c|c|c}{{\bar{A}}_1}^\top&\cdots&{{\bar{A}}_m}^{\top {}}\end{array}\right) [\upsilon ,\tau ] \right) . \end{aligned} \end{aligned}$$

  (B.1)

  The minors $\det (\left( \begin{array}{c|c|c}{{\bar{A}}_1}^\top&\cdots&{{\bar{A}}_m}^{\top {}}\end{array}\right) [\upsilon ,\tau ])$ which occur in this sum range over all $k\times k$ minors of the $4 \times k$ matrix $\left( \begin{array}{c|c|c}{{\bar{A}}_1}^\top&\cdots&{{\bar{A}}_m}^{\top {}}\end{array}\right) [[4], \tau ].$ This $4\times k$ matrix has full rank k since if it did not, we could add $4-k$ additional columns from $\left( \begin{array}{c|c|c}{{\bar{A}}_1}^\top&\cdots&{{\bar{A}}_m}^{\top {}}\end{array}\right)$ to get a rank-deficient $4\times 4$ matrix, contradicting our assumption that $\left( \begin{array}{c|c|c}{{\bar{A}}_1}^\top&\cdots&{{\bar{A}}_m}^{\top {}}\end{array}\right)$ is minor generic. Thus, $\det \left( \begin{array}{c|c|c}{{\bar{A}}_1}^\top&\cdots&{{\bar{A}}_m}^{\top {}}\end{array}\right) [\upsilon , \tau ] \ne 0$ for some $\upsilon \in \left( {\begin{array}{c}[4]\\ k\end{array}}\right) .$ Hence, the expressions in (B.1) are not zero, and setting  (B.1) to 0 we obtain a hypersurface in $\text {GL}_4$. Any choice of H lying outside the union of the finitely many hypersurfaces, obtained by varying over all $k, \sigma , \tau$ yields an arrangement $(A_1 H, \ldots , A_m H)$ satisfying the conclusion.

• (3) This statement follows from the first two. $\square$

1.2 B.2. Proof of Proposition 2.2, Part 1

Proof

Let $I = \langle g_1, \ldots , g_s \rangle .$ By definition, $g_1, \ldots ,g_s$ forming a Gröbner basis with respect to < means that

$$\begin{aligned} in_< (I) = \langle in_< (g_1), \ldots , in_< (g_s) \rangle .\end{aligned}$$

First, we verify that this monomial ideal in $R[x_1,\ldots , x_k]$ is radical. To see this, let $in_<(g_s) = x_{i_1} \cdots x_{i_l}$ and note that

$$\begin{aligned} in_< (I) = \displaystyle \bigcap _{j=1}^l \langle in_< (g_1) , \ldots , in_< (g_{s-1}), x_{i_j} \rangle . \end{aligned}$$

Iterating this argument, we obtain $in_< (I)$ as an intersection of prime ideals generated by subsets of the variables.

Now, to show that $I = \langle g_1, \ldots , g_s \rangle$ is radical, suppose that $f \in \sqrt{I},$ so that $f^n \in I$ for some positive integer n. We need to argue that $f \in I$. We have

$$\begin{aligned} in_<(f)^n = in_< (f^n) \in in_< (I) \quad \Rightarrow \quad in_< (f) \in in_< (I), \end{aligned}$$

where the first equality of leading terms uses the fact that R is a domain and the implication holds since $in_<(I)$ is radical. Thus, there exists $f_0 \in I \subseteq \sqrt{I}$ such that $in_< (f_0) = in_< (f).$ Now $f-f_0 \in \sqrt{I}$ is an element whose leading term is strictly smaller than $in_<(f).$ Replacing f with $f-f_0$ and iterating the argument, we obtain $f_0, \ldots , f_l \in I$ such that $f = f_0 + \cdots + f_l \in I$. $\square$

1.3 B.3. Proof of the Recognition Criterion: Proposition 2.3

Proof

A point $x \in \mathbb {P}^{n_1} \times \cdots \times \mathbb {P}^{n_k}$ may be represented in homogeneous coordinates by a point ${\hat{x}}$ in the affine space $\mathbb {C}^{n_1+1} \times \cdots \times \mathbb {C}^{n_k+1}.$ Consider the affine cone

$$\begin{aligned} {\hat{X}} = {{\,\mathrm{cl}\,}}\{ {\hat{x}} \in \mathbb {C}^{n_1+1} \times \cdots \times \mathbb {C}^{n_k+1} \text { s.t. }x \in X \}. \end{aligned}$$

This is an affine variety whose vanishing ideal is precisely the vanishing ideal of X.

Suppose that Conditions 1–3 are satisfied; we must show that $\langle f_1, \ldots , f_s \rangle$ is the vanishing ideal of X,  or equivalently that of ${\hat{X}}.$ Condition 3 and the Nullstellensatz [15, Ch. 4, §2] imply that $\langle f_1, \ldots , f_s \rangle$ is the vanishing ideal of the affine variety

$$\begin{aligned} \mathrm {V}_a(f_1, \ldots , f_s) = \{ {\hat{x}} \in \mathbb {C}^{n_1+1} \times \cdots \times \mathbb {C}^{n_k+1} \text { s.t. }f_1({\hat{x}})=\cdots = f_s({\hat{x}}) \}. \end{aligned}$$

Moreover, Conditions 2 and 3 together with standard properties of ideal quotients and saturation [15, Ch. 4, §4] imply $\langle f_1, \ldots , f_s \rangle$ is the vanishing ideal of

$$\begin{aligned} {{\,\mathrm{cl}\,}}\left( \mathrm {V}_a(f_1, \ldots , f_s) \setminus \mathrm {V}_a( \mathfrak {m}_{\mathbf {x}_1} \cap \cdots \cap \mathfrak {m}_{\mathbf {x}_k})\right) . \end{aligned}$$

Since affine varieties are uniquely determined by their vanishing ideals, it is now enough to observe the following equality, which holds whenever Condition 1 is satisfied:

$$\begin{aligned} {\hat{X}} = {{\,\mathrm{cl}\,}}\left( \mathrm {V}_a(f_1, \ldots , f_s) \setminus \mathrm {V}_a( \mathfrak {m}_{\mathbf {x}_1} \cap \cdots \cap \mathfrak {m}_{\mathbf {x}_k}) \right) . \end{aligned}$$

Conversely, we verify Conditions 1–3 when $\langle f_1, \ldots , f_s \rangle$ is the vanishing ideal of X:

• (1) $X \subset \mathrm {V}(f_1, \ldots , f_s)$ since each $f_i$ vanishes on all points of X. On the other hand, we have $X = \mathrm {V}(g_1, \ldots , g_s)$ for certain homogeneous polynomials $g_1, \ldots , g_s,$ all of which must be contained in $\langle f_1, \ldots , f_s \rangle$. If $f_1,\ldots , f_s$ vanish at a point, so must $g_1, \ldots , g_s$, and thus $\mathrm {V}(f_1, \ldots , f_s) \subset X.$

• (2) Let $f \in \langle f_1, \ldots , f_s \rangle : (\mathfrak {m}_{\mathbf {x}_1} \cap \cdots \cap \mathfrak {m}_{\mathbf {x}_k})^\infty .$ To show $f \in \langle f_1, \ldots , f_s \rangle ,$ it is enough to show that each of the homogeneous components of f vanish on X,  so suppose further that f is homogeneous. If X is empty, then f vanishing on X holds vacuously. Otherwise, for any point in X there exists some monomial of the form

  $$\begin{aligned} m(x) = x_{1, i_1} \, \cdots \, x_{k, i_k} \in \mathfrak {m}_{\mathbf {x}_1} \cap \cdots \cap \mathfrak {m}_{\mathbf {x}_k} \end{aligned}$$

  which does not vanish at that point. Since $m(x)^n f$ is in the vanishing ideal for some $n\ge 1,$ we see that that f must vanish at this point.

• (3) If $f^n (x) =0$ for some $n\ge 1$ and all $x\in X$, then $f(x) =0$ for all $x\in X.$

$\square$

Appendix C. Generators of $G_{{M^{m,1}_{\mathbf {A}, \mathbf {q}, \mathbf {p}}}}$

The Gröbner basis $G_{M_{\mathbf {A},\mathbf {q},\mathbf {p}}^{m,1}}$ of Proposition 5.2 contains elements in degree $3,4,5,6,7,8,9.$ Here, for completeness, we give explicit formulas for all of them.

Degree 3 3m elements—for $1 \le i \le m ,$ $1\le i_1 < i_2 \le 3, \, \det \Big ( \left( \begin{matrix}A_i q&p_i \end{matrix}\right) [ \{i_1, i_2 \} , :] \Big ).$

Degree 4 m elements—for $1 \le i \le m , \, \det \left( \begin{matrix}A_i[:,1]&A_i q&p_i \end{matrix}\right) .$

Degree 5 $9\, {m \atopwithdelims ()2}$ elements—for $1 \le i < j \le m ,$ $1 \le i_1 < i_2 \le 3,$ $1\le j_1 < j_2 \le 3,$

$$\begin{aligned}&\det \Big ( \left( \begin{matrix}A_i[:,1] \, p_i\end{matrix}\right) [\{i_1, i_2\} , :] \Big ) \cdot \det \Big ( \left( \begin{matrix}A_j q p_j\end{matrix}\right) [ \{j_1, j_2 \} , : ] \Big ) \\&\quad -\, \det \Big ( \left( \begin{matrix}A_j[:,1]\, p_j\end{matrix}\right) [\{j_1, j_2\}, :] \Big ) \cdot \det \Big ( \left( \begin{matrix}A_i q \, p_i\end{matrix}\right) [\{i_1, i_2\}, :] \Big ). \end{aligned}$$

Degree 6 $6\, {m \atopwithdelims ()2}$ elements—for $1 \le k_1 < k_2 \le 3,$ $1\le i , j \le m,$ $i\ne j,$

$$\begin{aligned} \sum _{l=1}^3 (-1) \cdot&\det \Big (\left( \begin{matrix}A_j q&p_j \end{matrix}\right) [\{ 1,2,3 \} \setminus l , :] \Big ) \, \cdot \Bigg ( p_i[k_1] \det \left( \begin{matrix}A_i [k_2, 1:2] A_j [l, 1:2] \end{matrix}\right) \\&- p_i[k_2] \det \left( \begin{matrix}A_i [k_1, 1:2] \\ A_j [l,1:2] \end{matrix}\right) \Bigg ) \\&+ \det \left( \begin{matrix}A_j[:,1]&A_j[:,2]&p_j \end{matrix}\right) \cdot \det \Big (\left( \begin{matrix}A_i q&p_i\end{matrix}\right) [ \{k_1, k_2\}, :] \Big ) . \end{aligned}$$

Degree 7 $\left( {\begin{array}{c}m\\ 2\end{array}}\right)$ elements of the form $q_4$ times a 2-focal, plus an additional $27 \left( {\begin{array}{c}m\\ 3\end{array}}\right)$ elements—for $1\le i<j < k \le m,$ $1\le i_1 < i_2\le 3,$ $1\le j_1 < j_2 \le 3,$ $1\le k_1 < k_2 \le 3,$

$$\begin{aligned}&\det \Big ( \left( \begin{matrix}A_i q p_i \end{matrix}\right) [\{i_1, i_2\}, :] \Big ) \cdot \Bigg ( p_j[j_1] p_k[k_1] \cdot \det \left( \begin{matrix}A_j [j_2, 1:2] \\ A_k [k_2, 1:2] \end{matrix}\right) \\&\quad - p_j[j_1] p_k[k_2] \cdot \det \left( \begin{matrix}A_j [j_2, 1:2]\\ A_k [k_1, 1:2] \end{matrix}\right) \\&\quad -p_j[j_2] p_k[k_1] \cdot \det \left( \begin{matrix}A_j [j_1, 1:2] \\ A_k [k_2, 1:2] \end{matrix}\right) \\&\quad + p_j[j_2] p_k[k_2] \cdot \det \left( \begin{matrix}A_j [j_1, 1:2]\\ A_k [k_1, 1:2] \end{matrix}\right) \Bigg )\\&-\det \Big ( \left( \begin{matrix}A_j q&p_j \end{matrix}\right) [\{j_1, j_2\}, :] \Big ) \cdot \Bigg ( p_i[i_1] p_k[k_1] \cdot \det \left( \begin{matrix}A_i [i_2, 1:2] \\ A_k [k_2, 1:2] \end{matrix}\right) \\&\quad - p_i[i_1] p_k[k_2] \cdot \det \left( \begin{matrix}A_i [i_2, 1:2] \\ A_k [k_2, 1:2] \end{matrix}\right) \\&\quad -p_i[i_2] p_k[k_1] \cdot \det \left( \begin{matrix}A_i [i_2, 1:2] \\ A_k [k_2, 1:2] \end{matrix}\right) \\&\quad + p_i[i_2] p_k[k_2] \cdot \det \left( \begin{matrix}A_i [i_1, 1:2] \\ A_k [k_1, 1:2] \end{matrix}\right) \Bigg ) \\&+\det \Big (\left( \begin{matrix}A_k q&p_k \end{matrix}\right) [\{k_1, k_2\}, :] \Big ) \cdot \Bigg ( p_i[i_1] p_j[j_1] \cdot \det \left( \begin{matrix}A_i [i_2, 1:2] \\ A_j [j_2, 1:2] \end{matrix}\right) \\&\quad - p_i[i_1] p_j[j_2] \cdot \det \left( \begin{matrix}A_i [i_2, 1:2] \\ A_j [j_1, 1:2] \end{matrix}\right) \\&\quad - p_i[i_2] p_j[j_1] \cdot \det \left( \begin{matrix}A_i [i_1, 1:2] \\ A_j [j_2, 1:2] \end{matrix}\right) \\&\quad + p_i[i_2] p_j[j_2] \cdot \det \left( \begin{matrix}A_i [i_1, 1:2]\\ A_j [j_1, 1:2] \end{matrix}\right) \Bigg ) \end{aligned}$$

Degrees 8 & 9 $q_4$ times $27 \left( {\begin{array}{c}m\\ 3\end{array}}\right)$ 3-focals and $q_4$ times $81 \left( {\begin{array}{c}m\\ 4\end{array}}\right)$ 4-focals, respectively.