The subspace \(n_1^\perp \cap n_2^\perp \) has dimension \(d - 1\), as \(n_1\) and \(n_2\) are linearly independent. Let \(q_1, \dots , q_{d-1}\) be a basis of it. By lemma 22.9, \(q_1 \vee \cdots \vee q_{d-1} = c\, C(n_1, n_2)\) for some \(c \in K\). The vectors \(q_i\) are linearly independent, so their join is nonzero (the remark after definition 18.3), and \(c \neq 0\). Take \(x_1 = c^{-1} q_1\) and \(x_i = q_i\) for \(2 \le i \le d - 1\).

By definition, \(C(a_i, b_i) = *(a_i \vee b_i)\) (definition 23.1), and the complement isomorphism \(*\) (definition 22.5) is a linear isomorphism. A linear isomorphism preserves and reflects linear independence.

Let \(e\) have ends \(u \neq v\). Then \(n_u\) and \(n_v\) are linearly independent, so \(C(p(e)) = C(n_u, n_v) \neq 0\) (the remark after definition 23.1).

Let \(u \neq v\). The first two coordinates of \(n_u\) and \(n_v\) form the matrix with rows \((1, t_u)\) and \((1, t_v)\), whose determinant \(t_v - t_u\) is not zero. So \(n_u\) and \(n_v\) are linearly independent. If \(d \ge 2\), then \(\hat n_v = (1, t_v, \dots , t_v^{d-1})\) is the point of the moment curve in \(K^d\) at \(t_v\), and the same argument shows that \(\hat n_u\) and \(\hat n_v\) are linearly independent. An infinite field has \(|V|\) distinct elements for every finite \(V\).

The hinge at an edge with ends \(u\) and \(w\) is \(C(n_u, n_w)\) (definition 23.5). It depends only on the normals at \(u\) and \(w\), so it does not change when \(u, w \neq v\). If \(v\) lies on no edge of \(G\), then no hinge changes, and so no hinge constraint changes (definition 19.3). So the space of infinitesimal motions does not change, nor does its subspace \(\operatorname {pinned}(w)\) of the motions \(S\) with \(S(w) = 0\).

Each edge of \(G'\) is an edge of \(G\) with the same ends, and it has the same hinge in \((G',p)\) as in \((G,p)\) (definition 23.13). So every infinitesimal motion of \((G,p)\) satisfies the hinge constraint at every edge of \(G'\) (definition 19.3), and \(Z(G,p) \subseteq Z(G',p)\). The rank form follows, as \(\operatorname {rank} R(G,p) = D|V| - \dim Z(G,p)\) (definition 19.7).

Let \(k = \operatorname {def}(\tilde G)\). Since \(\operatorname {rank} R(G,p) = D|V| - \dim Z(G,p)\) (definition 19.7), the claim is that \(\dim Z(G,p) = D + k\). The hypothesis is the inequality \(\dim Z(G,p) \le D + k\). The reverse inequality holds as the hinges at the edges whose ends differ are nondegenerate (lemma 20.15).

Let \((G,p)\) be a generic full-rank realization with normals \(n_v\). It is at the target rank. Each normal \(n_v\) is nonzero, since \(|V| \ge 2\) gives a body \(w \neq v\), and \(n_v\) and \(n_w\) are linearly independent. Let \(e\) be an edge with ends \(u \neq v\). Its hinge \(C(n_u, n_v)\) is nonzero (lemma 23.11). It is the meet of the hyperplanes \(n_u^\perp \) and \(n_v^\perp \), so it is \(x_1 \vee \cdots \vee x_{d-1}\) for a basis \(x_1, \dots , x_{d-1}\) of \(n_u^\perp \cap n_v^\perp \) (lemma 23.3 and the remark after it). So it lies in both panels. A loop at a body \(u\) gives no row of \(R(G,p)\), since its constraint involves only \(S(u) - S(u) = 0\). So we may replace its hinge by \(C(n_u, n_w)\) for a body \(w \neq u\), without changing the rank. As \(n_u\) and \(n_w\) are linearly independent, this hinge is nonzero (the remark after definition 23.1), and as above it lies in the panel of \(u\).

Let \((G, q_0)\) be a generic full-rank realization. As \(G\) has no loops, every edge has two distinct ends, so every hinge of \((G, q_0)\) is nondegenerate (lemma 23.11). By lemma 23.120 there is a polynomial \(Q_1\) in the normals with \(Q_1(q_0) \neq 0\) such that \(\operatorname {rank} R(G,q) \ge \operatorname {rank} R(G,q_0)\) at every \(q\) with \(Q_1(q) \neq 0\). Let \(Q_2\) be the product, over the ordered pairs \((u, v)\) of distinct bodies, of the \(2 \times 2\) minor of the first two coordinates of the normals \(n_u\) and \(n_v\). As \(d \ge 2\), these are coordinates of \(\hat n_u\) and \(\hat n_v\), so the panels of \((G, q)\) are nonparallel wherever \(Q_2(q) \neq 0\). At moment-curve normals with distinct parameters \(t_v\), each factor is \(t_v - t_u \neq 0\) (lemma 23.12), and an infinite field has distinct parameters for the finitely many bodies, so \(Q_2 \neq 0\). As \(K\) is infinite, the nonzero polynomial \(Q_1 Q_2\) does not vanish at some \(q\).

The panels of \((G, q)\) are nonparallel, so its normals are in general position (definition 23.10) and its hinges are nondegenerate (lemma 23.11). Its rank is at least that of \((G, q_0)\), the target rank, and at most the target rank (lemma 20.16). So \((G, q)\) is a nonparallel full-rank realization.

Part (2) of lemma 20.27 (KT Lemmas 3.3 and 3.6) applies, as \(G\) is not \(2\)-edge-connected. It gives a set \(V_1\) of vertices with \(\emptyset \neq V_1 \neq V(G)\) and the following properties. Let \(V_2 = V(G) \setminus V_1\). At most one edge joins \(V_1\) to \(V_2\); let \(C\) be the set of such edges. Then \(G[V_1]\) and \(G[V_2]\) are minimal \(k_1\)- and \(k_2\)-dof-graphs, with \(k = k_1 + k_2 + D - (D-1)\, |C|\). Both have fewer vertices than \(G\). So by hypothesis they have nondegenerate-hinge panel realizations \((G[V_1], p_1)\) and \((G[V_2], p_2)\), with \(\operatorname {rank} R(G[V_i], p_i) = D(|V_i| - 1) - k_i\).

Let \(p\) have the normals of \(p_i\) at the bodies of \(V_i\), and the hinges of \(p_i\) at the edges of \(G[V_i]\). Suppose that \(C = \{ e\} \), where \(e\) joins \(u \in V_1\) to \(v \in V_2\). The subspace \(n_u^\perp \cap n_v^\perp \) of \(K^{d+1}\) has dimension at least \(d - 1\), so it contains linearly independent vectors \(x_1, \dots , x_{d-1}\). Let \(p\) have the hinge \(x_1 \vee \cdots \vee x_{d-1} \neq 0\) (definition 18.3) at \(e\). Katoh and Tanigawa take the intersection of the two panels as this hinge, after they move \(p_2\) so that the panels are not parallel. Here any nonzero extensor of vectors in both panels serves. So \((G,p)\) has nonzero normals, and each of its hinges is nonzero and lies in the panels of the ends of its edge.

The hinges of \(p\) at the edges of \(G[V_i]\) are those of \(p_i\), so \(\operatorname {rank} R(G[V_i], p) = D(|V_i| - 1) - k_i\). By lemma 19.24,

by the formula for \(k\) and as \(|V(G)| = |V_1| + |V_2|\). As every hinge of \(p\) is nonzero and \(\operatorname {def}(\tilde G) = k\), lemma 20.16 gives \(\operatorname {rank} R(G,p) \le D(|V(G)| - 1) - k\).

Let \(V_1\), \(V_2\), \(C\), \(k_1\) and \(k_2\) be as in the proof of lemma 23.27. As \(G\) is simple, so are \(G[V_1]\) and \(G[V_2]\). By hypothesis they have generic full-rank realizations, with normals \(q_1\) and \(q_2\). For \(i = 1, 2\), there is a polynomial \(Q_i\) in the normals with \(Q_i(q_i) \neq 0\) such that \(\operatorname {rank} R(G[V_i], q) \ge D(|V_i| - 1) - k_i\) at all normals \(q\) with \(Q_i(q) \neq 0\) (lemma 23.120). Let \(Q_0\) be the product, over the pairs of distinct bodies \(u\) and \(v\), of the \(2 \times 2\) minor of the first two coordinates of \(n_u\) and \(n_v\). Where \(Q_0\) does not vanish, the normals are pairwise linearly independent. At normals on the moment curve each of these minors is a nonzero Vandermonde determinant (lemma 23.12), so \(Q_0\) is not the zero polynomial. Hence the product \(Q_1 Q_2 Q_0\) of three nonzero polynomials is not the zero polynomial. Since \(K\) is infinite, there are normals \(q_0\) at which it does not vanish. This is the argument by which nonempty Zariski-open sets of normals meet in the proof of lemma 23.112. Katoh and Tanigawa instead apply an isometry to one of the two realizations, which keeps its rank, so that no panel of the one is parallel to a panel of the other (KT p. 672).

Let \((G, q_0)\) be the panel-hinge framework on \(G\) with the normals \(q_0\), whose hinge at each edge is the intersection of the panels of its ends (definition 23.5). Its normals are pairwise linearly independent, so its hinges are nonzero (lemma 23.11), and \(\operatorname {rank} R(G[V_i], q_0) \ge D(|V_i| - 1) - k_i\) for \(i = 1, 2\). As in the proof of lemma 23.27, lemma 19.24 and lemma 20.16 give \(\operatorname {rank} R(G, q_0) = D(|V(G)| - 1) - k\).

An infinitesimal motion satisfies the hinge constraints at \(e_1\) and \(e_2\), so this is KT Lemma 5.3 (lemma 19.21).

The subspace \(n^{\perp }\) has dimension at least \(3\), so it contains linearly independent vectors \(a\), \(b\) and \(c\). Extend them by a vector \(w\) to a basis of \(K^4\), and take \(p_0 = a\), \(p_1 = q_0 = b\) and \(q_1 = c\). Then \(a \vee b\) and \(b \vee c\) are two of the extensors of KT Lemma 2.1 for this basis: the first omits \(c\) and \(w\), and the second omits \(a\) and \(w\). So they are linearly independent (lemma 18.9 and the remark after it).

By lemma 20.39, \(\operatorname {def}(\tilde G) = k = D(|V| - 1)\), so the target rank \(D(|V| - 1) - \operatorname {def}(\tilde G)\) is \(0\). As \(G\) has no edge, \(R(G,p)\) has no rows, and every realization of \(G\) is at the target rank. For the first realization, give every body the same nonzero normal; there is no hinge. For the second, take normals in general position, which exist as \(K\) is infinite (lemma 23.12).

Let the edge \(e\) of \(G\) join \(x\) and \(y\). As \(\operatorname {def}(\tilde G) = 1\), the target rank is \(D(2 - 1) - 1 = D - 1\). The rows of \(R(G,p)\) are the \(D - 1\) rows of \(e\), and they are linearly independent when the hinge at \(e\) is nondegenerate (lemma 19.9). So each realization needs only a nondegenerate hinge at \(e\). For the first, give \(x\) and \(y\) the same nonzero normal \(n\), and give \(e\) the extensor \(v_1 \vee \cdots \vee v_{d-1}\) of the paragraph after lemma 23.31. It lies in the panel with normal \(n\), and it is nonzero, as \(v_1, \dots , v_{d-1}\) are linearly independent. For the second, take normals in general position (lemma 23.12). Then the hinge \(C(n_x, n_y)\) at \(e\) is nonzero (lemma 23.11).

We follow Katoh and Tanigawa’s proof of their Lemma 5.3 (KT p. 670). Let the edges \(e\) and \(f\) of \(G\) join \(x\) and \(y\). Give \(x\) and \(y\) the same nonzero normal \(n\), so that their panels coincide. Give \(e\) and \(f\) two linearly independent extensors that lie in the panel with normal \(n\) (the form of lemma 23.31 for \(K^{d+1}\)). Both are nonzero, and each lies in the panels of \(x\) and \(y\). By lemma 23.30, the framework is infinitesimally rigid. So its space of motions is the \(D\)-dimensional space of trivial motions (lemma 19.18), and its rank is \(2D - D = D\) (definition 19.7). As \(\operatorname {def}(\tilde G) = 0\), this is the target rank.

By lemma 20.39, \(G\) is a graph of lemma 23.32, lemma 23.33 or lemma 23.34, and that lemma gives the realizations. In the last case the two edges of \(G\) join the same two vertices, so \(G\) is not simple.

For a multigraph \(H\), let \(P(H)\) say that \(H\) has a nondegenerate-hinge panel realization, and also a generic full-rank realization if \(H\) is simple. We prove \(P(G)\) for every minimal \(k\)-dof-graph \(G\) with at least one vertex, for all \(k\) at once, by induction on \(|V(G)|\) (theorem 21.56). The induction hypothesis is KT eq. (6.1): \(P(H)\) holds for every minimal \(k'\)-dof-graph \(H\) with at least one and fewer than \(|V(G)|\) vertices, for every \(k'\).

The base. If \(G\) has one or two vertices, then \(P(G)\) holds (lemma 23.36). A minimal \(k\)-dof-graph with two vertices has no edge, one edge, or two parallel edges, and the last case is KT Lemma 5.3 (lemma 23.30).

The four cases. Let \(|V(G)| \ge 3\). We follow Katoh and Tanigawa’s four cases (KT p. 671). If \(G\) is not \(2\)-edge-connected, then KT Lemma 6.1 gives \(P(G)\). Lemma 23.27 gives the nondegenerate-hinge realization, and lemma 23.28 gives the generic one when \(G\) is simple. If \(G\) has a proper rigid subgraph, then Case I gives \(P(G)\) (KT §6.2; lemma 23.109). It uses KT Lemma 6.2 when \(G\) is not simple, KT Lemma 6.3 when \(G\) is simple and has a proper rigid subgraph \(H\) with \(G/E(H)\) simple (lemma 23.100), and KT Lemma 6.5 otherwise.

In the remaining two cases \(G\) is \(2\)-edge-connected and has no proper rigid subgraph, so \(G\) is simple (lemma 21.57). If \(k {\gt} 0\), then Case II gives a generic full-rank realization of \(G\) (KT Lemma 6.8; lemma 23.135). If \(k = 0\), then Case III does (KT Lemmas 6.10 and 6.13; lemma 23.208). In both cases this realization is also a nondegenerate-hinge one (lemma 23.25), so \(P(G)\) holds.

The nonparallel realization. If \(G\) is simple, it has no loops, and \(V(G) \neq \emptyset \), so the generic full-rank realization that \(P(G)\) gives for it yields a nonparallel full-rank realization (lemma 23.26). This is the realization that the theorem states.

The deficiencies. The induction hypothesis is used at deficiencies other than \(k\), even when \(k = 0\). KT Lemma 6.1 applies it to the two sides of a cut of at most one edge, at deficiencies smaller than \(k\). Case I applies it to a proper rigid subgraph \(H\), a minimal \(0\)-dof-graph, and to the contraction \(G/E(H)\), a minimal \(k\)-dof-graph. In Case II, splitting off a vertex of degree two gives a minimal \((k-1)\)-dof-graph. Case III applies it to a minimal \(k'\)-dof-graph with \(k' \le D - 2\) (KT Claim 6.11, eq. (6.22); lemma 23.140). For general \(d\), Case III also distinguishes whether \(G\) is a cycle on at most \(d\) vertices or has a chain of length \(d\) (KT Lemma 6.13; lemma 21.68). The case \(d = 3\), \(k = 0\) of the theorem is theorem 23.45.

The bodies and the pairs of adjacent bodies are unchanged (vertexSet_embedEdges, embedEdges_isLink): the edge \(\iota (e)\) joins the bodies that \(e\) joins. As \(\iota \) is injective, \(G'\) has a loop, or two edges joining one pair of bodies, only if \(G\) does. A partition crosses exactly the images of the edges it crossed before (edgeSet_embedEdges), so it crosses as many edges and keeps its deficiency, and the largest of these values is unchanged.

Deleting an edge of a minimal \(k\)-dof-graph raises the deficiency: a base of \(M(\tilde G)\) of the deleted graph is independent in \(M(\tilde G)\) (lemma 20.24), it has the size of a base of \(M(\tilde G)\) if the deficiency stayed the same (KT eq. (2.4), theorem 20.12), so it is a base, and it meets the copies of the deleted edge by minimality, a contradiction. The same count shows that a \(k\)-dof-graph whose every edge deletion raises the deficiency is minimal. Relabelling commutes with deleting an edge and keeps every deficiency (lemma 23.42), so \(G'\) is minimal. The padded label set has more than \(D(|W| - 1)\) labels, and remark 23.40 applies.

The edges of \(G'\) are the edges \(\iota (e)\), for the edges \(e\) of \(G\), and \(\iota (e)\) joins the bodies that \(e\) joins. So the rows of the rigidity matrix at \(\iota (e)\) and at \(e\) are the rows of one hinge between the same two bodies (definition 19.7), and a motion satisfies the hinge constraint of \(\iota (e)\) exactly when it satisfies that of \(e\) (definition 19.3). In a panel-hinge framework the hinge at an edge is the meet of the panels of its ends, and the ends of \(\iota (e)\) are those of \(e\).

This is theorem 23.38 at \(d = 3\) and \(k = 0\).

By theorem 23.45, \(G\) has a nonparallel full-rank realization \((G,p)\), of rank \(D(|V| - 1)\). So \(\dim Z(G,p) = D|V| - \operatorname {rank} R(G,p) = D\) (definition 19.7). As \(\operatorname {def}(\tilde G) = 0\), this is the rank formula at \((G,p)\), in the form of proposition 23.19.

We follow Katoh and Tanigawa’s proof (KT p. 670). Let \(c = \operatorname {def}(\tilde G)\).

The realization. By lemma 20.34, \(G\) has a spanning subgraph \(G'\) that is a minimal \(c\)-dof-graph. It is simple, as a subgraph of \(G\), and it has the \(|V| \ge 2\) vertices of \(G\). By KT Theorem 5.5 (theorem 23.38), \(G'\) has a nonparallel full-rank realization \((G', q)\) (definition 23.21), with \(\operatorname {rank} R(G', q) = D(|V| - 1) - c\). Let \((G, p)\) be the panel-hinge framework on \(G\) with the same normals \(n_v\). Its panels are those of \((G', q)\), so they are nonparallel, and at each edge of \(G'\) it has the hinge of \((G', q)\). Katoh and Tanigawa first apply a projective transformation so that the panels of any two bodies meet in a \((d-2)\)-dimensional affine subspace. Infinitesimal rigidity is invariant under such a transformation. They then give each edge of \(G\) outside \(G'\) a hinge in the intersection of the panels of its ends. Here no transformation is needed, as the panels are nonparallel already. So the normals are in general position (definition 23.10), and each edge of \(G\) has two distinct ends, as \(G\) is simple. So the hinge \(C(n_u, n_v)\) of every edge \(uv\) of \(G\) is nonzero (lemma 23.11).

The rank. The restriction of \((G, p)\) to the edges of \(G'\) is \((G', q)\), so \(Z(G, p) \subseteq Z(G', q)\) (lemma 23.18). Hence

The hinges of \((G, p)\) are nondegenerate, so the upper bound on the rank of lemma 20.15, which proposition 23.19 uses, holds at \((G, p)\), and \(\operatorname {rank} R(G, p) \le D(|V| - 1) - c\). So \((G, p)\) is at the target rank.

Let \(c = \operatorname {def}(\tilde G)\). If \(|V| = 1\), then \(G\) has no edge, as it is simple, and \(c = 0\). So every realization of \(G\) has rank \(0 = D(|V| - 1) - c\), and its panels are nonparallel, as it has no two distinct bodies. If \(|V| \ge 2\), then the realization of theorem 23.48 is a nonparallel full-rank realization. In both cases the normals are in general position (definition 23.10), and each edge of \(G\) has two distinct ends, as \(G\) is simple, so every hinge is nondegenerate (lemma 23.11).

This is theorem 23.50 at \(d = 3\).

The realization is the one built in the proof of theorem 23.48. Let \(c = \operatorname {def}(\tilde G)\), and let \(G'\) be a spanning subgraph of \(G\) that is a minimal \(c\)-dof-graph (lemma 20.34). Let \((G', q)\) be a generic full-rank realization of \(G'\) (theorem 23.38), and let \((G, p)\) be the panel-hinge framework on \(G\) with the normals of \((G', q)\). These normals are in general position, so every hinge of \((G, p)\) is nondegenerate (lemma 23.11).

The restriction of \((G, p)\) to the edges of \(G'\) has the hinges of \((G', q)\). So it has the same motions as \((G', q)\), and hence the same row space (lemma 19.25). Its rows are among those of \(R(G, p)\), so \(\operatorname {rank} R(G, p) \ge \operatorname {rank} R(G', q) = D(|V| - 1) - c\). Conversely, the hinges of \((G, p)\) are nondegenerate, so \(\operatorname {rank} R(G, p) \le D(|V| - 1) - c\), also when the screw assignments are defined on a larger set of bodies (lemma 20.16).

We follow Katoh and Tanigawa (KT p. 670). Suppose first that \(G\) has an infinitesimally rigid panel-hinge realization with nondegenerate hinges (and nonparallel panels, which this direction does not use). Its body-hinge interpretation (definition 23.5) is an infinitesimally rigid body-hinge realization of \(G\) with the same hinges.

Conversely, let \((G, p)\) be an infinitesimally rigid body-hinge realization of \(G\) with nondegenerate hinges. As it is infinitesimally rigid, \(\dim Z(G, p) = D\) (lemma 19.18). As its hinges are nondegenerate, \(\dim Z(G, p) \ge D + \operatorname {def}(\tilde G)\) (lemma 20.15). Hence \(\operatorname {def}(\tilde G) \le 0\), and so \(\operatorname {def}(\tilde G) = 0\) (definition 20.3). By KT Theorem 5.6 (theorem 23.50), \(G\) has a panel-hinge realization with nonparallel panels at the target rank \(D(|V| - 1)\) (definition 23.21). It is infinitesimally rigid (lemma 19.18). Its normals are in general position (definition 23.10), and each edge of \(G\) has two distinct ends, as \(G\) is simple, so its hinges are nondegenerate (lemma 23.11).

We argue as in the proof of KT Lemma 5.3 (lemma 19.21). By the hinge constraints (definition 19.3), there are scalars \(\lambda _1\), \(\lambda _2\) and \(\lambda _3\) with \(S(u) - S(v) = \lambda _1 C(p(e_1))\), \(S(v) - S(w) = \lambda _2 C(p(e_2))\) and \(S(w) - S(u) = \lambda _3 C(p(e_3))\). The three differences sum to zero, so \(\lambda _1 C(p(e_1)) + \lambda _2 C(p(e_2)) + \lambda _3 C(p(e_3)) = 0\). By the linear independence, \(\lambda _1 = \lambda _2 = \lambda _3 = 0\), so \(S(u) = S(v) = S(w)\).

Let \(e_0, \dots , e_d\) be the standard basis of \(K^{d+1}\), and let \(n_i = e_i\) for \(i = 0, 1, 2\), which is possible as \(d \ge 2\). The joins \(e_a \vee e_b\) with \(a {\lt} b\) are linearly independent (the remark after definition 18.3). Up to sign, \(e_0 \vee e_1\), \(e_1 \vee e_2\) and \(e_2 \vee e_0 = -\, e_0 \vee e_2\) are three distinct ones among them, so they are linearly independent, and in particular nonzero. The extensors \(C(n_i, n_{i+1})\) are their images under the complement isomorphism \(*\) (definition 23.1). Since \(*\) is an isomorphism, they are linearly independent too. Lemma 23.4 records this.

We argue as for lemma 23.61, with \(n_i = e_i\) for \(0 \le i \le m - 1\); this uses \(m \le d + 1\). Up to sign, the joins \(n_i \vee n_{i+1}\) are the \(e_a \vee e_b\) with \(a {\lt} b\) for the pairs \(\{ i, i+1\} \) with \(0 \le i \le m - 2\) and the pair \(\{ 0, m-1\} \), the edges of a cycle on \(\{ 0, \dots , m-1\} \). These \(m\) pairs are distinct, as \(m \ge 3\). So the joins are linearly independent, and so are the extensors \(C(n_i, n_{i+1}) = *(n_i \vee n_{i+1})\) (lemma 23.4).

We follow Katoh and Tanigawa’s proof (KT p. 670), as for theorem 23.48. Let \(c = \operatorname {def}(\tilde G)\).

One body. Let \(|V| = 1\). Every edge of \(G\) is a loop, and the only partition of \(V\) has one part, so \(c = 0\). The rows of a loop vanish, so every realization of \(G\) has rank \(0 = D(|V| - 1) - c\). Give the body a nonzero normal \(n\). The subspace \(n^\perp \) contains \(d - 1\) linearly independent vectors \(x_1, \dots , x_{d-1}\). Their join \(x_1 \vee \cdots \vee x_{d-1}\) is nonzero (the remark after definition 18.3), and it lies in the panel with normal \(n\). Give every edge this hinge.

The realization. Let \(|V| \ge 2\). By lemma 20.34, \(G\) has a spanning subgraph \(G'\) that is a minimal \(c\)-dof-graph. By KT Theorem 5.5 (theorem 23.38), \(G'\) has a nondegenerate-hinge panel realization \((G', q)\), with normals \(n_v\), whether or not \(G'\) is simple. Its rank is \(D(|V| - 1) - c\). We extend \(q\) to a realization \((G, p)\) of \(G\). At each edge of \(G'\) we keep the hinge of \(q\). We give each other edge of \(G\), with ends \(u\) and \(v\), possibly equal, a nonzero extensor that lies in the panels with normals \(n_u\) and \(n_v\). Such an extensor exists. The subspace \(n_u^\perp \cap n_v^\perp \) of \(K^{d+1}\) is cut out by two linear equations, so it contains \(d - 1\) linearly independent vectors, and their join is such an extensor (exists_extensor_in_two_panels_grade). With the normals \(n_v\), \((G, p)\) is a hinge-coplanar panel realization of \(G\).

Katoh and Tanigawa first apply a projective transformation so that any two panels meet in a \((d-2)\)-dimensional affine subspace. Infinitesimal rigidity is invariant under such a transformation. They then choose each new hinge in that intersection. In homogeneous coordinates the transformation is not needed, since any two panels contain a common nonzero extensor.

The rank. The restriction of \((G, p)\) to \(G'\) (definition 23.13) is \((G', q)\). So \(Z(G, p) \subseteq Z(G', q)\) (lemma 23.18), and

The hinges of \((G, p)\) are nondegenerate, so the upper bound on the rank of lemma 20.15, which proposition 23.19 uses, holds at \((G, p)\), and \(\operatorname {rank} R(G, p) \le D(|V| - 1) - c\). So \((G, p)\) is at the target rank.

This is theorem 23.66 at \(d = 3\).

We argue as for theorem 23.56. A hinge-coplanar panel realization of \(G\) is a body-hinge realization of \(G\) whose hinges are nondegenerate, at every label and so at the edges of \(G\). So if \(G\) has an infinitesimally rigid one, then \(G\) has an infinitesimally rigid body-hinge realization with nondegenerate hinges at the edges.

Conversely, let \((G, p)\) be an infinitesimally rigid body-hinge realization of \(G\) with nondegenerate hinges at the edges. As it is infinitesimally rigid, \(\dim Z(G, p) = D\) (lemma 19.18). As its hinges at the edges are nondegenerate, \(\dim Z(G, p) \ge D + \operatorname {def}(\tilde G)\) (lemma 20.15). Hence \(\operatorname {def}(\tilde G) \le 0\), and so \(\operatorname {def}(\tilde G) = 0\) (definition 20.3). By KT Theorem 5.6 (theorem 23.66), \(G\) has a hinge-coplanar panel realization at the target rank \(D(|V| - 1)\). It is infinitesimally rigid (lemma 19.18).

This is lemma 23.30 for the body-hinge interpretation of \((G,p)\) (definition 23.5). An infinitesimally rigid framework on two bodies has rank \(D(2 - 1) = D\) (lemma 19.18).

The extensors lie in \(\bigwedge ^{d-1} K^{d+1}\), which has dimension \(\binom {d+1}{d-1} = D\) (the remark after lemma 18.9).

Let \(e_0, \dots , e_d\) be the standard basis of \(K^{d+1}\). The \(\binom {d+1}{2} = D\) joins \(e_s \vee e_t\) with \(s {\lt} t\) are linearly independent (the remark after definition 18.3). Choose \(m\) distinct pairs \(s_i {\lt} t_i\), and let \(a_i = e_{s_i}\) and \(b_i = e_{t_i}\). Then the joins \(a_i \vee b_i\) are linearly independent, and so are the extensors \(C(a_i, b_i)\) (lemma 23.4).

We argue as for KT Lemma 5.3 (lemma 19.21) and lemma 23.60, which are the cases \(m = 2\) and \(m = 3\) with distinct bodies. Let \(S\) be an infinitesimal motion. By the hinge constraints (definition 19.3), there are scalars \(\lambda _0, \dots , \lambda _{m-1}\) with \(S(v_i) - S(v_{i+1}) = \lambda _i C(p(f_i))\) for each \(i\).

Since the indices are taken modulo \(m\), the \(m\) differences sum to zero. Hence \(\sum _i \lambda _i C(p(f_i)) = 0\), and by the linear independence every \(\lambda _i\) is zero. So \(S(v_i) = S(v_{i+1})\) for every \(i\). If \(v_0, \dots , v_{m-1}\) are all the bodies, then \(S\) is trivial (definition 19.15).

The normals \(q_0\) need not be in general position. As \((G, q_0)\) is infinitesimally rigid on \(V(G)\) and its hinges are nondegenerate, some \(D(|V(G)| - 1)\) rows of \(R(G, q_0)\) are linearly independent (lemma 23.124). As in the proof of lemma 23.120, there is a polynomial \(Q\) in the normals with \(Q(q_0) \neq 0\) such that the same rows are linearly independent at every \(q\) with \(Q(q) \neq 0\). Off the zero set of another nonzero polynomial, the normals are in general position (definition 23.9). As \(K\) is infinite, some normals \(q\) are a non-root of both.

Then \(R(G, q)\) has \(D(|V(G)| - 1)\) linearly independent rows. Let \(W\) be the set of bodies. The screw assignments form a space of dimension \(D|W|\), so \(\dim Z(G, q) \le D|W| - D(|V(G)| - 1) = D(|W \setminus V(G)| + 1)\), and \((G, q)\) is infinitesimally rigid on \(V(G)\) (lemma 23.117). Its rank is then \(D(|V(G)| - 1)\), as in the proof of lemma 23.124. As \(\operatorname {def}(\tilde G) = 0\), it is at the target rank (remark 23.23), and its normals are in general position. So it is a generic full-rank realization of \(G\).

Let \(v_0, \dots , v_{m-1}\) and \(f_0, \dots , f_{m-1}\) be the vertices and the edges of \(G\), where \(f_i\) joins \(v_i\) and \(v_{i+1}\), with indices modulo \(m\) (definition 21.64). Let \(e_0, \dots , e_d\) be the standard basis of \(K^{d+1}\), and give the body \(v_i\) the normal \(e_i\), which is possible as \(m \le d + 1\). Up to sign, the hinge at \(f_i\) is \(C(e_i, e_{i+1})\), again with indices modulo \(m\) (definition 23.5). These \(m\) extensors are linearly independent, by lemma 23.62 and its proof, and so are the hinges.

So every infinitesimal motion is constant on \(\{ v_0, \dots , v_{m-1}\} = V(G)\) (lemma 23.77). As \(G\) is a \(0\)-dof-graph, \(\operatorname {def}(\tilde G) = 0\), and the realization is at the target rank \(D(m - 1)\) (remark 23.23). Any two distinct standard basis vectors are linearly independent, so the normals are in general position (definition 23.9). Hence the realization is a generic full-rank realization of \(G\).

The conditions \(S(v) = 0\) for \(v \in s\) do not involve the graph, and \(Z(G,p) \subseteq Z(G',p)\) (lemma 23.18).

Let \(v \in s\). Then \(Z_s(G,p) \subseteq \operatorname {pinned}(v)\), and \(\operatorname {pinned}(v)\) meets the space of trivial motions only in \(0\) (lemma 19.20). The trivial motions form a \(D\)-dimensional subspace of \(Z(G,p)\) (lemma 19.18). So \(Z(G,p)\) contains the direct sum of this subspace and \(Z_s(G,p)\).

Let \((G,p)\) be infinitesimally rigid, and let \(S \in Z_s(G,p)\). Then \(S\) is trivial, that is, constant (definition 19.15), and it vanishes at a body of \(s\). So \(S = 0\).

Conversely, let \(S\) be an infinitesimal motion, let \(v \in s\), and let \(T\) be the trivial motion with the value \(S(v)\) at every body. Then \(S - T\) is an infinitesimal motion, and it vanishes on \(s\) because \(S\) is constant on \(s\). So \(S - T \in Z_s(G,p) = 0\), and \(S = T\) is trivial.

Let \(\mathcal R\), \(\mathcal R_H\) and \(\mathcal R'\) be the row spaces of \(R(G,p)\), \(R(H,p)\) and \(R(G \setminus E(H), p)\). The rows of \(H\) and of \(G \setminus E(H)\) are rows of \(G\), so \(\mathcal R_H\) and \(\mathcal R'\) lie in \(\mathcal R\). A row of an edge of \(H\) depends only on the screws at the bodies of \(W\), so \(\pi _W\) maps it to \(0\). Hence \(\mathcal R_H \subseteq \mathcal R \cap \ker \pi _W\). The rank of \(R_W(G \setminus E(H), p)\) is \(\dim \pi _W(\mathcal R')\), which is at most \(\dim \pi _W(\mathcal R)\). The bound follows, since \(\dim (\mathcal R \cap \ker \pi _W) + \dim \pi _W(\mathcal R) = \dim \mathcal R\) by rank-nullity for \(\pi _W\) on \(\mathcal R\).

The contraction keeps the deficiency, \(\operatorname {def}(\widetilde{G/E(H)}) = \operatorname {def}(\tilde G) = k\) (lemma 29.22; by KT Lemma 3.5, lemma 21.9, \(G/E(H)\) is even a minimal \(k\)-dof-graph). It has \(|V(G)| - |V(H)| + 1\) vertices (lemma 21.51), so \(\operatorname {rank} R(G/E(H), p_2) = D(|V(G)| - |V(H)|) - k\). The right side of the identity is therefore \(D(|V(G)| - 1) - k\), the target rank of \(G\). This proves the lemma.

Katoh and Tanigawa attain the right side in their Lemmas 6.2 and 6.3 (pp. 673–675). There \((H, p_1)\) is a realization of \(H\) of rank \(D(|V(H)| - 1)\), and \(p\) has the hinges of \(p_1\) at the edges of \(H\) (KT eqs. (6.2) and (6.6)). Put the columns of the bodies of \(H\) first. The hinge constraint at an edge of \(H\) involves only the screws at bodies of \(H\), so \(R(G,p)\) has the block form of KT eq. (6.3):

The first block row consists of the rows of the edges of \(H\), and the second of the rows of the other edges. Its right block \(R_{V(H)}(G \setminus E(H), p)\) consists of these rows with the columns of \(V(H)\) deleted. Suppose that some rows of the first block row are linearly independent, and that some rows of the second are linearly independent after the columns of \(V(H)\) are deleted. Then all of them together are linearly independent. Indeed, in a vanishing linear combination only the rows of the second block row have nonzero entries in the columns of \(V(G) \setminus V(H)\), so their coefficients vanish, and then so do the others. Hence

(lemma 23.86).

When the rows of \(H\) have rank \(D(|V(H)| - 1)\), as here, this bound is an equality. A screw assignment that vanishes on \(V(H)\) satisfies the hinge constraints at the edges of \(H\). It satisfies those at the other edges exactly when its values at the other bodies lie in the null space of \(R_{V(H)}(G \setminus E(H), p)\). So the block pin \(Z_{V(H)}(G,p)\) (definition 23.82) has dimension \(D(|V(G)| - |V(H)|) - \operatorname {rank} R_{V(H)}(G \setminus E(H), p)\). Since \(\operatorname {rank} R(G,p) = D|V(G)| - \dim Z(G,p)\), the bound \(D + \dim Z_{V(H)}(G,p) \le \dim Z(G,p)\) (lemma 23.84) is the reverse inequality.

It remains to compare the second block with \(R(G/E(H), p_2)\). The bodies of \(G/E(H)\) meet \(V(H)\) only in \(r\), so deleting the columns of \(V(H)\) from \(R(G/E(H), p_2)\) deletes those of \(r\). By KT Lemma 5.1 (lemma 19.20), this does not change the rank. Katoh and Tanigawa note this at eq. (6.5), and lemma 19.35 states it in this form. In KT Lemma 6.2, \(p\) has the hinges of \(p_2\) at the edges not in \(H\). The row of such an edge in \(R(G,p)\) then differs from its row in \(R(G/E(H), p_2)\) only in the columns of \(V(H)\) (KT eq. (6.4)). So the second block has the rank of \(R(G/E(H), p_2)\), and the identity holds.

In KT Lemma 6.3, \(G\) and \(G/E(H)\) are simple, and \(p\) changes the hinges at the edges between \(V(H)\) and the other bodies (KT eq. (6.6)). With KT Lemma 5.1, KT Claim 6.4 shows that the second block still has at least the rank of \(R(G/E(H), p_2)\) (KT eq. (6.9); lemma 23.96, and lemma 23.98 at one point, below). The proof of the claim uses that \(p_1\) and \(p_2\) are generic, so that the rank is the largest over all choices of the panels. The general form of this argument is the assertion of lemma 23.112 on the largest rank, in section 23.7: where every hinge is nondegenerate, the largest rank is attained on a dense Zariski-open set of normals. Hence the rank of \(R(G,p)\) is at least the target rank. It is also at most the target rank, as the hinges are nondegenerate (lemma 20.16).

Let \(S\) be an infinitesimal motion of \((G,p)\). As \(H\) is a subgraph of \(G\), \(S\) is an infinitesimal motion of \((H,p)\) (lemma 23.18), so \(S\) is constant on \(V(H)\). Let \(f\) map every vertex of \(H\) to \(r\) and fix the other bodies. Then \(S(f(w)) = S(w)\) for every body \(w\), as \(r \in V(H)\). An edge \(e\) of \(G/E(H)\) is an edge of \(G\) not in \(H\). If its ends in \(G\) are \(u\) and \(v\), then its ends in \(G/E(H)\) are \(f(u)\) and \(f(v)\) (definition 21.3), and \(S(f(u)) - S(f(v)) = S(u) - S(v)\). So \(S\) satisfies the hinge constraint of \((G/E(H), p)\) at \(e\), because it satisfies that of \((G,p)\) (definition 19.3). Hence \(S\) is an infinitesimal motion of \((G/E(H), p)\), and it is constant on \(s_c\). The sets \(V(H)\) and \(s_c\) share the body \(r\) and cover \(V(G)\), so \(S\) is constant on \(V(G)\).

Let \(u\) and \(v\) be the ends of \(e\), and let \(C = C(n_u, n_v)\) (definition 23.1). For \(x \in \bigwedge ^{d-1} K^{d+1}\), write \(x_t\) for its coordinate at the \(t\)-th standard basis vector. The family consists of the forms

for all pairs of indices \(t\) and \(t'\). As a form in \(x\), each of them vanishes at \(x = C\). If \(C \neq 0\), say \(C_{t_0} \neq 0\), then the forms \(x \mapsto C_{t_0} x_{t'} - C_{t'} x_{t_0}\) with \(t' \neq t_0\) are \(D - 1\) linearly independent forms that vanish at \(C\). So they span the space of forms that vanish at \(C\), which has dimension \(D - 1\). Hence the family spans the same space as the rows of \(R(G,n)\) at \(e\) (definition 19.5).

The coordinates of these forms are linear in those of \(C\). The coordinates of the join \(n_u \vee n_v\) (definition 18.4) are the \(2 \times 2\) minors \(n_u(i)\, n_v(j) - n_u(j)\, n_v(i)\) with \(i {\lt} j\), which are polynomials of degree two in the normals. The complement isomorphism \(*\) maps each standard basis vector of \(\bigwedge ^2 K^{d+1}\) to a nonzero multiple of one of \(\bigwedge ^{d-1} K^{d+1}\) (the proof of definition 22.5). So the coordinates of \(C = *(n_u \vee n_v)\) are polynomials of degree two in the normals, and so are those of the forms.

Deleting the columns of \(t\) replaces some entries of a row by zeros, independently of the normals, so the rows of \(R(G,q)\) with these columns deleted again have entries that are polynomials in \(q\) (lemma 23.91). Take \(m\) rows that are independent after the deletion at \(q_0\). Some \(m \times m\) minor of these \(m\) rows, with the columns of \(t\) deleted, is nonzero at \(q_0\). Let \(Q\) be this minor, as a polynomial in the normals. Wherever \(Q(q) \neq 0\), the same \(m\) rows are independent after the deletion. This is the argument of lemma 23.120.

Take the normals of the given realization; general position is a condition on the normals alone. The given realization orients each edge of \(G_1\) one way or the other. Reversing an edge changes the sign of the extensor of its hinge, which is built from the normals of its ends in order (definition 23.5), and so does not change its hinge constraint. The two orientations therefore give the same infinitesimal motions, and so the same row space and rank (lemma 19.25).

Let \(N = D(|V(G_1)|-1) - k\), and let \(f\) map the bodies of \(V(H)\) to \(r\) and fix the others. Orient \(G \setminus E(H)\), and \(G_1\) as in lemma 23.94. By that lemma there are normals \(n\) in general position at which \(R(G_1, n)\) has rank \(N\). The bodies of \(G_1\) meet \(V(H)\) only in \(r\), so deleting the columns of \(V(H)\) keeps the rank \(N\) (lemma 19.35). A minimal \(k\)-dof-graph has no loops: every base of \(M(\tilde G_1)\) meets the copies of every edge (definition 20.8), and the copies of a loop lie in no base (Graph.loopless_of_isMinimalKDof). So the hinges of \(G_1\) at \(n\) are nonzero (lemma 23.11), and some \(N\) rows of \(R(G_1, n)\) are independent after the deletion.

Let \(q_0\) give each body \(v\) the normal \(n_{f(v)}\): the bodies of \(V(H)\) get the normal of \(r\), and the other bodies keep theirs. At \(q_0\) an edge of \(G \setminus E(H)\) from \(u\) to \(v\) has the hinge that the same edge of \(G_1\), from \(f(u)\) to \(f(v)\), has at \(n\). Its rows in the two matrices differ only in the columns of the bodies of \(V(H)\). So after the columns of \(V(H)\) are deleted, the rows of \(R(G \setminus E(H), q_0)\) are those of \(R(G_1, n)\), and \(N\) of them are independent. By lemma 23.93 there is a polynomial \(Q\) with \(Q(q_0) \neq 0\), so that \(Q \neq 0\), such that at every \(q\) with \(Q(q) \neq 0\) some \(N\) rows of \(R(G \setminus E(H), q)\) are independent after the deletion.

Let \(N = D(|s_c| - 1)\), and let \(f\) map every body of \(V(H)\) to \(r\) and fix the other bodies. As \(G/E(H)\) is a \(0\)-dof-graph, its target rank is \(N\). Its generic full-rank realization has normals \(n\) in general position at which \(R(G/E(H), n)\) has rank \(N\), so it is infinitesimally rigid on \(s_c\) (lemma 23.117). The bodies of \(G/E(H)\) meet \(V(H)\) only in \(r\). So deleting the columns of \(V(H)\) from \(R(G/E(H), n)\) deletes those of the one body \(r\), and by KT Lemma 5.1 (lemma 19.20) the rank stays \(N\) (lemma 19.35).

Now give each body \(v\) the normal \(q_0(v) = n_{f(v)}\), so that the bodies of \(V(H)\) get the normal of \(r\). This is the choice in Katoh and Tanigawa’s proof of Claim 6.4 that turns \(p'\) into \(p_2\). Let \(e\) be an edge of \(G \setminus E(H)\) with ends \(u\) and \(v\). At \(q_0\) its hinge is \(C(n_{f(u)}, n_{f(v)})\), the hinge of \(e\) in \(G/E(H)\) at \(n\), as the ends of \(e\) in \(G/E(H)\) are \(f(u)\) and \(f(v)\). Its two rows differ only in the columns of the bodies of \(V(H)\). So after the columns of \(V(H)\) are deleted, the rows of \(R(G \setminus E(H), q_0)\) are those of \(R(G/E(H), n)\), and they have rank \(N\).

Finally, \(G/E(H)\) is simple, so it has no loops. As \(n\) is in general position, the hinges of \(G/E(H)\) at \(n\), and so those of \(G \setminus E(H)\) at \(q_0\), are nondegenerate (lemma 23.11). So the forms of lemma 23.91 at \(q_0\) span the rows of every edge of \(G \setminus E(H)\), and some \(N\) of them are linearly independent after the deletion, as in lemma 23.110.

By KT Lemma 3.3 (lemma 20.25), \(H\) is a minimal \(0\)-dof-graph. By KT Lemma 3.5 (lemma 21.9), \(G/E(H)\) is a minimal \(k\)-dof-graph, with the vertex set \(s_c = (V(G) \setminus V(H)) \cup \{ r\} \). So the given realizations \((H, p_1)\) and \((G/E(H), p_2)\) have the ranks \(D(|V(H)| - 1)\) and \(D(|s_c| - 1) - k\). In Katoh and Tanigawa’s proof the induction hypothesis gives them, with algebraically independent coefficients, and KT eq. (6.6) combines them into a realization of \(G\). Its rank is the target rank by KT eq. (6.3) and Claim 6.4, as in the proof of lemma 23.87.

We choose instead one set of normals \(q_0\) for all the bodies, off the zero sets of three nonzero polynomials in the normals. First, some \(D(|V(H)| - 1)\) rows of \(R(H, p_1)\) are linearly independent. The rows can be chosen with entries that are polynomials in the normals (lemma 23.91). So one of the maximal minors of these rows is a polynomial \(Q_1\) that does not vanish at the normals of \(p_1\), and wherever \(Q_1(q) \neq 0\), the same rows of \(R(H, q)\) are linearly independent. This is the argument of lemma 23.120, later in this chapter (section 23.8).

Second, there is a nonzero polynomial \(Q_2\) such that wherever \(Q_2(q) \neq 0\), some \(D(|s_c| - 1) - k\) rows of \(R(G \setminus E(H), q)\) are linearly independent after the columns of \(V(H)\) are deleted (lemma 23.96, applied to \((G/E(H), p_2)\)). This is KT Claim 6.4 with KT eq. (6.9), in polynomial form.

Third, write \(q(a)_i\) for the \(i\)-th coordinate of the normal \(q(a)\), from \(i = 0\). Let \(P\) be the product, over the ordered pairs of distinct bodies \(a\) and \(b\), of the minors \(q(a)_0\, q(b)_1 - q(a)_1\, q(b)_0\). Wherever \(P(q) \neq 0\), any two normals are linearly independent, so the normals are in general position (definition 23.9). The polynomial \(P\) is not zero: at normals on the moment curve (lemma 23.12), each factor is a difference \(t_b - t_a \neq 0\). As \(K\) is infinite, the nonzero polynomial \(Q_1 Q_2 P\) has a non-root \(q_0\).

Let \((G, q_0)\) be the panel-hinge framework on \(G\) with the normals \(q_0\), in which every hinge is the meet of the panels of the ends of its edge, as in KT eq. (6.6). As \(Q_1(q_0) \neq 0\), some \(D(|V(H)| - 1)\) rows of the edges of \(H\) are linearly independent. As \(Q_2(q_0) \neq 0\), some \(D(|s_c| - 1) - k\) rows of the other edges are linearly independent after the columns of \(V(H)\) are deleted. By the block form of KT eq. (6.3), all these rows together are linearly independent (the proof of lemma 23.87). The sets \(V(H)\) and \(s_c\) share only the body \(r\) and cover \(V(G)\), so there are

of them. As \(P(q_0) \neq 0\), the normals \(q_0\) are in general position. The graph \(G\) has no loops, so every hinge of \((G, q_0)\) is nondegenerate (lemma 23.11). Hence the rank of \(R(G, q_0)\) is at most \(D(|V(G)| - 1) - k\) (lemma 20.16), and \((G, q_0)\) is a generic full-rank realization of \(G\).

A minimal \(k\)-dof-graph has no loops, as in the proof of lemma 23.96. As \(G\) is not simple, it has two edges \(e \neq f\) that join the same vertices \(a \neq b\). Let \(H'\) be the subgraph with the vertices \(a\) and \(b\) and the edges \(e\) and \(f\). It is a \(0\)-dof-graph (lemma 20.36; KT p. 673), and a proper rigid subgraph of \(G\) as \(|V(G)| \ge 3\). Let \(G' = G/E(H')\) be its contraction to \(a\). By KT Lemma 3.5 (lemma 21.9), \(G'\) is a minimal \(k\)-dof-graph with \(|V(G)| - 1\) vertices. So, by the hypothesis, \(G'\) has a nondegenerate-hinge panel realization \((G', p_2)\), of rank \(D(|V(G)| - 2) - k\).

As Katoh and Tanigawa do, we give \(a\) and \(b\) the panel of the body \(a\) in \((G', p_2)\), and every other body its panel in \((G', p_2)\). We choose two linearly independent extensors \(C_e\) and \(C_f\) in the panel of \(a\) (lemma 23.31, in the form for \(K^{d+1}\) given after it). Let \(p\) give \(e\) and \(f\) the hinges \(C_e\) and \(C_f\), and every other edge its hinge in \((G', p_2)\) (KT eq. (6.2)). Every hinge of \(p\) is nonzero. It lies in the panels of the ends of its edge: the ends of an edge in \(G'\) are its ends in \(G\) with \(b\) replaced by \(a\) (definition 21.3), and \(a\) and \(b\) have the same panel.

We compute the rank as in the proof of lemma 23.87 for KT Lemma 6.2. By KT Lemma 5.3 (lemma 23.30), the restriction of \((G,p)\) to \(H'\) is infinitesimally rigid, so the rows of \(e\) and \(f\) have rank \(D\). The other edges have their hinges in \((G', p_2)\). After the columns of \(a\) and \(b\) are deleted, their rows are those of \(R(G', p_2)\) with the columns of \(a\) deleted (KT eq. (6.4)). By KT Lemma 5.1, these have the rank of \(R(G', p_2)\) (KT eq. (6.5); lemma 19.35). By the block form of KT eq. (6.3),

(lemma 19.33). As every hinge is nonzero, the rank is at most this (lemma 20.16).

Let \(G'\) be a proper rigid subgraph of \(G\) whose vertex set is maximal under inclusion. The subgraph of \(G\) induced on \(V(G')\) contains \(G'\) and has the same vertices, so its deficiency is at most that of \(G'\), which is zero (definition 20.3). So we may take \(G'\) induced. Then an edge of \(G\) not in \(G'\) has at most one end in \(V(G')\), and \(G/E(G')\) has no loop. By the hypothesis, \(G/E(G')\) is not simple, so it has two edges \(e \neq f\) with the same ends. As \(G\) is simple, \(e\) and \(f\) are not parallel in \(G\), so one of these ends is the vertex \(r\) to which \(G'\) is contracted. So each of \(e\) and \(f\) has exactly one end in \(V(G')\), and they join a common vertex \(v \notin V(G')\) to two vertices \(a \neq b\) of \(G'\). Katoh and Tanigawa read off \(v\), \(e\) and \(f\) from the non-simplicity of \(G/E(G')\) directly (KT p. 676). This needs \(G'\) induced: otherwise an edge of \(G\) with both ends in \(V(G')\) could give a loop of \(G/E(G')\) instead.

Let \(G''\) be obtained from \(G'\) by adding \(v\) and the edges \(e\) and \(f\). Removing \(v\) from \(G''\) gives \(G'\), so \(\operatorname {def}(\tilde G'') \le \operatorname {def}(\tilde G') = 0\), by KT Lemma 4.4 (lemma 21.38). So \(G''\) is a rigid subgraph of \(G\) with more vertices than \(G'\). By the choice of \(G'\) it is not proper, so \(V(G'') = V(G)\). As \(G\) contains \(G''\) and has the same vertices, \(k = \operatorname {def}(\tilde G) \le \operatorname {def}(\tilde G'') = 0\) (definition 20.3). As \(G\) has vertices, the partition \(\{ V(G)\} \) has value \(0\), so \(\operatorname {def}(\tilde G) \ge 0\) and \(k = 0\). Let \(e'\) be an edge of \(G\) not in \(G''\). Then \(G - e'\) contains \(G''\) and has the same vertices, so its deficiency is zero, as is that of \(G\). This contradicts the minimality of \(G\) (the remark after definition 20.8), so \(G = G''\). Hence \(v\) has degree two in \(G\), with the neighbours \(a \neq b\), and \(G_v = G'\) is a minimal \(0\)-dof-graph (KT Lemma 3.3, lemma 20.25). It is simple, as a subgraph of \(G\).

By KT Claim 6.6 (lemma 23.104), \(G\) has a vertex \(v\) of degree two, joined to two distinct vertices \(a\) and \(b\), such that \(G_v\) is a simple minimal \(0\)-dof-graph. A proper rigid subgraph of \(G\) has at least two vertices and not all of them, so \(|V(G)| \ge 3\), and \(G_v\) has \(|V(G)| - 1 \ge 2\) vertices. So it has a generic full-rank realization, of rank \(D(|V(G)| - 2)\). Katoh and Tanigawa keep this realization, and give \(v\) a panel \(\Pi ^\circ \) that is nonparallel to the other panels and does not contain \(\Pi (a) \cap \Pi (b)\). Then the hinges \(\Pi ^\circ \cap \Pi (a)\) and \(\Pi ^\circ \cap \Pi (b)\) of \(va\) and \(vb\) are distinct (KT p. 676).

We choose all the normals at once, as in the proof of lemma 23.100, at a non-root \(q\) of the product of three nonzero polynomials. The first is a polynomial \(Q\) that does not vanish at the normals of this realization, such that \(R(G_v, q)\) has rank at least \(D(|V(G)| - 2)\) wherever \(Q(q) \neq 0\) (lemma 23.120). The polynomial \(Q\) is the minor by which the proof of lemma 23.112 gives the lower semicontinuity of the rank. The second is the polynomial \(P\) of the proof of lemma 23.100, so that the normals \(q\) are in general position.

The third is a \(3 \times 3\) minor \(\Delta \) of the matrix with the rows \(q(v)\), \(q(a)\) and \(q(b)\), which is not the zero polynomial as \(d + 1 \ge 3\). As \(\Delta (q) \neq 0\), the normals \(q(v)\), \(q(a)\) and \(q(b)\) are linearly independent. Since \(q(a)\) and \(q(b)\) are linearly independent, this says that the panel of \(v\) does not contain the intersection of the panels of \(a\) and \(b\), as Katoh and Tanigawa ask of \(\Pi ^\circ \).

Let \((G, q)\) be the panel-hinge framework on \(G\) with the normals \(q\). The edges of \(G_v\) have the same hinges in \((G, q)\) and in \((G_v, q)\), so their rows are those of \(R(G_v, q)\), and these vanish in the columns of \(v\). The hinges of \(va\) and \(vb\) are \(C(q(v), q(a))\) and \(C(q(v), q(b))\). They are linearly independent, because the joins \(q(v) \vee q(a)\) and \(q(v) \vee q(b)\) are (lemma 23.4). Indeed, if \(q(v)\), \(q(a)\) and \(q(b)\) are extended to a basis of \(K^{d+1}\), the two joins belong to the corresponding basis of \(\bigwedge ^2 K^{d+1}\) (the remark after definition 18.3). So, as in the proof of KT Lemma 5.3 (lemma 19.21), the rows of \(va\) and \(vb\) have rank \(D\) in the columns of \(v\). This is the block form of KT eq. (6.10).

Take \(D\) of these rows that are linearly independent in the columns of \(v\), and \(D(|V(G)| - 2)\) linearly independent rows of \(G_v\). In a vanishing linear combination of them, only the first have nonzero entries in the columns of \(v\), so their coefficients vanish, and then so do the others. Hence

and the rank is at most this (lemma 19.18). As the normals \(q\) are in general position, \((G, q)\) is a generic full-rank realization of \(G\).

The hypothesis is the induction hypothesis KT eq. (6.1) for the graphs with fewer vertices than \(G\). A proper rigid subgraph of \(G\) has at least two vertices and not all of them, so \(|V(G)| \ge 3\). We follow Katoh and Tanigawa’s three cases (KT p. 673). If \(G\) is not simple, then KT Lemma 6.2 gives a nondegenerate-hinge panel realization (lemma 23.102). Let \(G\) be simple. Suppose that some proper rigid subgraph \(H\) has a simple contraction \(G/E(H)\). Then \(H\) and \(G/E(H)\) are simple minimal \(0\)-dof-graphs with fewer vertices than \(G\) (KT Lemmas 3.3 and 3.5). The induction hypothesis gives generic full-rank realizations of both, and KT Lemma 6.3 gives one of \(G\) (lemma 23.100). Otherwise, KT Lemma 6.5 gives one (lemma 23.106). In both cases, \(G\) also has a nondegenerate-hinge panel realization (lemma 23.25).

For \(k = 0\) this is lemma 23.107. Let \(k {\gt} 0\). As there, \(|V(G)| \ge 3\), since a proper rigid subgraph of \(G\) has at least two vertices and not all of them. If \(G\) is not simple, then KT Lemma 6.2 gives a nondegenerate-hinge panel realization (lemma 23.102). Let \(G\) be simple. Suppose that some proper rigid subgraph \(H\) has a simple contraction \(G/E(H)\). Then \(H\) is a simple minimal \(0\)-dof-graph and \(G/E(H)\) a simple minimal \(k\)-dof-graph, both with fewer vertices than \(G\) (KT Lemmas 3.3 and 3.5). The hypothesis gives generic full-rank realizations of both, and KT Lemma 6.3 gives one of \(G\) (lemma 23.100), which is also a nondegenerate-hinge panel realization (lemma 23.25). Otherwise no proper rigid subgraph of \(G\) has a simple contraction, and KT Claim 6.6 gives \(k = 0\) (lemma 23.104), a contradiction.

As the hinges are nondegenerate, the forms at each edge span the same space as the rows of \(R(G,p)\) at that edge (the proof of lemma 23.91). So all the forms span the row space of \(R(G,p)\), whose dimension is \(\operatorname {rank} R(G,p) \ge N\), and some \(N\) of them are linearly independent.

The set \(U\) contains the normals of Katoh and Tanigawa’s nonparallel realizations of a simple graph, as these normals are in general position (the remark after definition 23.10 and lemma 23.11). The remark after the proof shows that the lemma fails outside \(U\). The set \(U\) is Zariski-open, as the coordinates of the hinge \(C(q(u), q(w))\) of an edge with ends \(u\) and \(w\) are polynomials in \(q\) (definition 23.1). Choose the rows of \(R(G,q)\) at each edge as in lemma 23.91. Their coordinates are polynomials of degree two in \(q\), and at every \(q \in U\) they span the row space of \(R(G,q)\).

So for \(q \in U\), the rank of \(R(G,q)\) is at least \(r\) if and only if some \(r\) of these rows are linearly independent at \(q\) (lemma 23.110), that is, if and only if some \(r \times r\) minor of the matrix of these rows does not vanish at \(q\). Each minor \(\Delta \) is a polynomial in \(q\). Hence the set of the \(q \in U\) with \(\operatorname {rank} R(G,q) \ge r\) is the intersection of \(U\) with the union of the finitely many sets \(\{ \Delta \neq 0\} \), and it is Zariski-open. If \(\operatorname {rank} R(G,q_0) \ge r\) at some \(q_0 \in U\), then some minor \(\Delta \) has \(\Delta (q_0) \neq 0\). The rank is then at least \(r\) on the open set \(U \cap \{ \Delta \neq 0\} \), which contains \(q_0\). As \(K\) is infinite, a nonzero polynomial has a non-root, and so has a product of nonzero polynomials. Hence any two nonempty Zariski-open subsets of \(K^N\) meet, and each of them is dense. Taking for \(q_0\) normals in \(U\) of the largest rank gives the assertion on the largest rank. The assertions on \(Z(G,q)\) are the same, as \(\dim Z(G,q) = D|W| - \operatorname {rank} R(G,q)\) (definition 19.7).

A screw assignment that vanishes on \(V(G)\) satisfies every hinge constraint, as both ends of every edge lie in \(V(G)\). So the block pin \(Z_{V(G)}(G,p)\) (definition 23.82) consists of all the screw assignments that vanish on \(V(G)\). Such an assignment takes an arbitrary value in \(K^D\) at each body of \(W \setminus V(G)\), so these assignments form a space of dimension \(D|W \setminus V(G)|\).

The normals \(q_0\) will do. The \(m\) rows lie in the row space of \(R(G,q_0)\), so

and \(|W| = |V(G)| + |W \setminus V(G)|\).

Fix a body \(v_0 \in V(G)\). A motion that vanishes on \(V(G)\) vanishes at \(v_0\), so \(Z_{V(G)}(G,p) \subseteq \operatorname {pinned}(v_0)\). By KT Lemma 5.1 (lemma 19.20), the hypothesis and lemma 23.114,

So the two subspaces are equal: a motion that vanishes at \(v_0\) vanishes on \(V(G)\). Every motion \(S\) is the sum of the trivial motion with value \(S(v_0)\) and a motion that vanishes at \(v_0\) (lemma 19.20), so \(S\) is constant on \(V(G)\). This is the criterion in the note after lemma 23.87, with \(s = \{ v_0\} \) and \(t = V(G)\).

Let \(W\) be the set of bodies. By lemma 23.115 and its proof, the normals \(q = q_0\) satisfy \(\dim Z(G,q) \le D(|W \setminus V(G)| + 1)\). By lemma 23.117, every infinitesimal motion of \((G,q)\) is then constant on \(V(G)\).

Choose the rows of \(R(G,q)\) at each edge as in lemma 23.91. As the hinges of \((G,q_0)\) are nondegenerate, these rows span the row space of \(R(G,q_0)\), so some \(N\) of them are linearly independent at \(q_0\). The matrix of their coordinates then has an \(N \times N\) minor that does not vanish at \(q_0\). Its entries are polynomials in the normals, and we let \(Q\) be this minor, as a polynomial in the normals.

Wherever \(Q(q) \neq 0\), the same \(N\) rows are linearly independent at \(q\). Each of them is a form \(S \mapsto \varphi (S(u) - S(w))\) at an edge with ends \(u\) and \(w\), where \(\varphi \) vanishes on the hinge of the edge at \(q\) (lemma 23.91). So it lies in the row space of \(R(G,q)\) (definition 19.5), and \(\operatorname {rank} R(G,q) \ge N\). This is the argument of lemma 23.112, with the minor kept.

By lemma 23.94 there are normals \(n\) in general position at which this framework has rank \(N\). As \(G_1\) has no loops, its hinges at \(n\) are nonzero (lemma 23.11). So lemma 23.120, at \(n\) with the bound \(N\), gives a polynomial \(Q\) with \(Q(n) \neq 0\), so that \(Q \neq 0\), such that the rank is at least \(N\) wherever \(Q\) does not vanish.

Let \(u\) and \(w\) be the ends of \(e\), and let \(C \neq 0\) be its hinge. The linear forms on \(K^D\) that vanish on \(C\) form a space of dimension \(D - 1\) (definition 19.5). The map \(S \mapsto S(u) - S(w)\) takes the screw assignments onto \(K^D\), as \(u \neq w\). So composing with it gives a space of linear forms on the screw assignments of dimension \(D - 1\) (lemma 19.9). The rows of \(R(G,q)\) at \(e\) span this space (lemma 23.91), and \(D - 1\) of them are linearly independent.

Fix a body \(w \in V(G)\). A motion that vanishes at \(w\) vanishes on \(V(G)\), as it is constant there. So \(\operatorname {pinned}(w) = Z_{V(G)}(G,p)\), and by KT Lemma 5.1 (lemma 19.20) and lemma 23.114,

Hence \(\operatorname {rank} R(G,p) = D|W| - \dim Z(G,p) = D(|V(G)| - 1)\). If the hinges are nondegenerate, the rows of \(R(G,p)\) span its row space (the proof of lemma 23.91), and \(D(|V(G)| - 1)\) of them are linearly independent.

These rows are the rows of \(R(G_1, q_1)\) at the indices \(f(i)\), which are distinct as \(f\) is injective. A subfamily of a linearly independent family is linearly independent.

The row at an index depends only on the ends of its edge and their normals (lemma 23.91). So the hypothesis holds, with \(f\) the identity, for indices at edges that \(G_1\) and \(G_2\) share with the same ends, when \(q_1\) and \(q_2\) agree at these ends. For instance, \(G\) and \(G_v^{ab}\) share the edges of \(G_v\) (lemma 21.12).

Suppose that \(\sum _i a_i \rho _i + \sum _j b_j \sigma _j = 0\), and let \(S\) be a screw assignment that vanishes off \(v\). Then \(\sigma _j(S) = 0\) for every \(j\), and hence \(\sum _i a_i \rho _i(S) = 0\). By the hypothesis on the \(\rho _i\), every \(a_i\) is \(0\). Then \(\sum _j b_j \sigma _j = 0\), and every \(b_j\) is \(0\).

We first make the count from a realization \((G_v^{ab}, q)\) with nondegenerate hinges whose rigidity matrix has rank at least \(N\), for any \(N\); the proof of lemma 23.135, in the next subsection, takes it with \(t = 1\). Let \(n_a = q(a)\) and \(n_b = q(b)\). The hinge \(C = C(n_a, n_b)\) of the edge \(ab\) of \((G_v^{ab}, q)\) is nonzero. As \(C\) is bilinear and \(C(n,n) = 0\) (definition 23.1),

So, up to sign, the edges \(vb\) and \(va\) of \((G, q_0)\) have the hinges \(C\) and \(-tC\), both nondegenerate as \(t \neq 0\). As \(v\) has degree two, every other edge of \(G\) is an edge of \(G_v\), and so of \(G_v^{ab}\) (lemma 21.12). Its ends differ from \(v\), so it has the same hinge, up to sign, in \((G, q_0)\) as in \((G_v^{ab}, q)\), and this hinge is nondegenerate.

The normals \(q_0\) and \(q\) differ only at \(v\), which lies on no edge of \(G_v^{ab}\). So \((G_v^{ab}, q_0)\) has the hinges and the infinitesimal motions of \((G_v^{ab}, q)\) (lemma 23.16), and so its row space (lemma 19.25). Some \(N\) rows \(\sigma _1, \dots , \sigma _N\) of \(R(G_v^{ab}, q_0)\) are linearly independent (lemma 23.110). Each of them lies in the row space of \(R(G, q_0)\). A row at an edge of \(G_v\) is, up to sign, a row of \(R(G, q_0)\) at that edge, as the edge has the same hinge in both frameworks up to sign. A row at the edge \(ab\) is, up to sign, a form \(S \mapsto \varphi (S(a) - S(b))\), where \(\varphi \) is a linear form on \(K^D\) that vanishes on \(C\). The forms \(S \mapsto \varphi (S(v) - S(b))\) and \(S \mapsto \varphi (S(v) - S(a))\) lie in the row spaces of the edges \(vb\) and \(va\) of \((G, q_0)\), as \(\varphi \) vanishes on their hinges, and

(lemma 23.161). This is the row form of Katoh and Tanigawa’s column operation, which adds the columns of \(v\) to those of \(a\) (KT eqs. (6.14)–(6.16)).

By lemma 23.122, some \(D - 1\) rows \(\rho _1, \dots , \rho _{D-1}\) of \(R(G, q_0)\) at \(vb\) are linearly independent. Up to sign, they are the forms \(S \mapsto \varphi _i(S(v) - S(b))\) for linearly independent forms \(\varphi _i\) on \(K^D\). On the screw assignments that vanish off \(v\), they are the forms \(S \mapsto \varphi _i(S(v))\), which are again linearly independent. The \(\sigma _j\) vanish there, as no edge of \(G_v^{ab}\) has \(v\) as an end. So the \(\rho _i\) and the \(\sigma _j\) are linearly independent together (lemma 23.127), and they lie in the row space of \(R(G, q_0)\). Hence (lemma 19.34)

Now let every infinitesimal motion of \((G_v^{ab}, q)\) be constant on \(V(G_v^{ab})\), whose vertices are those of \(G\) other than \(v\) (definition 21.10). Then \(R(G_v^{ab}, q)\) has rank \(N = D(|V(G)| - 2)\), by the dimension of its motion space (lemma 23.124), and

As the hinges of \((G, q_0)\) are nondegenerate, its rows span its row space. So some \(D(|V(G)| - 1) - 1\) of them are linearly independent (lemma 23.110). Katoh and Tanigawa note this count for \(k = 0\) on p. 680  [ KT11 ] .

By lemma 19.20, \(\dim Z(G,p) = D + \dim \operatorname {pinned}(v)\), which gives the first assertion. If \(v\) lies on no edge of \(G\), then \((G,n)\) and \((G,n')\) have the same subspace \(\operatorname {pinned}(v)\) (lemma 23.16). So the second assertion is the first, applied to \((G,n')\).

As \(v\) is a vertex of \(G\), we have \(V(G) = t \cup \{ v\} \). A screw assignment \(S\) is constant on \(t \cup \{ v\} \) if and only if it is constant on \(t\) and \(S(v) = S(w)\) for every \(w \in t\).

We follow Katoh and Tanigawa’s proof of KT Lemma 6.8 (pp. 677–679). The graph \(G\) is simple (lemma 21.57). As \(d \ge 2\), we have \(D \ge 3\), so \(G\) has a vertex \(v\) of degree two (lemma 21.50; KT Lemma 4.6). Its edges \(e_a\) and \(e_b\) join it to vertices \(a\) and \(b\), and \(a \neq b\) as \(G\) is simple.

Next, \(G_v^{ab}\) is simple, by KT Lemma 6.7 (lemma 21.58). If \(|V(G)| \ge 4\), this is its last assertion. Let \(|V(G)| = 3\). If an edge of \(G\) joined \(a\) and \(b\), then this edge, \(e_a\) and \(e_b\) would be all the edges of the simple graph \(G\). So \(G\) would be a triangle, and a \(0\)-dof-graph by the first assertion of KT Lemma 6.7, contradicting \(k {\gt} 0\). So no edge of \(G\) joins \(a\) and \(b\), and \(G_v^{ab}\) is simple, as in the proof of lemma 21.58.

By KT Lemma 4.8(ii) (lemma 21.53), \(G_v^{ab}\) is a minimal \((k-1)\)-dof-graph. Its vertices are the \(|V(G)| - 1 \ge 2\) vertices of \(G\) other than \(v\) (lemma 21.51). So the induction hypothesis gives a generic full-rank realization \((G_v^{ab}, q)\), with

which is KT eq. (6.11). Its normals are in general position, so the normals \(n_a = q(a)\) and \(n_b = q(b)\) are linearly independent.

Katoh and Tanigawa keep the hinges of \((G_v^{ab}, q)\) at the edges of \(G_v\). They give the edge \(vb\) the hinge \(q(ab)\) of the edge \(ab\), and the edge \(va\) a \((d-2)\)-dimensional affine subspace \(L\) of the panel \(\Pi (a)\) (KT eq. (6.12)). With the panel \(\Pi (a)\) also at \(v\), this is a panel-hinge realization \((G, p_1)\) for every such \(L\) (KT Claim 6.9). They add the columns of \(v\) to those of \(a\) (KT eqs. (6.14) and (6.15)). As \(p_1(vb) = q(ab)\), the rows of \(vb\) and of the edges of \(G_v\) then have, outside the columns of \(v\), the entries of \(R(G_v^{ab}, q)\) (KT eqs. (6.16) and (6.17)). The rows of \(va\) vanish outside the columns of \(v\), and they have rank \(D - 1\) there. So

Finally they rotate the panel of \(v\) slightly, which does not lower the rank, to make the realization nonparallel (KT Lemma 5.2, KT p. 679).

In a panel-hinge framework the hinge of an edge is the meet of the panels of its ends (definition 23.5), while Katoh and Tanigawa ask only that it lie in both panels. For the hinge of \(vb\) to be \(q(ab) = \Pi (a) \cap \Pi (b)\), the panel \(\Pi (v)\) must contain \(\Pi (a) \cap \Pi (b)\). It must also differ from \(\Pi (a)\), as otherwise the hinge of \(va\) would be degenerate. Then the hinge \(\Pi (v) \cap \Pi (a)\) of \(va\) is again \(\Pi (a) \cap \Pi (b)\), so that \(L = q(ab)\).

The normal \(n_a + n_b\) gives such a panel, as \(n_a\) and \(n_b\) are linearly independent. Let \(q_0\) be the normals with \(q_0(v) = n_a + n_b\) and \(q_0(w) = q(w)\) for every body \(w \neq v\), and let \((G, q_0)\) be the panel-hinge framework on \(G\) with these normals. As \(C\) is bilinear and \(C(n,n) = 0\) (definition 23.1), the edges \(vb\) and \(va\) of \((G, q_0)\) have, up to sign, the hinge \(C(n_a, n_b)\) of the edge \(ab\) of \((G_v^{ab}, q)\). Every other edge of \(G\) is an edge of \(G_v\), and so of \(G_v^{ab}\) (lemma 21.12). Its ends differ from \(v\), so it has the same hinge in both frameworks, up to sign.

We now count rows as Katoh and Tanigawa do, with a row operation in place of their column operation. The hinges of \((G_v^{ab}, q)\) are nondegenerate, as its normals are in general position (lemma 23.11). So the count of lemma 23.128, with \(t = 1\) and \(N = D(|V(G)| - 2) - (k - 1)\), shows that the hinges of \((G, q_0)\) are nondegenerate and that

Its proof places every row of \(R(G_v^{ab}, q)\) in the row space of \(R(G, q_0)\), a row at \(ab\) being the difference of a row at \(vb\) and a row at \(va\), and adds \(D - 1\) rows at \(vb\). These are linearly independent on the screw assignments that vanish off \(v\), where the rows of \(R(G_v^{ab}, q)\) vanish. For \(k = 0\), the graph \(G_v^{ab}\) would be a minimal \(0\)-dof-graph, and the same count would give one row less than the target rank (KT p. 680). Case III supplies the missing row (section 23.10).

The normals \(q_0\) need not be in general position. Instead of rotating a panel, we choose all the normals at once, as in the proof of lemma 23.100 and the passage after it. Every hinge of \((G, q_0)\) is nonzero, and \(R(G, q_0)\) has the rank above. So there is a polynomial \(Q\) in the normals with \(Q(q_0) \neq 0\) such that \(\operatorname {rank} R(G, q) \ge D(|V(G)| - 1) - k\) wherever \(Q(q) \neq 0\). This is lemma 23.120, and \(Q\) is the minor by which the proof of lemma 23.112 gives the lower semicontinuity of the rank. Let \(P\) be the polynomial of the proof of lemma 23.100, so that the normals \(q\) are in general position wherever \(P(q) \neq 0\). As \(K\) is infinite, the nonzero polynomial \(QP\) has a non-root \(q'\).

The rank of \(R(G, q')\) is then at least \(D(|V(G)| - 1) - k\), and the normals \(q'\) are in general position. As \(G\) has no loops, every hinge of \((G, q')\) is nondegenerate (lemma 23.11). So the rank is at most \(D(|V(G)| - 1) - k\) (lemma 20.16), and \((G, q')\) is a generic full-rank realization of \(G\).

We follow Katoh and Tanigawa  [ KT11 , p. 684 ] . The graph \(G_v\) is a subgraph of \(G\), so it is a minimal \(k'\)-dof-graph by KT Lemma 3.3 (lemma 20.25). It has the vertex \(a\), so the partition of \(V(G_v)\) into one part shows that \(k' \ge 0\) (definition 20.3). For the same reason \(\operatorname {def}(\widetilde{G_v^{ab}}) \ge 0\). By KT Lemma 4.3(i) (lemma 21.33), \(\operatorname {def}(\widetilde{G_v^{ab}}) \le \operatorname {def}(\tilde G) = 0\), so \(G_v^{ab}\) is a \(0\)-dof-graph.

By the forward half of KT Lemma 4.3(ii) (lemma 21.41), at \(k = 0\), \(M(\widetilde{G_v^{ab}})\) has a base \(B'\) with \(h = |\widetilde{ab} \cap B'| {\lt} D - 1\). As \(G_v^{ab}\) has the \(|V(G)| - 1\) vertices of \(G\) other than \(v\), KT eq. (2.4) (theorem 20.12) gives \(|B'| = D(|V(G)| - 2)\). The graph \(G_v\) is \(G_v^{ab}\) without the edge \(ab\) (lemma 21.12). So \(M(\tilde G_v)\) is the restriction of \(M(\widetilde{G_v^{ab}})\) to \(E(\tilde G_v)\) (lemma 20.24), in which \(B' \setminus \widetilde{ab}\) is an independent set of \(|B'| - h\) elements. As \(G_v\) and \(G_v^{ab}\) have the same vertices, KT eq. (2.4) gives

We assume, as in Case III, that the induction hypothesis KT eq. (6.1) holds for the graphs with fewer vertices than \(G\) (the note after the proof). The graph \(G_v\) is a minimal \(k'\)-dof-graph with \(k' = \operatorname {def}(\tilde G_v) \le D - 2\) (lemma 23.138). It is simple, as a subgraph of \(G\), and it has at least one and fewer than \(|V(G)|\) vertices. So the induction hypothesis gives a generic full-rank realization \((G_v, q_0)\) (definition 23.21), with

Its normals are in general position and \(G_v\) has no loops, so its hinges are nondegenerate (lemma 23.11). By lemma 23.120, there is a polynomial \(P\) in the normals with \(P(q_0) \neq 0\) such that \(\operatorname {rank} R(G_v, q) \ge D(|V(G_v)| - 1) - k'\) wherever \(P(q) \neq 0\). In particular \(P \neq 0\). As \(k' \le D - 2\), this is the bound of the lemma.

By lemma 23.122, \(R_e\) contains \(D - 1\) linearly independent forms \(r_1, \dots , r_{D-1}\). The space \(R_e\) has dimension \(D - 1\), as in the proof of that lemma, so they form a basis of \(R_e\). Suppose that no \(r_i\) lies in \(W + \operatorname {span}\{ r_j : j \neq i\} \). If \(\sum _j c_j r_j \in W\) with \(c_i \neq 0\), then dividing by \(c_i\) puts \(r_i\) in \(W + \operatorname {span}\{ r_j : j \neq i\} \). So \(\sum _j c_j r_j \in W\) only if every \(c_j\) is \(0\), and the images of the \(r_j\) in the quotient by \(W\) are linearly independent. Then \(\dim (W + R_e) = \dim W + (D - 1)\), contrary to the hypothesis.

We follow Katoh and Tanigawa  [ KT11 , p. 685 ] . The edges of \(G_v^{ab}\) are those of \(G_v\), with the same ends, and the new edge \(ab\) (lemma 21.12). The two frameworks have the same normals, so an edge of \(G_v\) has the same hinge, and the same rows, in both. Let \(W\) be the row space of \(R(G_v, q)\), and let \(R_{ab}\) be the span of the rows of \(R(G_v^{ab}, q)\) at \(ab\). Then the row space of \(R(G_v^{ab}, q)\) is \(W + R_{ab}\), and the two rank equations give

As \(k' \le D - 2 {\lt} D - 1\), the rows at \(ab\) raise \(\dim W\) by fewer than \(D - 1\).

The \(D - 1\) rows \(r_1, \dots , r_{D-1}\) of \(R(G_v^{ab}, q)\) at \(ab\) form a basis of \(R_{ab}\) (definition 19.5). The proof of lemma 23.143 applies to every basis of \(R_{ab}\). So some \(r_i\) lies in \(W + \operatorname {span}\{ r_j : j \neq i\} \). The rows of \(R(G_v, q)\) and the \(r_j\) with \(j \neq i\) then span \(W + R_{ab}\). So removing the row \(r_i\) does not lower the rank of \(R(G_v^{ab}, q)\), which is KT eq. (6.23).

The screw assignment \(S - S_{S(v)}\) has the value \(0\) at \(v\), so \(f(S - S_{S(v)}) = 0\). By linearity, \(f(S) = f(S_{S(v)})\).

Each value of \(\Phi (S)\) is a linear function of \(S\), so \(\Phi \) is linear. Let \(\Psi \) be the map with \(\Psi (S)(v) = S(v) - S(a)\) and \(\Psi (S)(u) = S(u)\) for \(u \neq v\). As \(a \neq v\), both maps keep the value at \(a\). Hence \(\Psi (\Phi (S))(v) = (S(v) + S(a)) - S(a) = S(v)\) and \(\Phi (\Psi (S))(v) = (S(v) - S(a)) + S(a) = S(v)\), and both composites keep the values at the other bodies. So \(\Psi \) is the inverse of \(\Phi \). As \(\Phi (S)(a) = S(a)\), we have

which is \(0\) when \(S(v) = 0\). So the composite depends only on \(S(v)\) (lemma 23.148).

Let us check that composing with \(\Phi \) is Katoh and Tanigawa’s column operation. Let \(f\) be a linear form on the screw assignments. On the screw assignments that vanish off \(v\), \(\Phi \) is the identity, so \(f \circ \Phi \) has the entries of \(f\) in the columns of \(v\). If \(S\) vanishes off \(a\), then \(\Phi (S)\) has the value \(S(a)\) at both \(v\) and \(a\). So the entries of \(f \circ \Phi \) in the columns of \(a\) are the sums of those of \(f\) in the columns of \(a\) and in the columns of \(v\). The entries in the other columns do not change. Replacing each row \(f\) of \(R(G, p_1)\) by \(f \circ \Phi \) therefore adds the columns of \(v\) to those of \(a\) (KT eqs. (6.14) and (6.15)). As \(\Phi \) is invertible, \(f \mapsto f \circ \Phi \) is an injective linear map, so it does not change the rank. The rows of \(va\) are forms \(S \mapsto x(S(v) - S(a))\), so they become forms \(S \mapsto x(S(v))\), with zero entries outside the columns of \(v\), as in KT eq. (6.15).

The hinge of \(e\) is \(C(q_0(u), q_0(w))\) in \((G, q_0)\) and \(C(q(u), q(w))\) in \((G', q)\) (definition 23.5), and these are equal. The rows at \(e\) are the forms \(S \mapsto x(S(u) - S(w))\) for the linear forms \(x\) on \(K^D\) that lemma 23.91 assigns to the hinge. They depend only on the hinge and on \(u\) and \(w\), not on the rest of the graph.

We follow Katoh and Tanigawa  [ KT11 , p. 685 ] . Let \(R_{ab}\) be the span of the rows at \(ab\). As \(a \neq b\), the map \(x \mapsto (S \mapsto x(S(a) - S(b)))\) is injective. It takes the linear forms on \(K^D\) that vanish on \(C(q(a), q(b))\) onto \(R_{ab}\) (definition 19.5). So every basis of \(R_{ab}\) consists of the rows at \(ab\) for one choice of \(r_1, \dots , r_{D-1}\). By lemma 23.145, for one such choice a row \(R(G_v^{ab}, q; (ab)i^*)\) lies in the span of \(W\) and the other rows at \(ab\). Write it as \(g - \sum _{j \neq i^*} \lambda _j R(G_v^{ab}, q; (ab)j)\) with \(g \in W\), and let \(\lambda _{i^*} = 1\). Then \(\sum _j \lambda _j R(G_v^{ab}, q; (ab)j) = g\) lies in \(W\).

The edges of \(G_v\) have the same rows in \(R(G_v, q)\) as in \(R(G_v^{ab}, q)\) (lemma 21.12). Writing \(g\) as a combination \(-\sum _{e, j} \lambda _{ej} R(G_v^{ab}, q; e_j)\) of these rows gives the relation, which is KT eq. (6.24) with \(\lambda _{(ab)j} = \lambda _j\), and \(\lambda _{i^*} = 1\) is KT eq. (6.25).

We follow Katoh and Tanigawa  [ KT11 , pp. 682–685 ] . Let \(m = |V(G_v^{ab})|\), which is at least \(1\), as \(a\) is a vertex of \(G_v^{ab}\). As the motions of \((G_v^{ab}, q)\) are constant on its vertex set, \(\operatorname {rank} R(G_v^{ab}, q) = D(m - 1)\), as in the proof of lemma 23.124. This is KT eq. (6.18). The edges of \(G_v\) are edges of \(G_v^{ab}\) with the same ends (lemma 21.12), and so with the same rows (lemma 23.91). So the row space of \(R(G_v, q)\) lies in that of \(R(G_v^{ab}, q)\), and \(\operatorname {rank} R(G_v, q) \le D(m - 1)\). Let \(k' = D(m - 1) - \operatorname {rank} R(G_v, q)\). Then \(\operatorname {rank} R(G_v, q) = D(m - 1) - k'\), which is KT eq. (6.22), and the hypothesis gives \(k' \le D - 2\). As \(G_v\) and \(G_v^{ab}\) have the same vertices, the hypotheses of lemma 23.145 hold.

So lemma 23.152 gives the forms, the index \(i^*\) and the scalars of KT eqs. (6.24) and (6.25), where \(r_j\) is the row \(S \mapsto r'_j(S(a) - S(b))\) for the linear forms \(r'_j\) on \(K^D\) of that lemma. The form \(\sum _j \lambda _j r_j\) is not zero, as the \(r_j\) are linearly independent and \(\lambda _{i^*} = 1\).

By KT eq. (6.24) (lemma 23.152), the combination is the zero form. So is its restriction to the screw assignments that vanish off \(u\).

We follow Katoh and Tanigawa  [ KT11 , pp. 690–691 ] . By KT eq. (6.43) (lemma 23.156), the combination of the rows of \(R(G_v^{ab}, q)\) in KT eq. (6.24) vanishes on the screw assignments that vanish off \(a\). Let \(S\) be such a screw assignment, with \(S(a) = y\). A row \(S \mapsto x(S(u) - S(w))\) at an edge with ends \(u, w \neq a\) vanishes at \(S\). As the edges of \(G\) at \(a\) are \(va\) and \(ac\), the edges of \(G_v^{ab}\) at \(a\) are \(ab\) and \(ac\) (definition 21.10). Their rows take the values \(r_j(y)\) and \(s_j(y)\) at \(S\). So

for every \(y \in K^D\), which is KT eq. (6.44).

We argue as in the proof of lemma 23.127. Suppose that \(\sum _i c_i \varphi _i + c\, \omega + \sum _j c'_j \psi _j = 0\). Every \(\psi _j\) vanishes on the screw assignments that vanish off \(v\), so \(\sum _i c_i \varphi _i + c\, \omega \) vanishes on them. By the first hypothesis, \(c\) and every \(c_i\) are \(0\). Then \(\sum _j c'_j \psi _j = 0\), and every \(c'_j\) is \(0\).

As \(C\) is bilinear and \(C(n, n) = 0\) (definition 23.1),

So \(vb\) has in \((G, q_0)\) the hinge that \(ab\) has in \((G_v^{ab}, q)\). The two rows with the same index are therefore the forms \(S \mapsto x(S(v) - S(b))\) and \(S \mapsto x(S(a) - S(b))\) for one form \(x\), which vanishes on \(C(q(a), q(b))\) (lemma 23.91). As \((S(v) - S(b)) - (S(a) - S(b)) = S(v) - S(a)\), their difference is \(S \mapsto x(S(v) - S(a))\). By lemma 23.149, the composite in the last claim is \(S \mapsto x(S(v))\).

The first claim holds as \(R(G_v^{ab}, q; (ab)j)\) is the form \(S \mapsto r_j(S(a) - S(b))\). Each \(r_j\) vanishes on \(C(q(a), q(b))\), and so does \(r\). Moreover \(r \neq 0\), as the \(r_j\) are linearly independent and \(\lambda _{i^*} = 1\). As in the proof of lemma 23.161, the hinge of \(vb\) in \((G, q_0)\) is \(C(q(a), q(b))\). So \(S \mapsto r(S(v) - S(b))\) is a row of \(R(G, q_0)\) at \(vb\) (definition 19.5). The identity holds as \((S(v) - S(b)) - (S(a) - S(b)) = S(v) - S(a)\).

This is Katoh and Tanigawa’s row operation, applied to the whole relation at once  [ KT11 , pp. 685–686 ] . In \((G, p_1)\), as in \((G, q_0)\), the hinge of \(vb\) is \(q(ab)\). So the rows \(R(G, p_1; (vb)j)\) of \(R(G, p_1)\) at \(vb\) are the forms \(S \mapsto r_j(S(v) - S(b))\), and they correspond to the rows \((ab)j\) of \(R(G_v^{ab}, q)\) (KT eq. (6.16)). The edges of \(G_v\) have the same rows in both matrices (KT eq. (6.13)). Katoh and Tanigawa add to the row \((vb)i^*\) the combination of the rows \((vb)j\) with \(j \neq i^*\) and of the rows of the edges of \(G_v\), with the coefficients of KT eq. (6.24) (KT eq. (6.26)). As \(\lambda _{i^*} = 1\), the result is the row of KT eq. (6.27):

as forms in \(S\), by KT eq. (6.24). We call this row \(w\). It reads \(S(a)\) as well as \(S(v)\): on the screw assignments that vanish off \(a\) it is the form \(S \mapsto -r(S(a))\), which is not zero. Its entries outside the columns of \(v\) vanish only after the column operation, which turns it into the form \(S \mapsto r(S(v))\) (lemma 23.149). This is the row \((r, 0)\) of KT eq. (6.28).

The extensor \(C(L)\) is not zero (definition 18.8). So the linear forms on \(K^D\) that vanish on \(C(L)\) form a space of dimension \(D - 1\) (definition 19.5), and \(r'_1, \dots , r'_{D-1}\) is a basis of it. A form vanishes on \(\operatorname {span} C(L)\) if and only if it vanishes at \(C(L)\) (the remark after definition 19.5). So \(r(C(L)) = 0\) if and only if \(r\) is a linear combination of the \(r'_i\), that is, if and only if the \(D\) forms are linearly dependent.

We follow Katoh and Tanigawa  [ KT11 , pp. 685–686 ] . Let \(W\) be the row space of \(R(G_v, q)\). The edges of \(G_v\) have the same hinges in \((G, p_1)\) as in \((G_v^{ab}, q)\), so \(W\) lies in the row space of \(R(G, p_1)\). The hinge of \(vb\) is \(q(ab)\), so the form \(S \mapsto r(S(v) - S(b))\) is a row at \(vb\), as in lemma 23.163. By KT eq. (6.24), the form \(S \mapsto r(S(a) - S(b))\) lies in \(W\). So their difference, the row \(w\) of KT eq. (6.27), the form \(S \mapsto r(S(v) - S(a))\) (lemma 23.161), lies in the row space of \(R(G, p_1)\).

Let \(r'_1, \dots , r'_{D-1}\) be linearly independent linear forms on \(K^D\) that vanish on \(C(L)\), so that the rows of \(va\) are the forms \(S \mapsto r'_i(S(v) - S(a))\) (definition 19.5). After the column operation (lemma 23.149), the rows of \(va\) and \(w\) are the forms \(S \mapsto r'_i(S(v))\) and \(S \mapsto r(S(v))\). They have entries only in the columns of \(v\). There they form the top-left \(D \times D\) block of KT eq. (6.29), with the rows \(r'_1, \dots , r'_{D-1}\) and \(r\). This block is nonsingular exactly when \(r\) is not a combination of the \(r'_i\), that is, when \(r\) does not vanish on \(C(L)\) (KT p. 690, lemma 23.165). The other rows of KT eq. (6.29) are the rows \((vb)j\) with \(j \neq i^*\) and the rows of the edges of \(G_v\). Outside the columns of \(v\) they have the entries of \(R(G_v^{ab} \setminus (ab)i^*, q)\), the matrix \(R(G_v^{ab}, q)\) without the row \((ab)i^*\). By KT eq. (6.24) that row lies in the span of the others, so KT eq. (6.23) holds:

Now suppose that \(r\) does not vanish on \(C(L)\). Then the \(r'_i\) and \(r\) span the linear forms on \(K^D\). So for every linear form \(x\) on \(K^D\), the form \(S \mapsto x(S(v) - S(a))\) is a combination of the rows of \(va\) and of \(w\), and it lies in the row space of \(R(G, p_1)\). Hence so does each row \((ab)j\) of \(R(G_v^{ab}, q)\):

In KT eq. (6.29) this is the row operation, with the rows of the top-left block, that clears the entries below it. So every row of \(R(G_v^{ab} \setminus (ab)i^*, q)\) lies in the row space of \(R(G, p_1)\), and we choose \(D(|V(G)| - 2)\) linearly independent ones. They vanish on the screw assignments that vanish off \(v\), as no edge of \(G_v^{ab}\) has the end \(v\) (as in the proof of lemma 23.157). On these screw assignments the rows of \(va\) and \(w\) are the forms \(S \mapsto r'_i(S(v))\) and \(S \mapsto r(S(v))\), which are linearly independent. By lemma 23.159, the \((D - 1) + 1 + D(|V(G)| - 2) = D(|V(G)| - 1)\) forms are linearly independent. They lie in the row space of \(R(G, p_1)\), so its rank is at least \(D(|V(G)| - 1)\) (lemma 19.34).

The extensors \(E_{ij}\) are those of lemma 18.9. By the remark after that lemma, they form a basis of \(\bigwedge ^{d-1} K^{d+1}\). At \(d = 3\), \(E_{ij} = \bar p_k \vee \bar p_l\), where \(k {\lt} l\) and \(\{ k, l\} = \{ 1, 2, 3, 4\} \setminus \{ i, j\} \). So the six joins are the six \(E_{ij}\), and they span \(\bigwedge ^2 K^4\).

The linear forms \(r\) and \(0\) agree on \(S\), hence on its span, which is \(\bigwedge ^{d-1} K^{d+1}\).

The map \(K^{d+1} \to K^m\) that sends \(x\) to \((x \cdot n_1, \dots , x \cdot n_m)\) is linear, and \(m {\lt} d + 1\). By rank–nullity its kernel has dimension at least \(d + 1 - m {\gt} 0\), so it contains a nonzero vector.

As \(K\) is infinite and \(P \neq 0\), there is a \(t \in K^\sigma \) with \(P(t) \neq 0\). At this \(t\) the vectors \(\bar p_1(t), \dots , \bar p_4(t)\) have a nonzero determinant, so the points are affinely independent (lemma 18.2).

Let \(P\) be the determinant of the \(4 \times 4\) matrix of polynomials with the rows \((P_{i1}, P_{i2}, P_{i3}, 1)\). Evaluation at \(t\) is a ring homomorphism \(K[\sigma ] \to K\), so it commutes with the determinant. It takes the row \((P_{i1}, P_{i2}, P_{i3}, 1)\) to \((p_i(t), 1) = \bar p_i(t)\) (definition 18.1). So \(P(t)\) is the determinant of \(\bar p_1(t), \dots , \bar p_4(t)\).

Let \(n_a, n_b, n_c\) be the first three standard basis vectors of \(K^4\), so that \(\Pi (a)\), \(\Pi (b)\) and \(\Pi (c)\) are the coordinate planes \(x_1 = 0\), \(x_2 = 0\) and \(x_3 = 0\) of \(K^3\). Let \(p_1 = 0\), \(p_2 = e_3\), \(p_3 = e_1\) and \(p_4 = e_2\), where \(e_1, e_2, e_3\) is the standard basis of \(K^3\). The point \(p_1\) lies on the three planes. The point \(e_3\) lies on \(\Pi (a)\) and \(\Pi (b)\) but not on \(\Pi (c)\), and \(e_1\) and \(e_2\) have the incidences required of \(p_3\) and \(p_4\) in the same way. Subtracting \(\bar p_1 = (0, 0, 0, 1)\) from the other rows of the matrix with the rows \(\bar p_1, \dots , \bar p_4\) leaves a permutation matrix, so its determinant is \(\pm 1\). Hence the points are affinely independent (lemma 18.2).

The linear map \(K^4 \to K^3\) that sends \(x\) to \((x \cdot n_a, x \cdot n_b, x \cdot n_c)\) has the linearly independent rows \(n_a\), \(n_b\), \(n_c\), so it has rank \(3\) and is onto. Let \(\bar p_2\), \(\bar p_3\) and \(\bar p_4\) be vectors that it sends to \((0, 0, 1)\), \((1, 0, 0)\) and \((0, 1, 0)\). Let \(\bar p_1\) be a nonzero vector orthogonal to \(n_a\), \(n_b\) and \(n_c\) (lemma 23.172, with \(m = 3\)). These vectors have the incidences of the claim.

Suppose that \(\sum _i g_i \bar p_i = 0\). The dot product with \(n_a\) gives \(g_3 = 0\), that with \(n_b\) gives \(g_4 = 0\), and that with \(n_c\) gives \(g_2 = 0\). Then \(g_1 \bar p_1 = 0\), and \(g_1 = 0\) as \(\bar p_1 \neq 0\). So the four vectors are linearly independent.

We follow Katoh and Tanigawa  [ KT11 , p. 690 ] . The forms \(r_1, \dots , r_{D-1}\) are linearly independent, and \(\lambda _{i^*} = 1\) (KT eq. (6.25), lemma 23.152). A linear combination of linearly independent vectors with a nonzero coefficient is not zero.

We argue as in the proof of lemma 23.167, with the roles of \(a\) and \(b\) exchanged (KT pp. 682 and 686). Let \(W\) be the row space of \(R(G_v, q)\), which lies in the row space of \(R(G, p_2)\). The hinge of \(va\) is \(q(ab)\), on which \(r\) vanishes, so the form \(S \mapsto r(S(v) - S(a))\) is a row at \(va\). By KT eq. (6.24), the form \(S \mapsto r(S(a) - S(b))\) lies in \(W\). So their sum, the form \(S \mapsto r(S(v) - S(b))\), lies in the row space of \(R(G, p_2)\).

Let \(r'_1, \dots , r'_{D-1}\) be linearly independent forms that vanish on \(C(L')\), so that the rows of \(vb\) are the forms \(S \mapsto r'_i(S(v) - S(b))\). Katoh and Tanigawa add the columns of \(v\) to those of \(b\) (KT p. 682). By lemma 23.149, with \(b\) in place of \(a\), this turns the rows of \(vb\) and the form \(S \mapsto r(S(v) - S(b))\) into the forms \(S \mapsto r'_i(S(v))\) and \(S \mapsto r(S(v))\). These make up the block \(M_2\), with the rows \(r'_1, \dots , r'_{D-1}\) and \(r\), which is nonsingular exactly when \(r(C(L')) \neq 0\) (lemma 23.165). The count in the proof of lemma 23.167, with \(b\) in place of \(a\), then gives the bound. In it the rows of the edges of \(G_v^{ab}\) vanish on the screw assignments that vanish off \(v\), as no edge of \(G_v^{ab}\) has the end \(v\) (as in the proof of lemma 23.157).

The edges of \(G_a^{vc}\) are \(vb\), the new edge \(vc\), and the edges of \(G\) that meet neither \(v\) nor \(a\). Those of \(G_v^{ab}\) are \(ac\), the new edge \(ab\), and the same edges of \(G\) (definition 21.10). The bijection \(\rho \) fixes the ends of an edge that meets neither \(v\) nor \(a\). It maps the ends \(v, b\) of \(vb\) to \(a, b\), and the ends \(v, c\) of \(vc\) to \(a, c\).

The normals \(q_\rho \) are those of \(q\), permuted, so they are in general position (definition 23.9). By lemma 23.183, an edge of \(G_a^{vc}\) with ends \(u\) and \(w\) corresponds to an edge of \(G_v^{ab}\) with ends \(\rho (u)\) and \(\rho (w)\). The two edges have the same hinge \(C(q(\rho (u)), q(\rho (w)))\), which is KT eq. (6.31).

For a screw assignment \(S\), let \(S_\rho \) agree with \(S\) except that \(S_\rho (a) = S(v)\) and \(S_\rho (v) = S(a)\). Then \(S_\rho (\rho (u)) - S_\rho (\rho (w)) = S(u) - S(w)\) for all vertices \(u, w\) of \(G_a^{vc}\). So if \(S\) is an infinitesimal motion of \((G_a^{vc}, q_\rho )\), then \(S_\rho \) is one of \((G_v^{ab}, q)\). Then \(S_\rho \) is constant on \(V(G_v^{ab})\), which is the image of \(V(G_a^{vc})\) under \(\rho \), and so \(S\) is constant on \(V(G_a^{vc})\). The last claim follows, as a realization of a \(0\)-dof-graph is at the target rank exactly when its infinitesimal motions are constant on the vertices (remark 23.23).

Let \(uw\) be an edge of \(G_a^{vc}\). As in the proof of lemma 23.185, the edge \(\rho (u)\rho (w)\) of \(G_v^{ab}\) has the same hinge, and every edge of \(G_v^{ab}\) arises in this way. For a form \(x\) that vanishes on this hinge, the row \(f \colon S \mapsto x(S(\rho (u)) - S(\rho (w)))\) of \(R(G_v^{ab}, q)\) satisfies \(f(S_\rho ) = x(S(u) - S(w))\). This proves the first claim. The map \(S \mapsto S_\rho \) is a linear automorphism, so \(f \mapsto (S \mapsto f(S_\rho ))\) is injective, and the two row spaces have the same dimension.

We follow Katoh and Tanigawa  [ KT11 , pp. 686–689 ] . Let \((G_a^{vc}, q_\rho )\) be the panel-hinge framework of lemma 23.185. In it the panel of \(v\) is \(\Pi (a)\), the hinge of \(vb\) is \(q(ab)\), the hinge of the new edge \(vc\) is \(q(ac)\), and the other edges keep their hinges (KT eq. (6.31)). Katoh and Tanigawa obtain \((G, p_3)\) from \((G_a^{vc}, q_\rho )\) as \((G, p_1)\) from \((G_v^{ab}, q)\), with \(a\), \(c\) and \(v\) in the roles of \(v\), \(a\) and \(b\). So \(va\) gets the hinge \(q(ac)\) of \(vc\), and \(ac\) gets \(L''\) (KT eqs. (6.32) and (6.33)).

Write the rows of \(R(G_v^{ab}, q)\) at \(ab\) and at \(ac\) as the forms \(S \mapsto r_j(S(a) - S(b))\) and \(S \mapsto s_j(S(a) - S(c))\), as in lemma 23.157. The rows of \(R(G, p_3)\) at \(vb\) and at \(va\) are then the forms \(S \mapsto r_j(S(v) - S(b))\) and \(S \mapsto s_j(S(v) - S(a))\). The edges of \(G\) other than \(va\), \(vb\) and \(ac\) have the same rows in both matrices (KT eq. (6.34)). Katoh and Tanigawa add to the row \((vb)i^*\) the other terms of KT eq. (6.24), with the rows at \(vb\) and \(va\) in place of those at \(ab\) and \(ac\) (KT eq. (6.40)). Call the result \(w_3\). For a screw assignment \(S\), let \(S'\) agree with \(S\) except that \(S'(a) = S(v)\). The edges of \(G_v\) other than \(ac\) do not meet \(a\), so KT eq. (6.24) at \(S'\) reads

where \(e\) runs over the edges of \(G_v\). Subtracting this from \(w_3(S)\) leaves

by KT eq. (6.44) (lemma 23.157). So \(w_3\) is the form \(S \mapsto r(S(a) - S(c))\).

Let \(r'_1, \dots , r'_{D-1}\) be linearly independent forms that vanish on \(C(L'')\), so that the rows of \(ac\) are the forms \(S \mapsto r'_i(S(a) - S(c))\). Katoh and Tanigawa add the columns of \(a\) to those of \(c\) (KT eqs. (6.35) and (6.36)). By lemma 23.149, with \(a\) and \(c\) in place of \(v\) and \(a\), this turns the rows of \(ac\) and \(w_3\) into the forms \(S \mapsto r'_i(S(a))\) and \(S \mapsto r(S(a))\). These make up the top-left block \(M_3\) of KT eq. (6.41), with the rows \(r'_1, \dots , r'_{D-1}\) and \(r\), up to sign. Katoh and Tanigawa write its last row as \(\sum _j \lambda _{(ac)j} s_j\), which is \(-r\). The block is nonsingular exactly when \(r(C(L'')) \neq 0\) (lemma 23.165).

Now suppose that \(r(C(L'')) \neq 0\). Then the \(r'_i\) and \(r\) span the linear forms on \(K^D\). So for every linear form \(x\) on \(K^D\), the form \(S \mapsto x(S(a) - S(c))\) lies in the row space of \(R(G, p_3)\). Hence so does every row of \(R(G_a^{vc}, q_\rho )\). Its rows at \(vb\) and at the edges of \(G\) that meet neither \(v\) nor \(a\) are rows of \(R(G, p_3)\). Its rows at \(vc\) are the forms

sums of a row of \(va\) and a form of this kind. By lemma 23.187, \(R(G_a^{vc}, q_\rho )\) has the rank \(D(|V(G)| - 2)\) of \(R(G_v^{ab}, q)\), and we choose \(D(|V(G)| - 2)\) linearly independent rows of it. They vanish on the screw assignments that vanish off \(a\), as \(a\) is not a vertex of \(G_a^{vc}\). On these screw assignments the rows of \(ac\) and \(w_3\) are the forms \(S \mapsto r'_i(S(a))\) and \(S \mapsto r(S(a))\), which are linearly independent. By lemma 23.159, the \((D - 1) + 1 + D(|V(G)| - 2) = D(|V(G)| - 1)\) forms are linearly independent. They lie in the row space of \(R(G, p_3)\), which gives the bound. Katoh and Tanigawa read the same bound off KT eq. (6.41), as \(D\) plus the rank of the block below \(M_3\), which KT eq. (6.23) gives.

We follow Katoh and Tanigawa  [ KT11 , pp. 690–691 ] . The form \(r\) is not zero (lemma 23.179). For \(d = 3\), the screw space \(K^D\) is \(\bigwedge ^2 K^4\), and the six joins \(\bar p_i \vee \bar p_j\) span it (lemma 23.169, from KT Lemma 2.1). If \(r\) vanished on all of them, then \(r\) would be zero (lemma 23.171).

This gives KT Claim 6.12: for some choice of the lines \(L \subset \Pi (a)\), \(L' \subset \Pi (b)\) and \(L'' \subset \Pi (c)\), one of \(M_1\), \(M_2\) and \(M_3\) is nonsingular. Indeed, let \(c\) be as at the start of this subsubsection, with \(q(a)\), \(q(b)\) and \(q(c)\) linearly independent. Let \(\bar p_1, \dots , \bar p_4\) be the vectors of lemma 23.177 for these normals. Any two of them are orthogonal to a common normal \(q(u)\), with \(u \in \{ a, b, c\} \). Let \(\bar p_i \vee \bar p_j\) be a join on which \(r\) does not vanish, and let \(n'\) be a vector orthogonal to \(\bar p_i\) and \(\bar p_j\) and linearly independent of \(q(u)\). Then \(\bar p_i \vee \bar p_j\) is a multiple of \(C(q(u), n')\) (lemma 22.16), so \(r\) does not vanish on \(C(q(u), n')\). This is a nonzero multiple of the extensor of the line in which \(\Pi (u)\) meets the plane with normal \(n'\) (the remarks after definition 23.1). So the block \(M_1\), \(M_2\) or \(M_3\) for this line is nonsingular, by lemma 23.165 and, for \(M_3\), KT eq. (6.44).

The lines are chosen after \(r\), and the proof needs this freedom. The extensors of three fixed lines span at most three of the six dimensions of \(\bigwedge ^2 K^4\), so a nonzero \(r\) can vanish on all three.