On Spaces of Infinitesimal Motions and Three Dimensional Henneberg Extensions
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Abstract
We investigate certain spaces of infinitesimal motions arising naturally in the rigidity theory of bar and joint frameworks. We prove some structure theorems for these spaces and, as a consequence, are able to deduce some special cases of a long standing conjecture of Graver, Tay and Whiteley concerning Henneberg extensions and generically rigid graphs.
Keywords
Frameworks Rigidity Henneberg movesMathematics Subject Classification
52C251 Introduction
1.1 Motivation
In the rigidity theory of bar and joint frameworks the Henneberg extensions play an important role. Let \(G=(V,E)\) be a simple finite graph and let \(n\) and \(k\) be positive integers, and suppose that \(X \subset V\) with \(X = n+k\) and \(F \subset E(X)\) with \(F =k\). We can form a new graph \(G'\) by deleting all the edges in \(F\) and adding a new vertex of degree \(n+k\) that is adjacent to all the vertices of \(X\). We say that \(G'\) is an n dimensional Henneberg kextension of \(G\) that is supported on the vertex set \(X\) and the edge set \(F\). A major stumbling block to a full understanding of the generic rigidity theory of three dimensional bar and joint frameworks is the lack of good sufficient conditions that ensure that a three dimensional Henneberg 2extension preserves generic rigidity. By contrast the planar situation is relatively well understood, and we have an extensive theory of planar rigidity stemming largely from Laman’s Theorem [7]. Laman’s result can be proved by an inductive argument based on two dimensional Henneberg extensions, so it is obviously of interest to see if similar lines of reasoning can be pursued in the three dimensional case.
Given a graph \(G\) and vertices \(i\) and \(j\) of \(G\), we say that \(ij\) is an implied edge in three dimensions if any flex of a generic three dimensional framework based on \(G\) is an infinitesimal isometry of the set of points corresponding to \(\{i,j\}\) (see Sect. 2 for the relevant background material). In particular, every edge of \(G\) is an implied edge, but there may be implied edges that are not edges of \(G\). An implied \(K_4\) is a set of implied edges of \(G\) that form a complete graph on 4 vertices. The following Conjecture of Graver, Tay and Whiteley remains open. For a discussion of this conjecture and many other related problems the reader should consult [4] (wherein it is referred to as the Henneberg Conjecture) or, for a more recent discussion, [5, Conjecture 2.7].
Conjecture 1
Let \(G=(V,E)\) be a generically 3isostatic graph and suppose that \(X\) is a subset of \(V\) such that \(X =5\). Let \(e \) and \(f\) be distinct edges whose vertices are in \(X\). Suppose that \(G\{e,f\}\) does not contain any implied \(K_4\) whose vertices are a subset of \(X\). Then the Henneberg extension that removes \(e\) and \(f\) and adjoins a vertex with neighbour set \(X\) results in a generically 3rigid graph.
A proof of this conjecture would be a significant step towards a better understanding of generic rigidity in 3space as it would allow us to characterise generically rigid graphs as precisely those that may be obtained from \(K_3\) by a sequence of allowable Henneberg \(k\)extensions, where \(k=0,1\) or \(2\).
Conjecture 1 has been proved in certain special cases. For example Graver [4, Sect. 5.5] has shown that if \(E(X) \{e,f\}\) contains a 5cycle then the conjecture is true. Jackson and Owen [6] have shown that if \(E(X)\) contains a 3cycle containing \(e\), and \(e\) and \(f\) are vertex disjoint then the conjecture is true.
In this article we investigate the infinitesimal dynamic properties of rigid frameworks that admit nonrigid Henneberg 2extensions. More precisely, we should say that the underlying graph admits a 2extension that is not generically rigid. As an application of our main result, we are able to prove Conjecture 1 in the case where \(E(X)\ge 7\)—generalising the result of Graver mentioned above. Note that, in the case where \(e\) and \(f\) are vertex disjoint, this result is already implied by the result of Jackson and Owen mentioned above. However, their proof does rely on the vertex disjointness of \(e\) and \(f\). Our result is therefore logically independent of theirs.
1.2 Structure of the Paper
In Sect. 2 we review some of the basic rigidity theory of frameworks—there are no new results in this section. However we present this material in a matrix algebraic setting that is appropriate to our later needs.
In Sects. 3, 4 and 5 we introduce and study the notion of \(p \)admissibility. This is the central concept of the paper and essentially captures the infinitesimal dynamic properties of any rigid framework that admits a Henneberg extension that is not generically rigid. We make use of a standard matrix identity known as the Sherman–Morrison formula to organise some of the messy polynomial equations that arise when considering the infinitesimal flexes of a framework. The main technical result of the paper is Theorem 15 which provides interesting geometric information about any possible counterexample to Conjecture 1. It is also, we believe, of independent interest as it seems to suggest a nonobvious connection between Euclidean rigidity and affine rigidity in the sense of [3].
Finally in Sect. 6, we apply the results of the previous sections to prove some special cases of Conjecture 1. We also point out a fundamental obstruction to a proof of the general case.
1.3 Notation
1.3.1 Matrices
We write \(\mathbb R^n\) for the space of \(n \times 1\) column vectors with real entries. Throughout this article, we will write \(\mathbb R^{m\times n}\) for the set of matrices with real entries that have \(m\) rows and \(n\) columns. For \(p \in \mathbb R^{m \times n}\), we write \(p_i\) for the \(i \)th column of \(p\). We write Open image in new window for the column vector all of whose entries are \(1\) and we write \(I\) for the identity matrix. We write \(m^\mathrm{T}\) for the transpose of the matrix \(m\). In particular, for \(u,p \in \mathbb R^n\) the dot product of \(u\) and \(p\) is \(u^\mathrm{T}p\).
1.3.2 Graphs
In the article graphs are simple undirected graph with no loops or parallel edges. If \(G=(V,E)\) is such a graph and \(E'\subset E \), then \(GE'\) denotes the graph obtained by removing all the edges in \(E'\) from \(G\). If \(E' \subset E\) then \(V(E)\) is the set of vertices in \(V\) spanned by \(E'\). If \(V'\subset V\) then \(E(V')\) is the set of edges in \(E\) spanned by \(V'\). \(K(X)\) denotes the complete graph with vertex set \(X, K_n\) denotes the complete graph with \(n \) vertices and \(K_{p,q}\) denotes the complete bipartite graph with vertex parts of order \(p\) and \(q\).
1.3.3 Miscellaneous
We write \(\{\{ \cdots \}\}\) to indicate a multiset. Statements such as “\(P(x)\) is true for almost all \(x\) in \(\mathbb R^3\)” mean that there is an open dense subset \(O \subset \mathbb R^3\) such that \(P(x)\) is true for all \(x \in O\).
2 Point Configurations and Motions
2.1 Infinitesimal Isometries of Euclidean Space
Lemma 2
Given an \(n\times n\) skew symmetric matrix \(a\), let \(\xi ^a \) be the vector field on \(\mathbb R^{n}\) defined by \(\xi ^a_x = ax\). The mapping \(a \mapsto \xi ^a\) is a linear isomorphism between the space of skew symmetric matrices and \(\mathcal I_0\).
Proof
Let \(\xi \in \mathcal I_0\). Let \(a_\xi \) be the \(n \times n \) matrix whose \(i\)th column is \(\xi _{e_j}\) where \(e_j\) is the \(j\)th standard basis vector of \(\mathbb R^{n}\). We leave it to the reader to verify that \(\xi \mapsto a_\xi \) is the inverse of the map defined in the statement. \(\square \)
Lemma 3
For any \(\xi \in \mathcal I \) there is a unique \(t \in \mathbb R^{n }\) and a unique skew symmetric \(a \in \mathbb R^{n \times n}\) such that \(\xi _x = t +ax\) for all \(x \in \mathbb R^{ n}\).
Proof
Let \(t = \xi _0\) and observe that \(t\) is the unique element of \(\mathbb R^{n}\) such that \(\xi t \in \mathcal I_0\). Now the lemma follows immediately from Lemma 2. \(\square \)
2.2 Point Configurations
Let \(n\) be a fixed positive integer. A \(k\)point configuration, \(p\), in \(\mathbb R^n\) is just an element of \(\mathbb R^{n\times k}\). We think of \(p_i\) as the \(i \)th point of the configuration.
Observe that any \(\xi \in \mathcal I\) induces an infinitesimal isometry \(u\), of \(p\), defined by \(u_i = \xi _{p_i}\). In this way, we can linearly embed \(\mathcal I \) in \(\mathbb R^{n \times k}\). We denote the image of \(\mathcal I\) in \(\mathbb R^{n \times k}\) by \(\mathcal I_p\). It is easily seen that this embedding is injective if and only if the affine span of \(p\) has dimension at least \(n1\). On the other hand, it is also easily seen that any infinitesimal isometry of \(p\) is induced by a unique (global) infinitesimal isometry of the affine span of \(p\). In particular, any infinitesimal isometry of \(p\) is induced by some infinitesimal isometry of \(\mathbb R^{n}\).
Lemma 4
Given \(p \in \mathbb R^{n \times k}\) and \(u \in \mathbb R^{n \times k}\), then \(u\) is an infinitesimal isometry of \(p\) if and only if there is some skew symmetric matrix \(a\in \mathbb R^{n\times n}\) and some \(t \in \mathbb R^{n}\) such that Open image in new window .
Proof
As we have just observed, \(u\) is an infinitesimal isometry of \(p\) if and only there is some infinitesimal isometry \(\xi \) of \(\mathbb R^{n}\) such that \(\xi _{p_i} = u_i\). Now the result follows immediately from Lemma 3. \(\square \)
We say that motions \(u\) and \(u'\) of \(p\) are \(p \)equivalent if \(uu' \in \mathcal I_p\). We also can extend this notion to subspaces of \(\mathbb R^{n\times k}\). Thus, linear subspaces \(S\) and \(S'\) of \(\mathbb R^{n\times k}\) are said to to be pequivalent if they have the same dimension and they have the same image under the projections \(\mathbb R^{n\times k} \rightarrow \mathbb R^{n\times k}/\mathcal I_P\). One readily checks that \(S\) and \(S'\) are \(p \)equivalent if and only if there is a linear isomorphism \(f:S \rightarrow S'\) such that \(uf(u) \in \mathcal I_p\) for all \(u \in S\).
The following notions will be important later on. We say that a vector field \(\xi \) on \(\mathbb R^n\) is a linear field if the function \(x \mapsto \xi _x\) is linear. We say that \(\xi \) is an affine field if the function \(x \mapsto \xi _x\) is an affine linear function. Given a motion \(u\) of a point configuration \(p\), we say that \(u\) is a linear motion, respectively an affine motion, if it is the restriction to \(p\) of a linear vector field, respectively an affine vector field. We remark that if \(p \in \mathbb R^{n \times (n+1)}\) is in general position then any \(u\in \mathbb R^{n \times (n+1)} \) is an affine motion of \(p\). On the other hand the affine motions of a general \(q \in \mathbb R^{n\times (n+2)}\) form a codimension \(n\) affine subspace of the vector space of all motions of \(q\).
2.3 Frameworks
In this section we review some elementary facts about frameworks. An \(n\) dimensional framework is a pair \((G,p)\) where \(G\) is a simple undirected graph with vertex set \(V \) and edge set \(E\) and where \(p\in \mathbb R^{n \times V}\). A \(G\)framework is a framework whose underlying graph is \(G\). For convenience, we will assume for the moment that \(V = \{1,\ldots ,k\}\).
We will be particularly concerned with frameworks whose underlying point configuration is generic. We say that \(p\) is generic if the multiset of its entries is algebraically independent over \(\mathbb Q\). We observe that generic configurations are in particular in general position. Also, we observe that if \(p\) is generic then every submatrix of \(p\) has maximal rank.
We say that a graph \(G\) is generically nrigid if \((G,p)\) is rigid for any generic configuration \(p\) in \(\mathbb R^{ n\times V}\). It can be shown that if \((G,p)\) is infinitesimally rigid for some (possibly nongeneric) \(p \in \mathbb R^{n \times V}\) then \(G\) is generically rigid. On the other hand, it can happen that for a generically rigid graph \(G\) there are certain (nongeneric) \(G\)frameworks that are not infinitesimally rigid. As mentioned in Sect. 1, a fundamental open problem in combinatorial rigidity theory is to find a good characterisation of generically \(n\)rigid graphs for \(n \ge 3\). A dimension counting argument suggests that \(n\) dimensional Henneberg extensions might preserve generic \(n\)rigidity. However, as is well known, this is not in general true—see Example 6 below.
3 \(p\)Admissible Subspaces
Definition 5
 (1)
\(S\cap \mathcal I_p = 0\).
 (2)
For almost all \(x \in \mathbb R^n\) there is some nonzero \(u \in S\) such that \(u \) is the restriction to \(p\) of some motion of the framework \((K_{k,1},p^x)\).
One may wonder why we restrict to an open dense subset of \(\mathbb R^n\) in the definition of \(p\)admissibility. We do this in order to avoid the complications that would arise from degenerate configurations if we had to check the condition for all \(x \in \mathbb R^n\). In all applications of this concept that we have in mind, it is sufficient that condition (2) above be satisifed on an open dense subset.
In order to motivate Definition 5 we consider the following situation. Let \((G,\rho )\) be an isostatic framework in \(\mathbb R^n\) and let \(p \in \mathbb R^{n\times k}\) be the first \(k\) columns of \(\rho \). We assume that \(V(G) = \{1,\ldots ,l\}\) for some \(l\ge k\). Let \(E' \subset E\) be a set of edges of \(G\) such that \(k = E'+n\) and such that \(V(E') \subset \{1,\ldots ,k\}\). Since \((G, \rho )\) is isostatic the framework \((GE',\rho )\) has a \(E'\) dimensional space of flexes \(\overline{S}\) such that any nontrivial element of \(\overline{S}\) restricts to a nontrivial motion of \(p\). Thus \(\overline{S}\) induces an \(E'\) dimensional subspace \(S\) of \(\mathbb R^{n\times k}\) such that \(S\cap \mathcal I_p = 0\). It is clear that \(S\) is \(p\)admissible if and only if almost every Henneberg \(E' \)extension (of the framework) that deletes \(E'\) and adjoins a vertex adjacent to \(\{1,\ldots ,k\}\) results in a nonrigid framework. Thus, we see the relevance of understanding \(p \)admissible subspaces of \(\mathbb R^{3 \times 5}\). In particular, any rigid generic framework that admits a \(2 \)extension that is not generically rigid will give rise to a two dimensional \(p\)admissible subspace of \(\mathbb R^{3\times 5}\) where \(p\) is some generic element of \(\mathbb R^{3 \times 5}\).
Example 6
Indeed, Conjecture 1 is equivalent to the assertion that, for \(p \in \mathbb R^{3 \times 5}\), the only \(p\)admissible subspaces that arise from the deletion of two edges from a generic isostatic framework whose vertices include \(p\) are those that are \(p\)equivalent to the one described in Example 6. There are however, examples of \(p\)admissible spaces that are not equivalent to the one described in Example 6.
Example 7
One consequence of Example 7 is that any proof of Conjecture 1 must implicity or explicitly demonstrate that this \(p\)admissible space cannot arise by deleting two edges from a generic isostatic framework. In fact the situation is even more complicated, as we will show below that there are many other essentially inequivalent examples of \(p\)admissible spaces for generic \(p \in \mathbb R^{3\times 5} \).
4 \(K_{n,1}\)Frameworks
Observe that \(K_{5,1}\) is the union of two copies of \(K_{3,1} \) that have one edge in common. Thus in order to understand flexes of \(K_{5,1}\)frameworks we should first understand the flexes of \(K_{3,1}\)framework and then consider flexes of two \(K_{3,1} \)frameworks with a common edge. In this section we will begin that programme by deriving some basic results concerning flexes of \(K_{n,1}\)frameworks in \(\mathbb R^n\).
Lemma 8
Suppose that \(q\) is an invertible matrix and that \(x\) does not belong to the affine span of the rows of \(q\). Then \(\mathcal P(x,q,v)\) is the unique vector such that \(( v, \mathcal P(x,q,v) )\) is a flex of the framework \((K_{n,1}, q^x)\).
Proof
Lemma 9
Proof
This follows easily by combining Eq. 5 with the definition of \(\mathcal L(x,q,v)\). \(\square \)
Note that Open image in new window if and only if \(x \) belongs to the linear subspace that is parallel to the affine span of the columns of \(q\).
While the right hand side of Eq. 6 may seem rather complicated still, it is worth noting that the matrix Open image in new window is just the matrix of the projection parallel to Open image in new window onto the orthogonal complement of \((q^{1}x)^\mathrm{T}\). Thus the geometry of \(\mathcal L\) is more transparent than that of \(\mathcal P\).
We also note that, given \(x, \mathcal L(x,q,v) \in x^\perp \) for all \(q\) and \(v\). Thus, if \(\pi \) is a fixed linear mapping on \(\mathbb R^n\) of rank \(n1\), then for almost all \(x, \mathcal L (x,q,v)\) is determined by \(x\) and \(\pi (\mathcal L(x,q,v))\). By contrast, it is not possible to infer that \(\mathcal P(x,q,v)\) belongs to any specific \(n1\) dimensional subspace of \(\mathbb R^{n}\) that depends only on \(x\).
4.1 On \(n+1\) Points in \(\mathbb R^n\)
We conclude this section by proving that there are essentially no nontrivial one dimensional \(p\)admissible spaces. The proof of this result is instructive as it shows how the algebra of quadratic forms that arise from the functions \(\mathcal L\) described above can lead to interesting geometric information about infinitesimal flexes of frameworks. This is an idea that will reoccur in later sections, hence one can view Proposition 10 as a warmup for the main technical results that we will see in Sect. 5.
Proposition 10
Let \(p \in \mathbb R^{ n \times (n+1) }\) be in general position. There is no one dimensional \(p\)admissible subspace of \(\mathbb R^{n\times (n+1)}\).
Proof
Proposition 10 immediately implies the well known fact (see, for example [4, Chap. 5]) that any Henneberg \(1\)extension of a generically \(n\)isostatic graph is also \(n\)isostatic. Indeed most proofs of that fact in the literature essentially amount to proofs of Proposition 10. Of course none of these previous proofs use the \(p\)admissibility concept explicitly. Typically some special position argument is used to demonstrate that for any nontrivial motion of \(n+1\) points in \(\mathbb R^n\) there is some placement of the point \(x\) which will not admit an extension of that flex to the new framework. We have included another proof here for the sake of completeness and also to illustrate that our matrix algebraic viewpoint provides another way to understand some well known geometric results from the literature.
5 \(p\)Admissibility for Five Points in \(\mathbb R^3\)
In this section we will use the results of the previous section to analyse two dimensional \(p\)admissible spaces where \(p\) is a five point configuration in \(\mathbb R^3\). Given that we are restricting our attention to such a special situation we find it appropriate to introduce the following special notational conventions. For \(p \in \mathbb R^{3 \times 5}, q\) is the matrix obtained by deleting the second and third columns of \(p\), whereas \(r\) is the matrix obtained by deleting the fourth and fifth columns of \(p\). Similarly for \(u \in \mathbb R^{3 \times 5}, v\) is the matrix obtained by deleting the second and third columns of \(u\), whereas \(w\) is the matrix obtained by deleting the fourth and fifth columns of \(u\).
Lemma 11
Let \(p,u \in \mathbb R^{3 \times 5}\). Then \(u\) is a linear motion of \(p\) if and only if there is some \(m \in \mathbb R^{3\times 3}\) such that \(v = mq\) and \(w = mr \). In particular, if \(q\) is invertible then \(u\) is a linear motion of \(p\) if and only if \(w = vq^{1}r\).
Proof
By definition \(u\) is a linear motion of \(p\) if and only if there is some \(m \in \mathbb R^{3 \times 3}\) such that \(u_i = mp_i\). \(\square \)
Lemma 12
Suppose that \(u,u'\in \mathbb R^{3 \times 5}\) are motions of \(p \in \mathbb R^{3 \times 5}\). Then \(u\) and \(u' \) are \(p\)equivalent if and only if there is some skew symmetric matrix \(a\in \mathbb R^{3 \times 3}\) and some \(t \in \mathbb R^3\) such that \(v + aq+t\mathbb {1}^\mathrm{T} = v'\) and \(w+ ar+t\mathbb {1}^\mathrm{T} = w'\)
Proof
This follows from the definition of \(p\)equivalence and Lemma 3. \(\square \)
Now we come to the main technical results of the paper. For the rest of this section we assume that both \(q\) and \(r\) are invertible. First we recharacterise \(p\)admissibility of a subspace \(S\) of \(\mathbb R^{3\times 5}\).
Proposition 13
Let \(S\) be a two dimensional subspace of \(\mathbb R^{3 \times 5}\) such that \(S\cap \mathcal I_p = 0\). Then \(S\) is \(p\)admissible if and only if the linear mapping \(h_x: S \rightarrow \mathbb R^{3}\) defined by \(h_x(u) = \mathcal P(x,q,v)  \mathcal P(x,r,w)\) has rank at most one for almost all \(x \in \mathbb R^3\).
Proof
Proposition 13 allows us to derive an interesting sufficient condition for \(p\)admissibility.
Proposition 14
 A1.

\(S\cap \mathcal I_p = 0\).
 A2.

Every \(u\in S\) is a linear motion of \(p\).
 A3.

For all \(u \in S\) we have \((q^\mathrm{T})^{1}\triangle (v^\mathrm{T}q) = (r^\mathrm{T})^{1}\triangle (w^\mathrm{T}r)\).
Proof
Property A3 of Proposition 14 is a rather mysterious looking property, however we will see in Theorem 20 below that it has an interesting consequence for any potential proof of Conjecture 1. Now we derive a necessary condition for \(p\)admissibility.
Theorem 15
 (1)
Every element of \(S\) is an affine motion of \(p\).
 (2)
\(S\) is \(p\)equivalent to a subspace \(S'\) where \(S' = Ez^\mathrm{T}\) for some two dimensional subspace \(E\) of \(\mathbb R^{3}\) and some \(z \in \mathbb R^{5}\).
Note that hypothetical conditions on \(p\) are, in particular, fulfilled by any generic \(p \in \mathbb R^{3 \times 5}\). The remainder of this section of the paper is devoted to proving Theorem 15. The strategy is as follows. Proposition 13 has allowed us to express \(p\)admissibility in terms of \(\mathcal P(,q,)\) and \(\mathcal P(,r,)\). Then a limiting argument (Lemma 16) allows us to obtain a condition on the functions \(\mathcal L(,q,)\) and \(\mathcal L(,r,)\) which is necessary for the space \(S\) to be \(p\)admissible. We then express this condition in terms of basis elements for the space \(S \) and this allows us to reformulate in terms of linear dependence properties of a certain pair of polynomial functions \(\mathbb R^2 \rightarrow \mathbb R^2\). We then derive an elementary characterisation of such pairs (Lemma 17 below). An application of this result allows us to deduce the conclusion of Theorem 15.
Lemma 16
For almost all \(x \in \mathbb R^3\) there is some nonzero \(u \in S\) such that \(\mathcal L(x,q,v)=\mathcal L(x,r,w)\)
Proof
It is clear that replacing \(u^i\) by \(u^i+t^i\mathbb {1}^\mathrm{T}\) leaves the function \(f^i\) unchanged—one can easily verify this algebraically. However the geometric intuition here is that \(t^i \mathbb {1}^\mathrm{T}\) represents an infinitesimal translation of a particular point configuration. But infinitesimal translations vanish in the limit that occurs in the definition of \(\mathcal L \) and therefore contribute nothing to \(f^i\). Now we will choose a particular \(t^i\) as follows: First choose some \(d \in \mathbb R^{3}\) so that \(\{ c,d \}\) is a basis for \(q_1^\perp \). By replacing \(u^i\) by \(u^i+t^i\mathbb {1}^\mathrm{T}\) for an appropriate \(t^i \in \mathbb R^3\) we can arrange that the column space of the matrix \(w^ir^{1}v^iq^{1}\) is spanned by \(d\).
Lemma 17
 (1)
There exist constants \(r, s \in \mathbb R\) such that \(( r,s) \ne (0,0)\) and \(rh_1 + sh_2 \equiv 0\). \((\)i.e. \(\{\{h_1,h_2\}\}\) is linearly dependent over \(\mathbb R)\).
 (2)
\(l_1 \equiv l_2 \equiv 0\).
 (3)
There exists some affine linear function \(m:\mathbb R^n \rightarrow \mathbb R\) such that \(q_i \equiv ml_i\) for \(i=1,2\).
Proof

\((1^{\prime })\quad \{\{g^1,g^2\}\}\) is linearly dependent over \(\mathbb R\).

\((2^{\prime })\quad (k^1)^\mathrm{T}x = (k^2)^\mathrm{T}x = 0\) for all \(x \) such that \((r^{1}x)^\mathrm{T}\mathbb {1}= 1\).

\((3^{\prime })\) There is some affine linear function \(m:\{x:(r^{1}x)^\mathrm{T}\mathbb {1}=1\} \rightarrow \mathbb R\) such that \(x^\mathrm{T}( w^ir^{1})^\mathrm{T}x = m(x)(k^i)^\mathrm{T}x\) for all \(x\) such that \((r^{1}x)^\mathrm{T}\mathbb {1}= 1\).
Suppose that \((2^{\prime })\) is true. So \(( w^ir^{1})^\mathrm{T}  ( v^iq^{1})^\mathrm{T} = 0\) for \(i=1,2\). Therefore, by Lemma 11, \({u}^i\) is a linear motion of \(p\) for \(i=1,2\). It follows easily from this that every element of \(S\) is a linear motion of \(p\). Remembering that we replaced \(u^i\) by \(u^i+t^i\mathbb {1}^\mathrm{T}\) earlier in the argument, we conclude that every element of our original \(S\) is in fact an affine motion of \(p\).
6 Applications to Rigidity Theory
The remainder of the paper will be concerned with applications of the results of Sects. 3, 4 and 5 to the rigidity theory of frameworks.
Lemma 18
Let \(p \in \mathbb R^{3\times 5}\) be generic and let \(S \) be a two dimensional \(p\)admissible subspace of \(\mathbb R^{3\times 5}\). Suppose that there is some three point subconfiguration of \(p\) such that that for every \(u \in S, u\) restricts to a space of isometries of that three point configuration. Then, up to a possible permutation of the points, \(S\) is \(p\)equivalent to one of the spaces described in Example 6 or Example 7.
Proof
We can use this to prove a special case of Conjecture 1. We introduce the following notation for the rest of the section. \(G=(V,E)\) is a generically 3isostatic graph, \(X =\{1,2,3,4,5\}\subset V, \{e,f\} \subset E(X)\) and \(G'\) is the 2extension of \(G\) supported on vertex set \(X\) and edge set \(\{e,f\}\). Moreover, given a framework \((G,\rho )\), let \(p \in \mathbb R^{3\times 5} \) be defined by \(p_i = \rho _i\) for \(i \in X\).
Corollary 19
and where the vertices of \(H\) are in \(X\). Then either \(G\{e,f\}\) contains an implied \(K_4\) whose vertices are in \(X\), or \(G'\) is generically 3rigid.
Proof
Let \((G,\rho )\) be a generic framework. Suppose that \(G'\) is not generically rigid. Then the nontrivial flexes of \((G\{e,f\},\rho )\) induce a \(p\)admissible subspace of \(\mathbb R^{3\times 5}\). Since \(H\) contains a triangle, we can apply Lemma 18 to conclude that this space is \(p\)equivalent to either Example 6 or 7. In the first case, it is clear that \(G\{e,f\}\) contains an implied \(K_4\). So it suffices to show that the space from Example 7 cannot arise from this framework. However this clearly follows from the genericity of \(p\) and the constraint imposed by the fourth edge of \(H\). \(\square \)
Our next result illustrates a fundamental difficulty that must be resolved directly or indirectly by any possible proof of Conjecture 1.
Theorem 20
For any generic \(p \in \mathbb R^{3 \times 5}\) there is a two dimensional \(p\)admissible subspace \(S\) of \(\mathbb R^{3 \times 5}\) such that \(S\) does not restrict to a space of isometries on any three point subconfiguration of \(p\).
Proof
Now suppose that \(S\) is one of the \(p\)admissible spaces constructed in Example 6 or 7. It is easy to see that for a given \(p\) there are only finitely many such spaces that consist of affine motions. We also note that if \(S\) and \(S'\) are \(p \)equivalent spaces and \(S\) consists of affine motions, then \(S'\) consists of affine motions. This follows from the fact that every trivial motion of \(p\) is also an affine motion. Therefore there must be many elements of \(M\) that are not \(p \)equivalent to either Example 6 or 7. By Lemma 18 this means that there are many elements of \(M\) that do not restrict to spaces of isometries for any three point subset of \(p\). \(\square \)
Contrast this situation with Proposition 10 which tells us that there are no one dimensional \(p\)admissible spaces for any generic \(p \in \mathbb R^{n\times (n+1)}\). As we observed after that proposition, this leads to an immediate understanding of Henneberg 1extensions. On the other hand any proof of Conjecture 1 must somehow (directly or indirectly) demonstrate that the motions whose existence is asserted by Theorem 20 cannot arise by deleting two edges from a generic framework. Even if we were able to do this, we would still face the difficulty of showing that the \(p \)admissible space of Example 7 cannot arise from deleting two edges from a generic isostatic framework. As far as the author is aware, even that special case remains open. Thus Theorem 20 and Example 7 identify a fundamental difficulty with Conjecture 1 that does not arise for the corresponding question about Henneberg 1extensions. In particular, it suggests that the typical ‘special position’ type arguments, that can be used in the planar case to show that certain Henneberg extensions preserve generic rigidity, may not be as fruitful in the case of 2extensions in three dimensional space.
Despite the somewhat negative observations above, we can obtain some positive new partial results toward Conjecture 1.
Lemma 21
Suppose that \(S\) is a two dimensional space of affine motions of a generic \(p \in \mathbb R^{3 \times 5}\) and suppose that \(S\) restricts to a space of isometries on each of five distinct edges of \(K_5\). Then there is some nonzero \(u \in S\) that is an infinitesimal isometry of \(p\).
Proof
Theorem 22
Suppose that \(E(X) \ge 7 \). Then either \(G \{e,f\}\) contains an implied \(K_4\) whose vertices are in \(X\), or \(G'\) is generically 3rigid.
Proof
 (1)
\(z_1,z_2,z_3\) are pairwise distinct, \(z_4= z_1\) and \(z_5=z_2\).
 (2)
\(z_1,z_2,z_3,z_4\) are pairwise distinct and \(z_5 = z_1\).
 (3)
\(z_1,z_2,z_3,z_4,z_5\) are pairwise distinct.
Theorem 22 generalises the result of Graver mentioned in the introduction. In his result, it is required that five of the edges in \(E(X) \{e,f\}\) form a cycle.
Notes
Acknowledgments
The author would like to thank Bob Connelly for some very helpful conversations on this topic and in particular for pointing out the application of the conic at infinity argument in the proof of Theorem 22. The author also expresses his thanks to Bill Jackson and John Owen for allowing him access to their unpublished manuscript [6]. Thanks are also due to one of the referees for several helpful suggestions regarding the exposition and structure of the paper. Finally I dedicate this article to my daughter Lily whose wakeful nights afforded me lots of time to think about Henneberg extensions.
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