On affine motions and bar frameworks in general position
aa r X i v : . [ m a t h . M G ] S e p On Affine Motions and Bar Frameworks in GeneralPosition
A. Y. Alfakih ∗† Department of Mathematics and StatisticsUniversity of WindsorWindsor, Ontario N9B 3P4CanadaYinyu Ye ‡§ Department of Management Science and EngineeringStanford UniversityStanford, California 94305USAOctober 9, 2018
AMS classification:
Keywords:
Bar frameworks, universal rigidity, stress matrices, points in generalposition, Gale transform.
Abstract
A configuration p in r -dimensional Euclidean space is a finite collection ofpoints ( p , . . . , p n ) that affinely span R r . A bar framework, denoted by G ( p ), in R r is a simple graph G on n vertices together with a configuration p in R r . Agiven bar framework G ( p ) is said to be universally rigid if there does not existanother configuration q in any Euclidean space, not obtained from p by a rigidmotion, such that || q i − q j || = || p i − p j || for each edge ( i, j ) of G .It is known [2, 6] that if configuration p is generic and bar framework G ( p )in R r admits a positive semidefinite stress matrix S of rank ( n − r − G ( p ) is universally rigid. Connelly asked [8] whether the same result holds trueif the genericity assumption of p is replaced by the weaker assumption of generalposition. We answer this question in the affirmative in this paper. ∗ E-mail: [email protected] † Research supported by the Natural Sciences and Engineering Research Council of Canada. ‡ E-mail:[email protected] § Research supported in part by NSF Grant GOALI 0800151 and DOE Grant de-sc0002009. Introduction A configuration p in r -dimensional Euclidean space is a finite collection of points( p , . . . , p n ) in R r that affinely span R r . A bar framework (or framework for short)in R r , denoted by G ( p ), is a configuration p in R r together with a simple graph G on the vertices 1 , , . . . , n . For a simple graph G , we denote its node set by V ( G )and its edge set by E ( G ). To avoid trivialities, we assume throughout this paperthat graph G is connected and not complete.Framework G ( q ) in R r is said to be congruent to framework G ( p ) in R r if config-uration q is obtained from configuration p by a rigid motion. That is, if || q i − q j || = || p i − p j || for all i, j = 1 , . . . , n , where || . || denotes the Euclidean norm. We say thatframework G ( q ) in R s is equivalent to framework G ( p ) in R r if || q i − q j || = || p i − p j || for all ( i, j ) ∈ E ( G ). Furthermore, we say that framework G ( q ) in R r is affinely-equivalent to framework G ( p ) in R r if G ( q ) is equivalent to G ( p ) and configuration q is obtained from configuration p by an affine motion; i.e., q i = Ap i + b , for all i = 1 , . . . , n , for some r × r matrix A and an r -vector b .A framework G ( p ) in R r is said to be universally rigid if there does exist aframework G ( q ) in any Euclidean space that is equivalent, but not congruent, to G ( p ). The notion of a stress matrix S of a framework G ( p ) plays a key role in theproblem of universal rigidity of G ( p ). Let G ( p ) be a framework on n vertices in R r . An equilibrium stress of G ( p ) is a realvalued function ω on E ( G ) such that X j :( i,j ) ∈ E ( G ) ω ij ( p i − p j ) = 0 for all i = 1 , . . . , n. (1)Let ω be an equilibrium stress of G ( p ). Then the n × n symmetric matrix S = ( s ij ) where s ij = − ω ij if ( i, j ) ∈ E ( G ) , i = j and ( i, j ) E ( G ) , X k :( i,k ) ∈ E ( G ) ω ik if i = j, (2)is called the stress matrix associated with ω , or a stress matrix of G ( p ). The followingresult provides a sufficient condition for the universal rigidity of a given framework. Theorem 1.1 (Connelly [5, 6], Alfakih [1])
Let G ( p ) be a bar framework in R r ,for some r ≤ n − . If the following two conditions hold: . There exists a positive semidefinite stress matrix S of G ( p ) of rank ( n − r − ).2. There does not exist a bar framework G ( q ) in R r that is affinely-equivalent,but not congruent, to G ( p ) .Then G ( p ) is universally rigid. Note that ( n − r −
1) is the maximum possible value for the rank of the stressmatrix S . In connection with Theorem 1.1, we mention the following result obtainedin So and Ye [11] and Biswas et al. [4]: Given a framework G ( p ) in R r , if there doesnot exist a framework G ( q ) in R s ( s = r ) that is equivalent to G ( p ), then G ( p )is universally rigid. Moreover, if G ( p ) contains a clique of r + 1 points in generalposition, then the existence of a rank-( n − r −
1) positive semidefinite stress matriximplies that framework G ( p ) is universally rigid, regardless whether the other non-clique points are in general position or not.Condition 2 of Theorem 1.1 is satisfied if configuration p is assumed to be generic(see Lemma 2.2 below). A configuration p (or a framework G ( p )) is said to be generic if all the coordinates of p , . . . , p n are algebraically independent over the integers.That is, if there does not exist a non-zero polynomial f with integer coefficients suchthat f ( p , . . . , p n ) = 0. Thus Theorem 1.2 (Connelly [6], Alfakih [2])
Let G ( p ) be a generic bar frameworkon n nodes in R r , for some r ≤ n − . If there exists a positive semidefinite stressmatrix S of G ( p ) of rank ( n − r − ). Then G ( p ) is universally rigid. The converse of Theorem 1.2 is also true.
Theorem 1.3 (Gortler and Thurston [10])
Let G ( p ) be a generic bar frame-work on n nodes in R r , for some r ≤ n − . If G ( p ) is universally rigid, then thereexists a positive semidefinite stress matrix S of G ( p ) of rank ( n − r − ). Connelly [8] asked whether a result similar to Theorem 1.2 holds if the genericityassumption of G ( p ) is replaced by the weaker assumption of general position. Aconfiguration p (or a framework G ( p )) in R r is said to be in general position if nosubset of the points p , . . . , p n of cardinality r +1 is affinely dependent. For example,a set of points in the plane are in general position if no 3 of them lie on a straightline.In this paper we answer Connelly’s question in the affirmative. Thus the followingtheorem is the main result of this paper. Theorem 1.4
Let G ( p ) be a bar framework on n nodes in general position in R r ,for some r ≤ n − . If there exists a positive semidefinite stress matrix S of G ( p ) ofrank ( n − r − ). Then G ( p ) is universally rigid. G ( p ) on n nodes in general position in R r for some r ≤ n −
2, where G is the ( r + 1)-lateration graph, admits a rank ( n − r − To develop the ingredients needed for the proof of our main result, we review thenecessary background on affine motions, stress matrices, and Gale matrices.An affine motion in R r is a map f : R r → R r of the form f ( p i ) = Ap i + b, for all p i in R r , where A is an r × r matrix and b is an r -vector. A rigid motion isan affine motion where matrix A is orthogonal.Vectors v , . . . , v m in R r are said to lie on a quadratic at infinity if there exists anon-zero symmetric r × r matrix Φ such that( v i ) T Φ v i = 0 , for all i = 1 , . . . , m. (3) Lemma 2.1 (Connelly [7]) Let G ( p ) be a bar framework on n vertices in R r . Thenthe following two conditions are equivalent:1. There exists a framework G ( q ) in R r that is equivalent, but not congruent, to G ( p ) such that q i = Ap i + b for all i = 1 , . . . , n ,2. The vectors p i − p j for all ( i, j ) ∈ E ( G ) lie on a quadratic at infinity. Lemma 2.2 (Connelly [7]) Let G ( p ) be a generic bar framework on n vertices in R r . Assume that each node of G has degree at least r . Then the vectors p i − p j forall ( i, j ) ∈ E ( G ) do not lie on a quadratic at infinity. Therefore, under the genericity assumption, Condition 2 in Lemma 2.1 does nothold. Consequently, Theorem 1.2 follows as a simple corollary of Theorem 1.1.Note that Condition 2 in Lemma 2.1 is expressed in terms of the edges of G . Anequivalent condition in terms of the missing edges of G can also be obtained usingGale matrices. This equivalent condition turns out to be crucial for our proof ofTheorem 1.4.To this end, let G ( p ) be a framework on n vertices in R r . Then the following( r + 1) × n matrix A := (cid:20) p p . . . p n . . . (cid:21) (4)4as full row rank since p , . . . , p n affinely span R r . Note that r ≤ n −
1. Let¯ r = the dimension of the null space of A ; i.e., ¯ r = n − − r. (5) Definition 2.1
Suppose that the null space of A is nontrivial, i.e., ¯ r ≥ . Any n × ¯ r matrix Z whose columns form a basis of the null space of A is called a Gale matrix of configuration p . Furthermore, the i th row of Z , considered as a vector in R ¯ r , iscalled a Gale transform of p i [9]. Let S be a stress matrix of G ( p ) then it follows from (2) and (4) that A S = 0 . (6)Thus S = Z Ψ Z T , (7)for some ¯ r × ¯ r symmetric matrix Ψ, where Z is a Gale matrix of p . It immediatelyfollows from (7) that rank S = rank Ψ. Thus, S attains its maximum rank of¯ r = ( n − − r ) if and only if Ψ is nonsingular, i.e., rank Ψ = ¯ r .Let e denote the vector of all 1’s in R n , and let V be an n × ( n −
1) matrix thatsatisfies: V T e = 0 , V T V = I n − , (8)where I n − is the identity matrix of order ( n − E ij , i = j , denote the n × n symmetric matrix with 1 in the ( i, j )th and ( j, i )th entries and zeros elsewhere,and let E ( y ) = P ( i,j ) E ( G ) y ij E ij where y ij = y ji . In other words, the ( k, l ) entry ofmatrix E ( y ) is given by E ( y ) kl = k, l ) ∈ E ( G ) , k = l,y kl if k = l and ( k, l ) E ( G ) . (9)Then we have the following result. Lemma 2.3 (Alfakih [2]) Let G ( p ) be a bar framework on n vertices in R r and let Z be any Gale matrix of p . Then the following two conditions are equivalent:1. The vectors p i − p j for all ( i, j ) ∈ E ( G ) lie on a quadratic at infinity.2. There exists a non-zero y = ( y ij ) ∈ R ¯ m such that: V T E ( y ) Z = , (10) where ¯ m is the number of missing edges of G , V is defined in (8), and E ( y ) isdefined in (9). here is the zero matrix of dimension ( n − × ¯ r . Condition 2 of Lemma 2.3 can be easily understood if a projected Gram matrixapproach is used for the universal rigidity of bar frameworks (see [2] for details).5
Proof of Theorem 1.4
The main idea of the proof is to show that Condition 2 of Lemma 2.3 does not holdunder the general position assumption, and under the assumption that G ( p ) admitsa positive semidefinite stress matrix of rank ( n − r − Lemma 3.1
Let G ( p ) be a framework on n nodes in general position in R r andlet Z be any Gale matrix of configuration p . Then any ¯ r × ¯ r submatrix of Z isnonsingular. Proof.
For a proof see e.g., [1]. ✷ Let ¯ N ( i ) denote the set of nodes of graph G that are non-adjacent to node i ;i.e., ¯ N ( i ) = { j ∈ V ( G ) : j = i and ( i, j ) E ( G ) } , (11) Lemma 3.2
Let G ( p ) be a framework on n nodes in general position in R r . Assumethat G ( p ) has a stress matrix S of rank ( n − − r ) . Then there exists a Gale matrix ˆ Z of G ( p ) such that ˆ z ij = 0 for all j = 1 , . . . , ¯ r and i ∈ ¯ N ( j + r + 1) . Proof.
Let G ( p ) be in general position in R r and assume that it has a stressmatrix S of rank ¯ r = ( n − − r ). Let Z be any Gale matrix of G ( p ), then S = Z Ψ Z T for some non-singular symmetric ¯ r × ¯ r matrix Ψ. Let us write Z as: Z = (cid:20) Z Z (cid:21) , (12)where Z is ¯ r × ¯ r . By Lemma 3.1, Z is non-singular. Now letˆ Z = (ˆ z ij ) = Z Ψ Z T . (13)Then ˆ Z is a Gale matrix of G ( p ). This simply follows from the fact that the matrixobtained by multiplying any Gale matrix of G ( p ) from the right by a non-singular¯ r × ¯ r matrix, is also a Gale matrix of G ( p ). Furthermore, S = Z Ψ Z T = Z Ψ [ Z T Z T ] = [ Z Ψ Z T ˆ Z ] . In other words, ˆ Z consists of the last ¯ r columns of S . Thus ˆ z ij = s i,j + r +1 . By thedefinition of S we have s i,j + r +1 = 0 for all i = j + r + 1 and ( i, j + r + 1) E ( G ).Therefore, ˆ z ij = 0 for all j = 1 , . . . , ¯ r and i ∈ ¯ N ( j + r + 1). ✷ emma 3.3 Let the Gale matrix in (10) be ˆ Z as defined in (13). Then the systemof equations (10) is equivalent to the system of equations E ( y ) ˆ Z = . (14) here is the zero matrix of dimension n × ¯ r . Proof.
System of equations (10) is equivalent to the following system ofequations in the unknowns, y ij ( i = j and ( i, j ) E ( G )) and ξ ∈ R ¯ r : E ( y ) ˆ Z = e ξ T , (15)Now for j = 1 , . . . , ¯ r , we have that the ( j + r + 1 , j )th entry of E ( y ) ˆ Z is equal to ξ j .But using (9) and Lemma 3.2 we have( E ( y ) ˆ Z ) j + r +1 ,j = n X i =1 E ( y ) j + r +1 ,i ˆ z ij = X i : i ∈ ¯ N ( j + r +1) y j + r +1 ,i ˆ z ij = 0 . Thus, ξ = 0 and the result follows. ✷ Now we are ready to prove our main theorem.
Proof of Theorem 1.4
Let G ( p ) be a framework on n nodes in general position in R r . Assume that G ( p )has a positive semidefinite stress matrix S of rank ¯ r = n − − r . Then deg( i ) ≥ r + 1for all i ∈ V ( G ), i.e., every node of G is adjacent to at least r + 1 nodes (for a proofsee [1, Theorem 3.2]). Thus | ¯ N ( i ) | ≤ n − r − r − . (16)Furthermore, it follows from Lemmas 3.2, 3.3 and 2.3 that the vectors p i − p j for all ( i, j ) ∈ E ( G ) lie on a quadratic at infinity if and only if system of equations(14) has a non-zero solution y . But (14) can be written as X j : ∈ ¯ N ( i ) y ij ˆ z j = 0 , for i = 1 , . . . , n, where (ˆ z i ) T is the i th row of ˆ Z . Now it follows from (16) that y ij = 0 for all( i, j ) E ( G ) since by Lemma 3.1 any subset of { ˆ z , . . . , ˆ z n } of cardinality ≤ ¯ r − y . Hence the vectors p i − p j ,for all ( i, j ) ∈ E ( G ), do not lie on a quadratic at infinity. Therefore, by Lemma2.1, there does not exist a framework G ( q ) in R r that is affinely-equivalent, but notcongruent, to G ( p ). Thus by Theorem 1.1, G ( p ) is universally rigid. ✷ eferences [1] A. Y. Alfakih. On dimensional rigidity of bar-and-joint frameworks. DiscreteAppl. Math. , 155:1244–1253, 2007.[2] A. Y. Alfakih. On the universal rigidity of generic bar frameworks.
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