1. Combinatorial Rigidity Jack Graver, Brigitte Servatius, Herman Servatius. Jenny Stathopoulou December 2004

2. Infinitesimal Rigidity Introduction We say that a framework (V, E, p) is generic if all frameworks corresponding to points in a neighborhood of P=p(V) in are rigid or not rigid. A set of points P in m-space is said to be generic if each framework (V, E, p) with P=p(V) is generic. We define a graph (V, E) to be rigid (in dimension m) if the frameworks corresponding to the generic embeddings of V into m-space are rigid. We say that a framework (V, E, p) in m-space is generically rigid if the graph (V, E) is rigid in dimension m. ℝmℝm

3. Infinitesimal Rigidity Introduction Is (V, E) generically rigid (generically independent) in dimension m? In m ≥3, the only technique is: for all graphs (V, E) chose a generic embedding p of V into and check if the framework (V, E, p) is infinitesimal rigid or infinitesimal independent. ℝmℝm

4. Infinitesimal Rigidity Basic Definitions V(E) = { i| ∃ j єV with (i,j) є E or (j,i)єE} K(U) = { (i,j)| i< j and {i,j} ⊆ U} p embedding of V into ℝmℝm p = (p 11, p 12, …, p 1m,…,p i1, p i2,…, p im,…, p n1, p n2,…, p nm ) p = (p 1, …, p i, …, p n )

5. Infinitesimal Rigidity Basic Definitions Let p be an embedding of V into and P denote set p(V). The set P is in general position and p is a general embedding if, for any q-element subset Q of P with q<(m+1) the affine space spanned by Q has dimension q-1. (e.g.: if q=2, Q spans a line; if q=3, Q spans a plane.) We may also measure the length of the edges by evaluating the rigidity function ρ: defined by (where the ij’th coordinate of ρ(p) is the square of the length of the edge ij in K.). We define the rigidity matrix for the embedding p, by ρ´(p)=2R(p). R(p) is an n(n-1)/2 by nm matrix whose entries are functions of the coordinates of p as a point in. ℝmℝm ℝ mn ￫ ℝ |E(K)| ρ(p) ij = (p i – p j ) 2 ℝmnℝmn

6. Infinitesimal Rigidity Basic Definitions e.g n=4 and m=2 p = (p 1, p 2, p 3, p 4 ) p 11 -p 21 p 21 -p 22 p 21 -p 11 0 0 0 0 0 p 11 -p 31 p 12 -p 32 0 0 p 31 -p 11 p 32 -p 12 0 0 p 11 -p 41 p 12 -p 42 0 0 0 0 p 41 -p 11 p 42 -p 12 0 0 p 21 -p 31 p 22 -p 32 p 31 -p 21 p 32 -p 22 0 0 0 0 p 21 -p 41 p 22 -p 42 0 0 p 31 -p 21 p 32 -p 22 0 0 0 0 p 31 -p 41 p 32 -p 42 p 41 -p 31 p 42 -p 32

7. Infinitesimal Rigidity Basic Definitions R(p) = p 1 -p 2 p 2 -p 1 0 0 p 1 -p 3 0 p 3 -p 1 0 p 1 -p 4 0 0 p 4 -p 1 0 p 2 -p 3 p 3 -p 2 0 0 p 2 -p 4 0 p 4 -p 2 0 0 p 3 -p 4 p 4 -p 3 Vector in. Corresponding to edge(2,4) of K ℝmnℝmn n

8. Infinitesimal Rigidity Independence and the Stress Space If E is independent with respect to one generic embedding into m-space, then it is independent with respect to all generic embeddings into m-space. If E is independent with respect to all generic embedding into m-space, then it is generically independent for dimension m. If E is dependent with respect to one m-dimensional embedding of V, then E is generically dependent for dimension m and is, in fact, dependent with respect to all embeddings of V into m-space.

9. Infinitesimal Rigidity Independence and the Stress Space An edge set E ⊆ K is independent with respect to p if the corresponding set of rows ( in R(p)) is independent as a set of vectors in An edge set will be dependent if and only if the corresponding set of rows of R(p) satisfies a non-trivial dependency relation. where is the row corresponding to the edge (i,j), and is a scalar and some ≠ 0. For each iєV(E) :, over K(V(E)). (*) ℝ mn ∑ (i,j)єE s ij r ij = 0 r ij s ij ∑ i≠j s ij (p i - p j ) = 0

10. Infinitesimal Rigidity Independence and the Stress Space S 13 =1 P 3 = (0,1) S 34 =1 P 4 = (1,1) P 1 = (0,0) S 12 =1 P2= (1,0) S 14 = -1 S 23 = -1 For i=1: 1(-1,0) + 1(0,-1) + (-1)(-1,-1) = 0 For i=2 : 1(1,0) + (-1)(1,-1) + 1(0,-1) = 0 … We conclude that for this p the set K is dependent !

11. Infinitesimal Rigidity Independence and the Stress Space P 3 = (0,1)P 4 = (1,1) P 1 = (0,0)P2= (1,0) In order that is satisfied at we must have : ∑ i≠j s ij (p i - p j ) = 0 p1p1 s 12 (p 1 -p 2 ) + s 13 (p 1 -p 3 ) + s 14 (p 1 -p 4 ) = 0  0 + s 13 (0,-1) + s 14 (-1,-1) = 0   s 13 = s 14 =0 Similarly at : s 23 = s 24 =0 and it follows that s 34 = 0 p2p2

12. Infinitesimal Rigidity Independence and the Stress Space Let E ⊆ K and consider the vector space of all functions from E to the reals. For s є, s is called a set of stresses for E and is the stress on the edge (i,j). A set of stresses for E, not all zero, which satisfy the equations in (*) is called a non-trivial resolvable set of stresses for E, S (E). We consider the linear transformation where and S (E) = ker( ), where = 0 if (i,j) is not in E. Thus, E will be independent with respect to p if and only if the kernel of is trivial. ℝEℝE ℝEℝE s i j T |E ℝ E ￫ ( ℝ m ) V (T |E (s)) i = ∑ j≠i s ij (p i -p j )T|ET|E T|ET|E s ij

13. Infinitesimal Rigidity Independence and the Stress Space e.g. P 3 = (0,1) P 1 = (0,0) P 4 = (x,y) P 2 = (1,0) S 24 S 34 S 14 S 13 S 12 S 32 Where x≠0, y≠0 and x+y ≠1 T(s) 1 = ( -s 12 - xs 14, -s 13 – ys 14 ) T(s) 2 = ( s 12 +s 23 + (1- x)s 24, -s 23 – ys 24 ) T(s) 3 = ( -s 23 – xs 34, s 13 + s 23 +(1- y)s 34 ) T(s) 4 = ( xs 14 +( x -1)s 24 + xs 34, ys 14 + ys 24 +(y-1)s 34 )

14. Infinitesimal Rigidity Independence and the Stress Space dim (ker( )) + dim(im( )) = dim(domain ( )). But dim(domain( )) = |E|. The space im( ) is a subspace of, so dim(im( )) + dim(im( ) ) = mn. E is independent if and only if dim(im( ) ) =mn -|E|+ dim( S (E)). T |E ℝ mn T |E

15. Infinitesimal Rigidity Infinitesimal Motions and Isometries Let V = {1,…,n}, p mapping V into, E be an edge set of (V,K) and consider the framework (V, E, p). Then uє is an infinitesimal motion of (V, E, p) if. uє ,for all (i,j) in E. * = 0 The set of infinitesimal motions of the framework is the orthogonal complement of the subspace spanned of the R(p) which correspond to the edges of E, or, equivalently the orthogonal complement of the image. Denote the space of infinitesimal motions of E by V (E). ℝmℝm (ℝm)V(ℝm)V (u i – u j )* (p i – p j ) = 0 (ℝm)V(ℝm)V (u 1, …, u n ) (0, …, 0, p i -p j, 0,..., 0, p j -p i, 0,..., 0) T |E

16. Infinitesimal Rigidity Infinitesimal Motions and Isometries Denote V (K(V(E))) the space of infinitesimal isometries of V(E) by, D (E). Since E ⊆ K(V(E)), the orthogonal complement of the space spanned by the rows of R(p) corresponding to E contains the orthogonal complement of the space spanned by the rows of R(p) corresponding to K(V(E)). A framework is rigid if V (E) = D (E).

17. Infinitesimal Rigidity Infinitesimal Motions and Isometries By an isometry of, we mean a ‘1-1’ function from to which preserves the distance between pairs of points. e.g. a vector field U : is an infinitesimal isometry of if, (U(x) – U(y)) * (x-y) = 0, for all x,y є then it is clear that uє defined by =, for i= 1,..,n is an infinitesimal motion for (V, E, p), for all E ⊆ K. For any subset S of, a function U: S is a infinitesimal isometry of S if (U(x) – U(y)) * (x-y) = 0, for all x,yєS. Isometries: direct ( translations, rotations), opposite (reflections). ℝmℝm ℝmℝm ℝmℝm ℝmℝm ℝmℝm ℝmℝm ℝmℝm (ℝm)V(ℝm)V uiui U(pi)U(pi) ℝmℝm ℝmℝm

18. Infinitesimal Rigidity Infinitesimal and Generic Rigidity Let X be the set of p for which the set of E is independent. If p ∉ X then p is called generic embedding. We define the dependency number of E as the dim( S (E)), denoted by dn (E). We define the degree of freedom of E to be the dim( V (E)) – dim( D (E)), denoted by df (E). If a framework (V(E), E, p) is infinitesimal rigid for some embedding p then E is generically rigid for dimension m.

19. Infinitesimal Rigidity Infinitesimal and Generic Rigidity P 5 = (-1,0) P 4 = (0,0) P 3 = (1,0) P 2 = (1,1) P 1 = (0,2) P 6 = (-1,1) Let W be any infinitesimal motion of the framework. The restriction of W to is an infinitesimal isometry of that set. U( ) = U( ) = 0. Let = (x, y) then ((x,y) –(0,0))*((-1,0)-(0,0))= 0  x=0 = (0,a). Similarly = (0,b). From =(x,y)  y=a, x=-a, b=3a. And form  a = 3b. We conclude that a=b=c=0.  Infinitesimal rigid { p 1, p 4 } p1p1 p4p4 U(p5)U(p5) U(p5)U(p5)U(p3)U(p3)U(p6)U(p6) U(p2)U(p2) See Appendix for examples

20. Let S be any set; an operator ≺. ≻ mapping the power set of S onto the power set of S is called a closure operator if the following conditions are satisfied : C1: if T ⊆ S then T ⊆≺ T ≻. C2: if R ⊆ T ⊆ S then ≺ R ≻ ⊆≺ T ≻. C3: if T ⊆ S then ≺≺ T ≻≻ ⊆≺ T ≻. A matroid consists of a finite set S. A matroid closure operation ≺. ≻ on S satisfies the additional condition: C4: if T ⊆ S and s,t є(S-T) then s є ≺ T U {t} ≻ if and only if t є ≺ T U {s} ≻. Infinitesimal Rigidity Rigidity Matroids

21. Let S be a finite set of vectors from some vector space and, for any T ⊆ S let ≺ T ≻ = span(T) ∩S. Then ≺. ≻ is a matroid closure operator on S. The matroid closure operation on S is an associated concept of dependency. We say that a T ⊆ S is independent ( with respect to the matroid) if, for every sєT, s ∉≺ T -{s} ≻. Let a ≺. ≻ be a matroid closure operator on a finite set S. Let T ⊆ S and sєS; we say that s is independent of T if s is not in ≺ T ≻. Infinitesimal Rigidity Rigidity Matroids

22. Let V and the embedding p of V. Let K=K(V) and let ≺. ≻ to be the closure operator of this embedding. Then, for all E ⊆ K, ≺ E ≻⊆ K(V(E)); furthermore E is rigid if and only if ≺ E ≻ = K(V(E)). Let V and the embedding p of V. Let K=K(V) and let ≺. ≻ to be the closure operator of this embedding. Then ≺. ≻ satisfies : C5: if E,F ⊆ K and |V(E)∩V(F)|<m then ≺ E ∪ F ≻ ⊆ K(V(E)) ∪ K(V(F)). C6: if E,F ⊆ K are rigid and |V(E)∩V(F)| ≥ m then E ∪ F is rigid. A matroid A m on K=K(V) whose closure operator satisfies the C5 and C6 is called m-dimensional abstract rigidity matroid of p. Infinitesimal Rigidity Rigidity Matroids

23. Appendix ( Figure 1)

24.  At ( a ) is rigid because the “chain” of the three rods is pulled taught. If the slightest slack is permitted, (b), the framework is no longer rigid.  At ( a ), assigning the zero vector to all points except to one of the two points on the “chain” (to this points let’s assign any non-zero vector perpendicular to the “chain”), there is an infinitesimal flex of that framework. So the framework ( a ) is rigid but not infinitesimal rigid and hence no generically rigid.  At (c ), there is a crossing that has no significance and adds no constrains. We think of the rods as being able to slide across one another. The framework ( c ) is no rigid because the three vertical rods are equal in length and parallel and hence permit a horizontal shear. If the uniform length or parallelism is destroyed, as in framework becomes rigid.  Framework (b) is neither rigid nor generically rigid; framework (c ) is not rigid but generically rigid; while (d) is both rigid and generically rigid.

25. Appendix We say that a framework is strongly rigid, if all solutions to the corresponding system of quadratic equations correspond to congruent frameworks; a framework is rigid if all solutions to the corresponding system in some neighborhood of the original solution come from congruent solutions. Clearly, strongly rigid implies rigid. The triangle is strongly rigid and hence rigid, while the rectangle is rigid but not strongly rigid. We flip one of the triangles ( of the next rectangle (a) ) over the diagonal. The framework obtained, (b), is not congruent to the original framework but is a solution to the system of quadratic equations (a)(b)

26. Appendix  Strong rigidity  rigidity  Infinitesimal rigidity  rigidity  Infinitesimal rigidity  generic rigidity Figure 2 (a) Rigid, NO Strong Rigid, NO Infinitesimal Rigid, NO Generically Rigid (b) Rigid, NO Strong Rigid, NO Infinitesimal Rigid, Generically Rigid (c) Rigid, Strong Rigid, NO Infinitesimal Rigid, Generically Rigid (d) Rigid, Strong Rigid, NO Infinitesimal Rigid, NO Generically Rigid

27. Τhe End