Projective background of the infinitesimal rigidity of frameworks

Pro jectiv e bac kground of the i nfinitesimal rigi dit y of fra m ew orks Iv an Izmestiev ∗ April 16, 2008 Abstract W e present pro ofs o f tw o classical theorems. The first one, due to Darb oux and Sauer, states that infinitesimal rigidity is a pro jective inv ariant; the other one esta blishes relations (infinitesimal Pogor elov maps) betw een the infinitesimal motions of a Euclidean framework and of its hyperb olic a nd spherical images. The ar guments use the sta tic formulation of infinitesimal r ig idity . The dualit y betw een statics and kinematics is establis he d through the principles of v ir tual work. A geo metric approach to statics, due essen- tially to Grassma nn, makes b o th theorems straightforward. Besides, it provides a simple der iv ation of the formulas b oth for the Darb oux- Sauer corr esp ondence a nd for the infinitesima l Pogorelov maps . 1 In tro duction 1.1 Infinitesimal rigidity A fr amework is a collectio n of bars joined toget her at their ends by un iv ersal join ts. A f ramew ork is called rigi d , if it cannot b e flexed at the join ts without d eforming th e bars; or, equiv alen tly , if it can b e mo v ed only as a rigid b o d y . The mathematical formalization of this is straigh tforward: a framew ork is a collection of p oin ts with distances b et ween some pairs of them fixed; rigidit y means that the p oin ts cannot b e mov ed w ith ou t c hanging one of those distances. A framew ork is infinitesimal ly rigid if its no des cannot b e mo v ed in such a mann er that the lengths of the bars remain constan t in the first order. In practice, an infin itesimally flexible framework allo ws a certain amount of mo v emen t, even if it is rigid in the ab o v e sense. A cla ssical result on infi n itesimal rigidity is the Legendre-Cauc hy-Dehn theorem, [LegII], [Cau13], [Deh16]: Every c onvex 3-dimensio nal p olyhe dr on ∗ Researc h for this article was supp orted by the DFG Researc h Unit 565 “Polyhedral Surfaces”. 1 is infinitesimal ly rigid. The theorem can b e restated in th e language of framew orks: The fr amew ork consisting of the v ertices, edges and all face diagonals of a co nv ex p olyhedr on is infinitesimally r igid. In fact, “all face diagonals” is redu ndant : it suffices to triangulate the faces arbitrarily , with- out adding new v ertices. S ee [Whi84b], wher e this is generalized to h igher dimensions. F or more information on differen t concepts of rigidit y and an ov erview of results in this area , s ee the survey article [C on93]. Another classical but undeservedly lit tle kno wn result is the pro j ective in v ariance of infinitesimal rigidit y . F or d iscrete structures it w as first noticed b y Rankine in 1863; it was pro ved by Darb oux for smo oth surfaces and by Liebmann and Sauer f or framewo rks, see Section 3.3. Closely related to the pro jectiv e in v ariance is the fact, disco vered by Pogo relo v, th at a Euclidean framew ork can b e turned into a h yp erb olic or spherical one, resp ecting th e infinitesimal r igidity . The pr esen t pap er con tains pro ofs of these tw o pr op erties of infi nitesimal rigidit y . The id ea b ehind the pr o ofs is not n ew, bu t we hop e th at a m o dern exp osition migh t b e useful. Our interest wa s stim ulated by new applications that the pr o jectiv e prop erties of infin itesimal rigidit y foun d in recen t ye ars, [Sc h05], [Sch06], [Fil]. There are further manifestations of the p ro jectiv e nature of the in- finitesimal rigidit y , suc h as its r elations with polarit y [Whi87, Whi89] and Maxw ell’s theorem on p r o jected p olyhedr a [Whi82 ]. No w let us state the tw o theorems in a p recise wa y . 1.2 Darb oux-Sauer c orresp ondence A fr amew ork P in the Euclidean space E d can b e mapp ed b y a pro jectiv e map Φ : R P d → R P d to another framew ork Φ( P ). Here we assume that an affine em b edd ing of E d in to R P d is fixed and that Φ maps n o vertex of P to infinity . F ramew orks P and Φ( P ) are cal led pr oje ctively e quiv alent . Theorem 1 (Darb oux-Sauer correspondence) L et P and P ′ b e two pr oje ctively e quivalent fr ameworks in E d . Then P ′ is infinitesimal ly rigid iff P is infinitesimal ly rigid. Mor e over, the numb e r of de gr e es of fr e e dom of P and P ′ c oincide. By the n um b er of degrees of freedom of a framework we mean the dimension of the s pace of its infinitesimal m otions mo d u lo trivial ones. An infinitesimal motion is called trivial if it can b e extended to an infi nitesimal motion of E d (equiv alen tly , if it mov es P as a rigid b od y). More sp ecifically , let Φ b e a pro jectiv e map s u c h that P ′ = Φ( P ). T hen Φ induces a bijection Φ kin b et w een the space of infi nitesimal motions of P and the space of infi nitesimal motions of P ′ that maps tr ivial motio ns to trivial ones. W e call the map Φ kin the (kinematic) Darb oux-Sauer c orr esp ondenc e . 2 1.3 Infinitesimal Pogorelo v maps Here is a simple wa y to d escrib e the infi nitesimal Po gorelo v map. Consid er a framew ork P that is contai ned in a disk D d ⊂ E d . When the interior of D d is view ed as Klein mo del of th e h yp erb olic space H d , the Euclidean framew ork P turn s into a hyp erb olic framework P H . P ogorelo v p ro v ed that P H is infinitesimally rigid iff P is; moreo v er, there is a canonical w a y to asso ciate to ev ery infinitesimal moti on of P an infinitesimal motion of P H (with trivial motions going to trivial ones). This asso ciation is cal led the infinitesimal Pogo relo v map. No w let’s b e formal. Make the follo wing iden tifications: E d = { x ∈ R d +1 | x 0 = 1 } ; (1) H d = { x ∈ R d +1 | x 0 > 0 , k x k 1 ,d = 1 } ; (2) S d = { x ∈ R d +1 | k x k = 1 } , (3) where k · k denotes the Euclidean norm, and k · k 1 ,d denotes the Minko wski norm of signature (+ , − , . . . , − ) in R d +1 . The pro jection from the origin of R d +1 defines the m ap s Π H : D d → H d ; (4) Π S : E d → S d , (5) where D d is the op en unit disk in E d ⊂ R d +1 cen tered at (1 , 0 , . . . , 0). T o a framework P in E d there corresp ond framew orks P H = Π H ( P ) an d P S = Π S ( P ) in H d and S d . Note that P H is defined iff P ⊂ D d . Theorem 2 (Infinitesimal Po gorelo v maps) L et P b e a fr amework in E d . Then the fol lowing ar e e qu ivalent: • the Euclide an fr amewor k P is infinitesimal ly rigid; • (for P ⊂ D d ) the hyp e rb olic fr amework P H is infinitesimal ly rigid; • the spheric al fr amewo rk P S is infinitesimal ly rigid. Mor e over, fr ameworks P , P H , and P S have th e same numb er of de gr e es of fr e e dom. Again, b oth statemen ts of the theorem follo w from the fact that there exist bijections b et we en infi nitesimal motions of f r amew orks P , P H , and P S that map trivial motions to trivial ones. These bijections are called the infinitesimal Po gor elov maps . 3 1.4 Plan of t he pap er Section 2 con tains p reliminary material. The focu s here is on the equ iv- alence b et w een infinitesimal rigidit y and static rigidit y expressed in Theo- rem 3. This theorem is a direct consequence of principles of virtual work (Lemma 2.8). Section 3 dev elops “pro jectiv e statics” and “pro jectiv e kinematics”. The goal is to defin e motions and loads w ithin pr o jectiv e geometry , whic h mak es the p r o jectiv e inv ariance of infinitesimal rigidity straigh tforw ard. Geometric description of Da rb oux-Sauer corresp ondence is deriv ed. With infinitesimal rigidit y d efined in p r o jectiv e terms , it is not hard to relate the kinematic s of frameworks P , P H , and P S . This is d one in Sec- tion 4, where f orm ulas for the infin itesimal P ogorelo v m aps are also derive d. 1.5 Ac kno wledgemen ts I am grateful to W alter Whiteley for inspiring discussions, and to F ran¸ cois Fillastre and Jean-Ma rc Sc hlenker for usefu l remarks. 1.6 Examples Let us illustrate Theorem 1 with some examples. Among the framewo rks with a giv en combinatorics, the infin itesimally flexib le ones sometimes hav e a nice geomet ric description. By Theorem 1, the description can alwa ys b e made in pr o jectiv e terms. Example 1 Blasc hke [Bla20] and Liebmann [Lie2 0 ] pro v ed the follo wing: Let P b e a framew ork com binatorially equiv alen t to the s k eleton of the o ctahedron. Color the triangles sp anned b y the edges of P blac k and w h ite s o that neighb ors hav e differen t colors. The framew ork P is infinitesimally flexible iff the planes of the four blac k triangles in tersect, ma yb e at infinit y . As a corollary , the p lanes of four white triangles intersec t iff the planes of the four blac k ones d o. Figure 1 sho ws t w o configurations satisfying this condition. At the left is an example from [W u n65]. It is obtained from a straigh t an tipr ism o v er a regular triangle by rotating one of the b ases by 90 ◦ . It is ea sy to see that the h orizon tal shaded tr iangle is cut by the planes of the other three shaded triangles along its medians. Hence the four shaded planes intersect at a p oint . The righ t-hand example is due to Liebmann and is also depicted in [Glu75]. Here the p oin ts A , B , C , and D are assumed to lie in one plane. Since eac h of the four shaded planes con tains o ne of the lines AB or C D , they all p ass through the in tersection p oin t of these lines. 4 A D B C Figure 1: Examples of in finitesimally flexible o ctahedra. Left: ant iprism t wisted by 90 ◦ . Righ t: the points A , B , C , D lie in one plane. Example 2 C onsider the planar framework at the left of Figure 2. The lines m atc hin g the v ertices of the tw o tr iangles are p arallel. This implies that the velocit y field represen ted by arro ws is an infinitesimal moti on. Hence, infinitesimally flexible will b e any framew ork where the thr ee m atc hin g lines are concurrent. In fact, this is a necessary and sufficien t cond ition: The planar fr amew ork on the right hand side of Figure 2 is in- finitesimally flexible iff the three lines a , b , c in tersect. Note that this condition is equiv alen t to the framew ork b eing a pro jection of a skeleto n of a 3-p olytop e, so that th e statemen t is a sp ecial case of Maxw ell’s theorem, [Whi82]. a b c Figure 2: Th e framewo rk at the left is infinitesimally flexible. T he frame- w ork at the righ t is infin itesimally flexible iff th e lines a , b , c intersec t. Example 3 W alter Whiteley [Whi85] shows ho w to derive f rom Th eorem 1 the follo w ing stat emen t: Let P b e a fr amew ork in the Eu clidean sp ace E d with combi- natorics of a bipartite graph. If all of the v ertices of P lie on a non-degenerate qu adric, then P is infinitesimally flexible. 5 Assume that all of the v ertices of P lie on the sphere. Mo ve all the white v ertices to wa rds the cen ter of the sphere, and all the blac k v ertices in the opp osite direction, see Figure 3. It is easy to see that the distances b et ween white and blac k vertice s don’t c hange in the fir st order. Th us P is infi nites- imally flexible. Since an y non-degenerate quadric is a pro jectiv e image of the sph er e, P is also infinitesimally flexible when it is inscrib ed in a quadr ic. Figure 3: The framew ork at the left is in finitesimally flexible. Due to the pro jectiv e inv ariance of infinitesimal rigidit y , the framew ork at th e righ t is also infinitesimally flexible. The question ab out rigidity of bipartite frameworks w as studied in [BR80]. A c haracterization of infinitesimally flexible complete b ip artite framew orks is giv en in [Whi84a ]. 2 Infinitesimal and static rigidit y 2.1 F ramew orks Let ( V , E ) b e a graph w ith v ertex set V and edge set E . W e denote the v ertices by letters i, j, . . . , and an edge joinin g the vertice s i and j by ij . Definition 2.1 A framew ork in E d with gr aph ( V , E ) is a map P : V → E d , i 7→ p i such that p i 6 = p j whenever ij ∈ E . In other w ords, a f ramew ork is a straight- line dra wing of a graph in E d , w ith self-in tersections (ev en non-transv erse ones) allo wed. The motiv ation for studying framew orks comes fr om mechanica l link ages; namely , the edges of a framew ork should b e considered as r igid b ars, and the vertices as un iv ersal join ts. Throughout th e pap er w e assume that the ve rtices ( p i ) i ∈V of the frame- w ork span the s p ace E d affinely . This is n ot a crucial restriction: if the 6 framew ork lies in an affine s ubspace of E d , then its in fi nitesimal motions can b e decomp osed into the direct su m of infinitesimal motions insid e span { p i } and arbitrary displacemen ts in d irections orthogonal to sp an { p i } . Remark W e use d ifferen t notations for the Euclidean space E d and for the v ector space R d . In f ormally sp eaking, E d consists of p oin ts, R d consists of v ectors. W e obtain E d from R d b y “forgetting” the origin. The tangen t space at every p oin t of E d is R d with the standard scalar pro duct. Also, ev ery pair of p oin ts p , p ′ in E d determines a vecto r p ′ − p ∈ R d . 2.2 Infinitesimal rigidity A con tinuous motion of the framewo rk P is a family P ( t ) of framew orks ( t ranges ov er a neigh b orho o d of 0) su c h that P (0) = P an d the length of ev ery bar does n ot dep end on t : k p i ( t ) − p j ( t ) k = const ij for ev ery ij ∈ E . (6) If P ( t ) is d ifferen tiable, then differen tiating (6) at t = 0 yields h p i − p j , ˙ p i − ˙ p j i = 0 f or ev ery ij ∈ E . This motiv ates the follo wing definition. Definition 2.2 A v elo cit y fi eld on the fr amework P is a map Q : V → R d , i 7→ q i . A velo city field on P is c al le d an infin itesimal motion of P iff h p i − p j , q i − q j i = 0 for every ij ∈ E . (7) Since the conditions (7) are linear in Q , infinitesimal motions of th e framew ork P f orm a vec tor space. Denote this vect or space by Q mot . Let { Φ t } b e a differentiable family of isometries of E d suc h that Φ 0 = id. The ve ctor field on E d giv en by Q ( x ) = d Φ t ( x ) dt     t =0 is called an infinitesimal isometry of E d . An infinitesimal m otion of P that is the restriction of an infinitesimal isometry of E d is called trivial . The space of trivial infinitesimal motions of P is denoted by Q triv . Definition 2.3 The fr amework P is c al le d in finitesimally rigid iff al l its infinitesimal motions ar e trivial. The dimension of the quotient sp ac e Q mot / Q triv is c al le d the numb er of kinematic degrees of freedom of the fr amework P . 7 The fr amew ork P is called rigid iff every contin uous motion P ( t ) has the form Φ t ◦ P with Φ t a con tin uous family of isometries of R d . Intuition suggests that an infi nitesimally rigid framew ork should b e r igid. This is true [Con80, R W81], but not straig htfo rward since not eve ry con tin uous motion can b e reparametrized in to a sm o oth one. In the opp osite d irection, rigidit y do es not imply infin itesimal rigidit y . An y of th e framew orks on figur es 1–3 can serve as an example. 2.3 Static rigidit y In the statics of rigid b o dy , a force is represented as a line-b ound v ector. A collect ion of forces d o es not n ecessarily reduce to a single force. Definition 2.4 A force i s a p air ( p, f ) with p ∈ E d , f ∈ R d . A system of forces is a formal sum P i ( p i , f i ) that may b e tr ansforme d ac c or ding to the fol lowing rules: 0. a for c e with a zer o ve ctor is a zer o for c e: ( p, 0) ∼ 0; 1. f or c es at the same p oint c an b e add e d and sc ale d as usual: λ 1 ( p, f 1 ) + λ 2 ( p, f 2 ) ∼ ( p, λ 1 f 1 + λ 2 f 2 ); 2. a for c e may b e move d along its line of action: ( p, f ) ∼ ( p + λf , f ) . In E 2 , an y system of forces is equiv alen t either to a single force or to a so called “couple” ( p 1 , f ) + ( p 2 , − f ) with p 1 − p 2 ∦ f . Definition 2.5 A load on the fr amework P i s a map F : V → R d , i 7→ f i . A lo ad is c al le d an equilibr ium load iff the system of for c es P i ∈V ( p i , f i ) is e quivalent to a zer o for c e. A rigid b o d y resp onds to an equ ilibrium load by interior stresses that cancel the forces of the load. Definition 2.6 A stress on the fr amework P is a map Ω : E → R , ij 7→ ω ij . 8 The str ess Ω is said to resolv e the lo ad F iff f i + X j ∈V ω ij ( p j − p i ) = 0 for al l i ∈ V , (8) wher e we assume ω ij = 0 for al l ij / ∈ E . W e denote the v ector space of equilibrium loads by F eq , and the vecto r space of resolv able loads b y F res . It is ea sy to see that only an equilibrium load can b e r esolv ed: F res ⊂ F eq . Definition 2.7 The fr amewo rk P is c al le d s tatical ly r igid iff every e q u ilib- rium lo ad on P c an b e r esolve d. The dimension of the quotient sp ac e F eq / F res is c al le d the numb er of static degrees of freedom of the fr amework P . 2.4 Relation b etw een infinitesimal and static rigidit y Define a pairin g b et w een velocit y fi elds and loads on th e framework P : h Q, F i = X i ∈V h q i , f i i . (9) Clearly , this pairing is non-d egenerate, thus it indu ces a dualit y b etw een the space of vel o cit y fields and th e space of loa ds. The follo win g theorem p r o vides a link b etw een statics and k in ematics of framew orks. Theorem 3 The p airing (9) induc es a duality Q mot / Q triv ∼ = ( F eq / F res ) ∗ b etwe en the sp ac e of non-trivial infinitesimal motions and the sp ac e of non- r esolvable e quilibriu m lo ads. In p articular, a fr amework is infinitesimal ly rigid iff it is static al ly rigid. F or an i nfinitesimal ly flexible fr amework, the numb er of kinematic de- gr e es of fr e e dom is e qual to the nu mb er of static de gr e es of fr e e dom. Pr o of T his follo ws from Lemma 2.8 and f rom the canonical isomorphism ( V 1 /V 2 ) ∗ ∼ = V ⊥ 2 /V ⊥ 1 for an y p air of vect or su bspaces V 1 ⊃ V 2 of a space V .  Lemma 2.8 (Principles of virtual work) U nder the p airing (9) , 1. the sp ac e of i nfinitesimal motions is the orth o gonal c omplement of the sp ac e of r esolvable lo ads: Q mot = ( F res ) ⊥ ; 9 2. the sp ac e of trivial infinitesimal motions is the ortho gonal c omplement of the sp ac e of e quilibrium lo ads: Q triv = ( F eq ) ⊥ . Pr o of The sp ace of resolv able loads is spanned by the loads ( F ij ) ij ∈E with comp onent s f ij i = p i − p j f ij j = p j − p i f ij k = 0 for k 6 = i, j. The orthogonalit y condition h Q, F ij i = 0 is equiv alent to h q i − q j , p i − p j i = 0. Th us Q ∈ ( F res ) ⊥ iff Q is an infinitesimal motion, and the first principle is pro v ed. Let us pro v e that Q triv ⊃ ( F eq ) ⊥ . Let Q b e a v elocity field that anni- hilates ev ery equilibrium lo ad. T he load F ij defined in the previous p ara- graph is an equilibrium load for ev ery i, j ∈ V (with ij not necessarily in E ). The equations h Q, F ij i = 0 imply that Q infinitesimally preserv es pairwise distances b et w een the p oin ts ( p i ) i ∈V . Therefore Q can b e extended to an infinitesimal isometry of E d , that is Q ∈ Q triv . Let us pro v e Q triv ⊂ ( F eq ) ⊥ . Let Q b e the restriction of an infinitesimal isometry of E d . W e ha v e to show that h Q, F i = 0 for every equ ilibr ium load F . Sin ce th e s ystem of forces P i ∈V ( p i , f i ) corresp ondin g to F is equiv alent to zero, there is a sequence of trans f ormations as in Definition 2.4 that leads fr om P i ( p i , f i ) to 0. It is n ot hard to sho w that the num b er h Q, F i remains unc hanged after eac h transformation (if a force ( p ′ , f ′ ) with a new application p oin t p ′ app ears, then we s ubstitute for q ′ in the expr ession h q ′ , f ′ i the v elo cit y v ector of our global infinitesimal isometry). Since F v anishes at the end, we ha v e h Q, F i = 0 also at the b eginning.  Corollary 2.9 dim F eq = d |V | −  d + 1 2  Pr o of Due to L emma 2.8, dim F eq = d |V | − dim Q triv = d |V | −  d +1 2  .  Let Φ : E d → E d b e an affine isomorphism. The framew ork P ′ = Φ ◦ P is called affinely e qu i valent to P . Corollary 2.10 Infinitesimal rigidity is an affine invariant. Mor e over, for any two affinely e quivalent fr ameworks ther e is a c anonic al bije ction b etwe en their i nfinitesimal motions that r estricts to a bi je ction b etwe en trivial in- finitesimal motions. Explicitly, let Φ : x 7→ Ax + b b e an affine isomorphism of E d , written in an orthonorma l c o or dinate system. Then the map tha t r elates infinitesimal motions of P with infinitesimal motions of Φ ◦ P is ( A ∗ ) − 1 . 10 Pr o of S tatic rigidity is affinely inv arian t in a s tr aigh tforward w ay . Defini- tions in Section 2.3 use only the affine stru cture of E d , and not the metric structure. Giv en an affine isomorph ism Φ : x 7→ Ax + b , the tr an s formation of forces Φ stat : f 7→ Af maps equilibrium loa ds to equilibriu m ones and resolv able to r esolv able. In order to obtain a transformation Φ kin of v elocit y fields, it suffices to require that h Φ kin ( q ) , Φ stat ( f ) i = h q , f i f or any q , f . This implies the form ula (Φ kin ) − 1 : q 7→ A ∗ q .  An alternativ e pro of of the affine inv ariance of infin itesimal rigidit y can b e foun d in [Bla2 0]. 2.5 Rigidit y matrix The rigidity matrix is a standard to ol for computing infi nitesimal motions and the num b er of d egrees of freedom of a framew ork. Definition 2.11 The rigidit y matrix of a fr amework P is an E × V matrix with ve ctor entries: R = ij i 0 B B B B @ . . . · · · p i − p j · · · . . . 1 C C C C A . It has the p attern of the e dge-vertex incidenc e matrix of the gr aph ( V , E ) , with p i − p j on the interse ction of the r ow ij and the c olumn i . Note that the ro w s of R are exactly the loads ( F ij ) ij ∈E that span the space F res , see the pro of of Lemma 2.8. The follo wing prop osition is just a reform ulation of the fir st principle of virtual w ork (Lemma 2.8, fi rst part), together with its p ro of. Prop osition 2.12 Consider R as the matr ix of a map ( R d ) V → R E . Then the fol lowing holds: k er R = Q mot ; im R ⊤ = F res . Corollary 2.13 The fr amework is infinitesimal ly rigid i ff rk R = d |V | −  d + 1 2  . Pr o of Ind eed, rk R = d |V | − dim k er R which is equal to d |V | − dim Q mot b y Prop osition 2.12. By definition, the framew ork is infi nitesimally rigid iff Q mot = Q triv . Since Q mot ⊂ Q triv and dim Q triv =  d +1 2  , the prop osition follo ws .  11 3 Pro jectiv e in terpretation of rigidit y In this sectio n w e pro ve Theorem 1. F or that, the static formulatio n of the infinitesimal rigidity turns out to b e the most suitable. Th e pro of amounts to redefining a force in p ro jectiv e terms, compatibly with Defin ition 2.4. This is done in Section 3.1. In the same s ection w e obtain f orm ulas describing the corresp ondence b et w een the loads in tw o pr o jectiv ely equiv alen t f ramew orks. In Section 3.2 w e derive fr om these f orm ulas of static corresp ond ence form u- las of kinematic corresp ondence, u sing the d ualit y from Section 2.4. Finally , w e introdu ce p ro jectiv e analogs of n otions of kinematics from Section 2.2. Recall that we identify the Euclidean space E d with the affine h yp erplane { x 0 = 1 } of R d +1 . Th is induces an affin e em b edding o f E d in to R P d . W e write p oints of R P d as equiv alence cla sses [ x ] of p oints of R d +1 \ { 0 } . 3.1 Pro ject ive statics Definition 3.1 A pro jectiv e f ramew ork with gr aph ( V , E ) is a map X : V → R P d , i 7→ [ x i ] such that [ x i ] 6 = [ x j ] whenever ij ∈ E . An affine em b eddin g of E d in to R P d asso ciates a pro jectiv e framew ork to ev ery E u clidean framework. Definition 3.2 A for c e applie d at a p oint [ x ] ∈ R P d is a de c omp osable bive ctor divisible thr ough x . Th us ev ery force at [ x ] can b e written as x ∧ y ∈ Λ 2 R d +1 . Let ( p, f ) b e a force in the sense of Definition 2.4, i.e. p ∈ E d ⊂ R d +1 , f ∈ T p E d ∼ = R d = { x 0 = 0 } . Asso ciate with ( p, f ) the bive ctor p ∧ f . Prop osition 3.3 The extension of the map ( p, f ) 7→ p ∧ f (10) by line arity is wel l- define d and establishes an isomorphism of systems of for c es on E d with Λ 2 R d +1 . Pr o of Th e extension is wel l-defined since the equiv alence relations from Definition 2.4 are resp ected b y the map (10) . Let us prov e that the map is sur jectiv e. It suffices to show that any decomp osable b ivecto r x ∧ y ∈ Λ 2 R d +1 is an image of a sys tem of forces. If the plane spann ed by x and y is not con tained in R d , then there is a point p ∈ span { x, y } ∩ E d , and hence x ∧ y = p ∧ f for an appropriate f ∈ R d . 12 Otherwise, x, y ∈ R d . In this case represen t x as x 1 + x 2 with x 1 , x 2 / ∈ R d . Then the su m x 1 ∧ y + x 2 ∧ y corresp ond s to a force couple. T o pr o v e the injectivit y , it su ffices to sho w th at the space of systems of forces on E d has dimension at most  d +1 2  = dim Λ 2 R d +1 . This follo ws from an easy fact that any force can b e written as a linear com b ination of forces from the set { ( p i , p i − p j ) | i < j } , wher e p 0 , . . . , p d is a set of affinely indep end en t p oin ts in E d .  Due to P r op osition 3.3, the follo wing d efinitions are compatible with definitions of S ection 2.3. Definition 3.4 A lo ad on a pr oje ctive fr amework X is a map G : V → Λ 2 R d +1 , i 7→ g i , wher e g i is a for c e at [ x i ] . A lo ad is c al le d an e quilib rium lo ad iff P i ∈V g i = 0 . Definition 3.5 L et X b e a pr oje ctive fr amework with gr aph ( V , E ) . Denote by E or the set of oriente d e dges: E or = { ( i, j ) | ij ∈ E } . A str ess on X is a map W : E or → Λ 2 R d +1 , ( i, j ) 7→ w ij such that w ij ∈ Λ 2 span { x i , x j } and w ij = − w j i . The str ess W is said to r esolve the lo ad G iff for al l i ∈ V we have g i = X j w ij . Prop osition 3.6 L e t P and P ′ b e two fr ameworks in E d ⊂ R P d such that P ′ = Φ ◦ P , wher e Φ : R P d → R P d is a pr oje ctive map. Then ther e is an isomorph ism b etwe en the sp ac es of e quilibrium lo ads on P and P ′ that maps r esolvable lo ads to r esolvable ones. Pr o of Cho ose a represen tativ e M ∈ GL ( R d +1 ) for Φ and den ote by X and X ′ the pr o jectiv e framew orks asso ciated to P and P ′ . The map M ind u ces a map M ∗ : Λ 2 R d +1 → Λ 2 R d +1 . Being a linear iso- morphism, M ∗ maps equilibrium loads on X to equilibrium loads on X ′ , and resolv able ones to resolv able ones. Due to Pr op osition 3.3, the (pro jectiv e) loads on X , resp ectiv ely X ′ , nicely co rresp ond to (Euclidean) loads on P , resp ectiv ely P ′ . This yields the desired isomorphism b et ween the spaces of loads on P and P ′ .  13 Denote by Φ stat the isomorph ism b etw een the spaces of loads on P and P ′ constructed in the p ro of of Prop osition 3.6. It consists of a family of isomorphisms Φ stat p : T p E d → T Φ( p ) E d , p ∈ E d . Since the construction in v olv es th e choic e of a representati v e M , the iso- morphism Φ stat is determined only up to scaling. The next t w o p rop ositions describ e Φ stat explicitly . Prop osition 3.7 L e t L ⊂ E d b e the hyp e rplane that is sent to infinity by the pr oje ctive map Φ . Then Φ stat p ( f ) = h L ( p ) h L ( p + f ) · (Φ( p + f ) − Φ( p )) , if p + f / ∈ L, (11) wher e h L denotes the sig ne d distanc e to the hyp erplane L . In other wor ds: to obtain Φ stat p ( f ) , map the applic ation p oint and the endp oint of the ve ctor ( p, f ) by the map Φ , and sc ale the r esulting ve ctor by the pr o duct of distanc es of these p oints fr om the hyp erpla ne L . If ( p, f ) is suc h that p + f ∈ L , then (11) con tains an in determinacy . In this case the map Φ stat p can b e extended to f by con tin uit y or by linearit y . Pr o of Consider P as a p ro jectiv e framew ork. Cho ose a represent ativ e M ∈ GL ( R d +1 ) of Φ. The vec tor Φ stat p ( f ) is un iquely determined b y the equation M ∗ ( p ∧ f ) = Φ( p ) ∧ Φ stat p ( f ) (12) and the condition Φ stat p ( f ) ∈ R d . Denote x ′ := M ( p ). Then w e hav e x ′ = λ · Φ( p ) . It is n ot h ard to see that λ = c · h L ( p ) for some constant c indep endent of p . Thus w e ha v e M ∗ ( p ∧ f ) = M ∗ ( p ∧ ( p + f )) = c 2 h L ( p ) h L ( p + f ) · Φ( p ) ∧ Φ( p + f ) = c 2 h L ( p ) h L ( p + f ) · Φ( p ) ∧ (Φ( p + f ) − Φ( p )) . Cho osing M so that c = 1, w e obtain (11) fr om (12).  Prop osition 3.8 Φ stat p = h 2 L ( p ) · d Φ p , (13) wher e dΦ p is the differ ential of the map Φ at the p oint p . 14 Pr o of This follo ws fr om (11) b y replacing f with tf and taking th e limit as t → 0. There is also a sim p le d irect p ro of. F rom the pro of of Prop osition 3.6, it is immed iate that the v ectors Φ stat p ( f ) and dΦ p ( f ) are collinear for ev ery f . Since Φ stat p and dΦ p are linear maps, this implies Φ stat p = λ ( p ) · dΦ p , (14) and it remains to determine the fun ction λ ( p ). C onsider t w o arb itrary p oin ts p 1 and p 2 that are not mapp ed to infin ity by Φ and the forces p 2 − p 1 at p 1 and p 1 − p 2 at p 2 . Since these forces are in equilibrium, so must b e their images. Thus we hav e λ ( p 1 ) λ ( p 2 ) = k d Φ p 2 ( p 1 − p 2 ) k k d Φ p 1 ( p 1 − p 2 ) k . (15) T o compute the right hand side, restrict the map Φ to the lin e p 1 p 2 . In a co ordinate system with the origin at the int ersection p oin t of p 1 p 2 with th e h yp erp lane L , this restriction take s the form x 7→ c/x . Since th e deriv ativ e of c/x is prop ortional to x − 2 , and x is prop ortional to h L , (14) and (15) imply (13) (we can forget abou t c b ecause Φ stat is defined up to scaling).  3.2 Pro ject ive kinem atics Prop osition 3.9 L e t P and P ′ b e two fr ameworks in E d ⊂ R P d such that P ′ = Φ ◦ P , wher e Φ : R P d → R P d is a pr oje ctive map. Then ther e is an isomorph ism Φ kin b etwe en the infinitesimal motions of P and P ′ that ma ps trivial infinitesimal motions to trivial ones. The map Φ kin c onsists of a family of isomorphisms Φ kin p : T p E d → T Φ( p ) E d given by Φ kin p = h − 2 L ( p ) · (dΦ − 1 p ) ∗ , (16) wher e h L denotes the signe d distanc e to the hyp erplane L sent to infinity by Φ . Pr o of This is a d irect consequence of Theorem 3, Prop osition 3.6 an d for- m ula (13).  F or the sak e of completeness and for the reason of curiosit y , let us find the pro jectiv e counterparts to the notions of kinematics. Let X b e a framew ork in R P d with graph ( V , E ). Definition 3.10 A ve lo c ity ve ctor at a p oint [ x ] ∈ R P d is an element of the ve ctor sp ac e (Λ 2 R d +1 ) ∗ / Λ 2 x ⊥ , wher e x ⊥ ⊂ ( R d +1 ) ∗ denotes the ortho gonal c omplement of x . 15 Here is the motiv atio n for this defi n ition. Lemma 3.1 1 F or a pr oje ctive fr amework, the ve ctor sp ac e of velo cities a t [ x ] is dual to the ve ctor sp ac e of for c es at [ x ] . Pr o of Indeed, th e space of forces at x is x ∧ R d +1 ⊂ Λ 2 R d +1 . F or a subsp ace W of a v ector space V , there is a canonical isomorphism W ∗ ∼ = V ∗ /W ⊥ . Since ( x ∧ R d +1 ) ⊥ = Λ 2 x ⊥ , the p r op osition follo ws.  Definition 3.12 A velo city field ( τ i ) i ∈V on a pr oje ctive fr amework X is c al le d an infinitesimal motion iff h x i ∧ x j , τ i − τ j i = 0 for every ij ∈ E . (17) An infinitesimal motion is c al le d trivial iff ther e exists τ ∈ (Λ 2 R d +1 ) ∗ such that τ i = τ + Λ 2 x ⊥ i for al l i ∈ V . Note that the difference τ i − τ j ∈ (Λ 2 R d +1 ) ∗ / ( x i ∧ x j ) ⊥ is well- defined sin ce Λ 2 x ⊥ i ⊂ ( x i ∧ x j ) ⊥ ⊃ Λ 2 x ⊥ j . Let us establish a corresp ondence with the n otions of Section 2.2. Recall that the Euclidean space E d is id en tified with the hyperp lane { x 0 = 1 } ⊂ R d +1 . C onsider a framew ork P in E d as a pro jectiv e framewo rk X with x i = p i . T o any classical v elocity v ector q ∈ T p E d asso ciate a p ro jectiv e v elocity vect or τ ∈ ( p ∧ R d +1 ) ∗ giv en by h p ∧ y , τ i := h y , q i for eve ry y ∈ T p E d . (18) (The angle brack ets at the right hand sid e mean the scalar pro d uct in T p E d .) Con v ersely , for ev ery τ ∈ (Λ 2 R d +1 ) ∗ / Λ 2 p ⊥ consider the (well-defined) co v- ector p y τ ∈ p ⊥ ⊂ ( R d +1 ) ∗ . Restrict p y τ to R d , identify R d with T p E d b y parallel translatio n, and ident ify T p E d with T ∗ p E d using the scalar pro d uct. Denote the resu lt by q : q := ( p y τ | R d ) ∗ . (19) It is n ot hard to see that (18 ) and (19) define an isomorph ism from T p E d to (Λ 2 R d +1 ) ∗ / Λ 2 x ⊥ and its in verse. Lemma 3.1 3 L et P b e a fr amework on E d , and let X b e the c orr esp onding pr oje ctive fr amework on R P d . L et Q b e a velo city field on P , and let T b e the v e lo city field on X asso ciate d to Q via (18) . Then T is an infinitesimal motion of X if and only if Q is an infinitesimal motion of P , and T is trivial if and only if Q is trivial. Pr o of Equation (18) implies h p i − p j , q i − q j i = h p i − p j , q i i − h p i − p j , q j i = −h p i ∧ p j , τ i − τ j i . 16 Therefore T satisfies (17) iff Q satisfies (7). Assume that T is trivial: τ i = τ + Λ 2 x ⊥ i for some τ . Th en for ev ery i, j ∈ V w e ha ve h q i − q j , p i − p j i = h p i y τ − p j y τ , p i − p j i = 0 , whic h implies th at Q can b e extended to an infinitesimal isometry of E d . Th us if T is trivial, so is Q . T o prov e the in v erse imp lication, compute the dimensions of the sp aces of trivial motions. F or the Eu clidean framew ork P it is equal to  d +1 2  (recall that ( p i ) i ∈V affinely span E d b y assump tion). F or the corresp ond in g pro jectiv e fr amew ork it is equal to the r ank of th e map (Λ 2 R d +1 ) ∗ → M i ∈V (Λ 2 R d +1 ) ∗ / Λ 2 p ⊥ i . Since th e v ectors ( p i ) i ∈V span R d +1 , this map is injectiv e, so its rank is equal to dim(Λ 2 R d +1 ) ∗ =  d +1 2  .  3.3 Remarks Grassmann in tro duced in his b ook of 1844 “Die lineale Ausdehnun gslehre” the biv ector represen tation of forces acting on a rigid b o dy (in terms of what w e now call Grassmann co ordinates). A go o d accoun t on that is giv en in [Kle04]. As Klein remarks, “This b ook... is written in a st yle that is extraordinarily obscure, so that for d ecades it w as not considered nor un- dersto o d. Only when similar trains of thought came from other sources w ere they recognized b elatedly in Grassmann ’s b o ok.” Once sp elled out, the biv ector represen tation of forces readily implies the p ro jectiv e inv ariance of stati c rigidit y . Apparently , this w as observ ed b y Rankine in [Ran63], where he w r ites “...theorems disco v ered b y Mr. Sylv ester ... obvio usly giv e at once the solutio n of the question”. Unfortu- nately , we don’t kn ow which theorems are mean t; p robably this is something similar to P rop osition 3.3. An exp osition of these elegan t b ut un fortunately little kn o w n ideas, along with additional r eferences, can b e found in [CW82], [Whi85]. It seems that the obser v ation of Rankine w asn’t given muc h att ent ion, b ecause th e next ment ion of the pro jectiv e inv ariance of s tatic rigidit y I am aw are of is 1920 in the pap er [Lie20] of Liebmann. Liebmann pro v es it only for f ramew orks with |E | = d |V | −  d +1 2  that conta in d pairwise connected join ts. In this case the rigidity matrix can b e red u ced to a square matrix by fixing the p ositions of these d joints. Infinitesimal or static rigidity is then equiv alent to v anishing of the d eterminan t of this square matrix. Liebmann sho ws that the determin ant is m ultiplied w ith a non-zero facto r when the framework u ndergo es a pr o jectiv e transf orm ation. T his argumen t can probably b e extended to the general case, but d o esn’t s eem to pro du ce 17 a corresp ondence b et ween the loads or v elocity fi elds of t w o pr o jectiv ely equiv alen t framew orks . Sauer in [Sau35b] giv es a pro of of the pr o jectiv e in v ariance of static rigidit y using Grassmann co ordinates of forces and finds form ula (11). In [Sau35a], Sauer pr o v es th e pro j ectiv e in v ariance of in finitesimal rigidit y in an indep enden t wa y . F or smo oth surfaces in R 3 , the pro jectiv e inv ariance of in finitesimal r igid- it y is pro ved by Darb oux [Dar93]. Other pro ofs can b e found in W underlic h [W un82] for fr amew orks, and V olk o v [V ol74] for smooth m an if olds . The asso ciation Φ 7→ Φ stat as describ ed by form ulas (11) and (13) f ails to b e a functor. Namely , the equati on (Φ ◦ Ψ) stat p = Φ stat Ψ( p ) ◦ Ψ stat p holds only up to a constan t factor, b ecause the definition of Φ stat in v olv ed c ho osing a r epresen tativ e M ∈ GL ( R d +1 ) of Φ ∈ P GL ( R d +1 ). If one w ould lik e to ha v e fu nctorialit y , one s h ould c ho ose the matrix M in S L ± ( R d +1 ) = { M ∈ GL ( R d +1 ) | det M = ± 1 } . When d is even, there are tw o p ossibilities, but they lead to the same map Φ stat due to ( − x ) ∧ ( − y ) = x ∧ y . Cho osing M in S L ± ( R d +1 ) changes the form u las (11) and (13) b y a constan t factor that dep ends on Φ . It wo uld b e in teresting to know whether this factor has a geometric meaning. Notions of s tatics clearly ha ve a homolog ical fla v or: equilibriu m loads are kind of cycles, resolv able loa ds are kind of b oundaries. This is ea sy to formalize; in th e pro jectiv e in terpretation of Section 3.1 we ha v e a c hain complex M E Λ 2 span { x i , x j } δ − → M V x i ∧ R d +1 ǫ − → Λ 2 R d +1 with appr op r iately defined maps, so that k er ǫ consists of equilibrium loads, and im δ of resolv able loads on f ramew ork X . F or kin ematics, there is a dual co c hain complex M E (Λ 2 R d +1 ) ∗ / ( x i ∧ x j ) ⊥ d ← − M V (Λ 2 R d +1 ) ∗ / Λ 2 x ⊥ i ι ← − (Λ 2 R d +1 ) ∗ with d = δ ∗ , ι = ǫ ∗ . T he maps δ and d can b e expressed thr ough the rigidit y matrix defin ed in Section 2.5. There exist higher-dimens ional generalizations of statics, see [TWW95], [TW00], [Lee96]. By dualit y they are related to the algebra of w eights [McM93], [McM96], and to the combinato rial in tersection cohomology [Bra06]. Algebraic prop erties of the arising c h ain complexes can b e u sed to pro v e deep theorems on the com b inatorics of simplicial p olytop es [Sta80], [Kal87], [McM93]. 18 4 Infinitesimal P ogorelo v maps 4.1 Pro of of Theorem 2 The definitions of fr ameworks in H d and S d rep eat Definition 2.1. Let us define infi nitesimal m otions. Definition 4.1 L et P b e a fr amework in H d or S d with gr aph ( V , E ) . A velo city field is a c ol le ction ( q i ) i ∈V of tangent v e ctors at the vertic es of the fr amework: q i ∈ T p i H d , r esp e ctively q i ∈ T p i S d . A velo city field ( q i ) is c al le d an infinitesimal motion of P iff d dt     t =0 dist( p i ( t ) , p j ( t )) = 0 for every family ( p i ( t )) such tha t p i (0) = p i , ˙ p i (0) = q i . An infinitesimal motion is c al le d trivial iff it is g ener ate d b y a differ en- tiable family of isometries of H d , r esp e ctively S d . Recall that we ident ify H d and S d with sub sets of R d +1 according to (2) and (3). The follo wing lemma is str aigh tforward. Lemma 4.2 A velo city field ( q i ) i ∈V is an infinitesimal motion of P iff h p i − p j , q i − q j i = 0 for every ij ∈ E . Her e h· , ·i deno tes the M inkowski or the Euclide an sc alar pr o duct in R d +1 , ac c or ding to whether P i s a hyp er- b olic or a sp heric al fr amework. The tangent sp ac e at p is identifie d with a ve ctor subsp ac e of R d +1 . Em b edd ings (2) and (3) allo w to asso ciate with every fr amew ork P in H d or S d a pr o jectiv e framewo rk X . Exactly as in the Euclidean case, formulas (18) and (19) define a natural bij ection b et ween vel o cit y fi elds on framew orks P and X . Lemma 4.3 L et P b e a fr amework in H d or S d , and let X b e the c orr e- sp onding pr oje ctive fr amewor k. L et Q b e a velo city field on P , and let T b e the velo city fie ld on X asso ciate d with Q . Then T is an infinitesimal motion of X if and only if Q is an infinitesimal motion of P , and T i s trivial if and only if Q is trivial. Pr o of Due to Lemma 4.2, th e arguments f rom the pro of of Lemma 3.13 can b e applied.  Theorem 2 n o w follo w s from L emmas 3.13 and 4.3. 19 4.2 Computing P ogorelo v maps Let P b e a Euclidean f ramew ork w ith asso ciate d hyp erb olic and spherical framew orks P H and P S . Our pro of of Theorem 2 sho ws that there are nat- ural bijections b et we en v elocity fi elds on P , P H , and P S that map in finites- imal motions to infi n itesimal motio ns and resp ect the trivialit y pr op ert y . Let u s denote the v ector fields associated with Q = ( q i ) b y Q H = ( q H i ) and Q S = ( q S i ). Prop osition 4.4 The velo city fields Q , Q H , and Q S ar e r elate d by the e qua- tions q i = pr( p 1 − k p i − c k 2 · q H i ); q i = pr( p 1 + k p i − c k 2 · q S i ) , with k · k denoting the Euclide an sc alar pr o duct. Her e c ∈ E d ⊂ R d +1 is the p oint with c o or dinates (1 , 0 , . . . , 0) (th e “tangent p oint” of E d , H d , and S d ), and pr : R d +1 → R d is the pr oje ction ( x 0 , x 1 , . . . , x d ) 7→ ( x 1 , . . . , x d ) . Pr o of F or brevit y , let u s omit the index i and denote by p H the v ertex of framew ork P H corresp ondin g to the verte x p of P . Let τ b e a v elocit y v ector at the vertex [ p ] in the underlying pro jectiv e framework. By defi nition, w e ha v e q = ( p y τ | T p E d ) ∗ ; q H = ( p H y τ | T p H H d ) ∗ . Since p H y τ ∈ p ⊥ , w e ha v e q H = ( p H y τ ) ∗ , where this time α 7→ α ∗ denotes the isomorphism R d +1 → ( R d +1 ) ∗ induced b y the Mink o wski scalar pro du ct. Also, it is not h ard to sh o w that q = pr(( p y τ ) ∗ ). F rom p H = p 1 − k p i − c k 2 · p w e obtain the fir st form ula of the prop osition. The formula connecting q with q S is prov ed similarly , replacing the Minko wski sca lar pro duct in R d +1 with the Euclidean one.  4.3 Remarks A differen t deriv ation of the formulas of Prop osition 4.4 can b e found in [SW]. In addition to infinitesimal P ogorelo v maps there are finite Po gorelo v maps, [Pog7 3]. Th ey asso ciate with a pair of isometric hypers urfaces P 1 , P 2 in E d pairs P H 1 , P H 2 and P S 1 , P S 2 of isometric hypers urfaces in the hyp erb olic and in th e spherical space, resp ectiv ely . Liebmann in [Lie20] pro v es the pro jectiv e inv ariance of static rigidit y for a certain class of framew ork s , see Section 3.3. After develo ping statics and 20 kinematics in an arb itrary Ca yley metric (whic h was also done by Lindemann [Lin74]), h e pro v es that the static rigidit y of a fr amework do esn ’t dep end on the choice of the metric. In the smo oth case, V olk ov [V ol74 ] prov es th at a map b et ween Rieman- nian manifolds that send s geod esics to geo desics maps in finitesimally flexible h yp ersu rfaces to infinitesimally flexible ones. Since pro jectiv e maps of E d to itself and gnomonic pro jections of the Euclidean space to the spherical and h yp er b olic spaces send geo d esics to geod esics, V olk o v’s theorem includes Darb oux’ and P ogorelo v’s as sp ecial cases. There also exist infi nitesimal P ogorelo v maps to framewo rks in the d e Sitter s p ace d S d (the one-sheeted hyp erb oloid {k x k 1 ,d = − 1 } with the metric induced by the Mink o wski metric). The metric on d S d is Lorentzia n of con- stan t curv ature 1. 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