Higher Order Flexibility of Octahedra. Hellmuth Stachel. Institute of Geometry, Vienna University of Technology.

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1 Higher Order Flexibility of Octahedra Hellmuth Stachel Institute of Geometry, Vienna University of Technology Abstract More than hundred years ago R. Bricard determined all continuously exible octahedra. On the other hand, also the geometric characterization of rst-order exible octahedra has been well known for a long time. The objective of this aer is to analyze the cases between, i.e., octahedra which are innitesimally exible of order n > but not continuously exible. We rove exlicit necessary and sucient conditions for the orders two, three and even for all n < 8, rovided the octahedron under consideration is not totally at. Any order 8 imlies already continuous exibility, as the conguration roblem for octahedra is of degree eight. Key Words: Innitesimal exibility, exible olyhedra, octahedra MS 994: 5225, 53A7 Introduction Let F be a framework in the Euclidean d-sace E d with vertex set V and edge set E, i.e., V = f ; : : : ; v g; i 2 R d 8i 2 I := f; : : : ; vg and E n (i; j) j i < j; (i; j) 2 I 2 o : For each edge i j of F the Euclidean length l ij obeys () f ij ( i ; j ) := k i j k l ij = 8(i; j) 2 E: We suose l ij > for all edges of F. Denition: The framework F = (V; E) is innitesimally exible of order n (in the classical sense) if and only if for each k 2 I there is a olynomial function (2) x k (t) := k + t x k; + : : : + t n x k;n ; n ; such that the relacement of i and j in () by x i (t) and x j (t), res., gives functions obeying (3) f ij (x i (t); x j (t)) = kx i (t) x j (t)k l ij = o(t n ) 8(i; j) 2 E; i.e., with a zero of multilicity n+ at t =, and

2 in order to exclude trivial exes, the vectors x ; ; : : : ; x v; are not the velocity vectors of the vertices ; : : : ; v, res., under any motion of F as a rigid body. The v-tuel of functions (x (t); : : : ; x v (t)) is called a nontrivial n-th-order ex of F. Suose, the framework F is given by its combinatorial structure E and by the lengths l ij of its edges. Then () denes a system of e := #E quadratic equations (4) ( i j ) ( i j ) = l 2 ij 8(i; j) 2 E for the d v unknown coordinates of the vertices. After substituting (2) in (4) the comarison of coecients of t; t 2 ; : : : ; t n gives rise to the following systems of linear equations, each for all (i; j) 2 E, for the unknown x k;r 2 R d, k = ; : : : ; v and r 2 f; : : : ; ng: (5) (6) (7) (8) ( i j ) (x i; x j; ) = ; ( i j ) (x i;2 x j;2 ) = 2 (x i; x j; ) (x i; x j; ); ( i j ) (x i;3 x j;3 ) = (x i; x j; ) (x i;2 x j;2 ); ( i j ) (x i;4 x j;4 ) = (x i; x j; ) (x i;3 x j;3 ) 2 (x i;2 x j;2 ) (x i;2 x j;2 ); and so on. The evd matrix M of coecients on the left side of these systems is always the same. It is called rigidity matrix of F (cf. []). If (ex ;r ; : : : ; ex v;r ) is any articular solution of the r-th linear system, r 2 f; : : : ; ng, then also (9) x k;r := ex k;r + c + k ; T = ; for k = ; : : : ; v with constant c 2 R d and any skew symmetric dd matrix solves this system. In articular, the \velocity vectors" x ; ; : : : ; x v; of the vertices of F have to solve the homogeneous system (5). Therefore innitesimal exibility of order can be characterized by the rank condition d(d + ) () rk(m) < vd : 2 Any nontrivial solution of (5) denes the right side in the inhomogeneous system (6), and then 2 nd -order exibility of F is equivalent to the solvability of this system. This can be reeated ste by ste for (7), (8), : : : to gure out the order of exibility for any given framework. 2 Flexible Octahedra In 898 Raoul Bricard [4] roved that there are exactly three tyes of exible octahedra 2 O in the Euclidean 3-sace E 3, the octahedra with line symmetry, those with lanar symmetry and nally a articular third tye which admits also two at ositions. In this \classical" denition the cases with trivial velocity vectors x ; = = x v; = o are excluded. This is a roer restriction since R. onnelly and H. Servatius [6] resented 994 a framework which admits an analytical ex, but only under the assumtion of vanishing initial velocities. Already a few years earlier I. Sabitov [9] roosed to denote the order of an innitesimal ex by a air (m; n) of numbers, the smallest and the highest exonent of t showing u in (2). Note also Remark 2 in [3]. 2 An octahedron is called exible when its -skeleton can be exibly embedded in E 3. During the induced analytical self-motion edges are allowed to ass through one another. 2

3 Let ( ; 2 ), (q ) and (r ; r 2 ) denote the airs of oosite vertices of O. We suose that for all i; j; k 2 f; 2g the vertices i q j r k are non-collinear. Then for the rst of Bricard's tyes all airs of oosite vertices are symmetric with resect to a line. In the second case two airs are symmetric with resect to a lane which asses through the two remaining vertices. The at congurations of the third tye have edges which are tangent to three concentric circles (see [5], Fig. 297) or which form three arallelograms (see [5], Fig. 298 or [2], Fig. 4). For further references see [2] where a new roof for the uniqueness of Bricard's octahedra was given. Actually there are two additional but degenerate cases 3 : One consists of a two-fold covered four-sided yramid ( = 2 ), which trivially exes like a sherical four-bar mechanism. At the fth tye two airs of oosite vertices, e.g. q ; r ; r 2, are located on a line l. Hence this octahedron consists of two at four-sided yramides which can rotate indeendently about the common \basis line" l. Both degenerate cases have also non-degenerate congurations which in general are rigid: In the rst case this conguration consists of a double yramid with a lanar equator q r r 2 and the remaining vertices ; 2 being symmetric with resect to the equator lane (cf. [2], footnote and case.2.). The octahedra treated in [6] are of this tye. In the second case the equator q r r 2 is located on a one-sheet hyerboloid of revolution while and 2 lie on the hyerboloid's axis (cf. [2], case.2.2). There are octahedra which at the same time belong to all three Bricard tyes and whose exions can also continuously blend into the trivial motions listed above as degenerate cases ([2],. 53). Octahedra made from cardboard can also look exible when they are innitesimally exible or when they admit two isometric and suciently adjacent congurations. Suggestions for roducing such 'exible' models can be found in W. Wunderlich's aer [5]. Here he also roves by kinematic means a geometric characterization of octahedra with rst-order innitesimal exibility (German: \Wackeloktaeder"). This is the equivalence \(i), (ii)" in the following Theorem : For an octahedron O with four non-colanar q ; r ; r 2 and 6= 2 any two of the following four statements are equivalent: (i) O is innitesimally exible of rst order. (ii) There are two oints s ; s 2 such that q r s and 2 r 2 s 2 are two tetrahedra of Mobius tye which at the same time are mutually inscribed and circumscribed. (iii) There are four airwise edge-disjoint faces of O, which san concurrent lanes. (iv) There is a second-order surface assing through the vertices ; 2 and through the sides of the skew quadrangle q r r 2. The quadrules of edge-disjoint faces mentioned in (iii) are f q r, 2 r, r 2, 2 q r 2 g as well as f 2 r 2, q r 2, 2 q r, r g. The concurrence of the lanes sanned by one of these quadrules imlies already the concurrence of the other four. The common oints are exactly s and s 2 showing u in statement (ii). 3 The author exresses his thanks to I. Sabitov for the valuable discussion held recently in this concern. 3

4 2 q 3 Φ Figure : First-order exible octahedron O with velocity vectors orthogonal to The statement (iv) (see Fig. ) can be found in [],. 36, and in [2],. 44. The equivalence \(i), (iv)" shows that one can build a rst-order exible octahedron from any given ve vertices (comare orollary in Section 4). This is since the sides of the skew quadrangle q r r 2 together with the vertex dene the second-order surface of statement (iv) uniquely. The surface is the geometric locus for the sixth vertex 2. For a lanar quadrangle q r r 2 it is sucient to choose 2 in the ane san of this quadrangle. We will rove later with equation (23) that in the skew case the velocity vectors of the vertices can be secied orthogonally to the surface. 3 onditions for n-th-order exibility We change the notation used in the revious sections. We see the octahedron O as a susension with the equator and the two oles q. Now we subdivide the edge set E of O into the equator f g and the set of edges i q j, i 2 f; : : : ; 3g and j 2 f; 2g. The latter form a biartite sub-framework O of O. This is the reason why 4

5 we can take over some of the arguments used in [4] for the analysis of a lanar biartite framework. Let an n-th-order ex of O be given by () i (t) = i + i; t + + i;n t n and q j (t) = q j + q j; t + + q j;n t n for i = ; : : : ; 3 and j = ; 2. From now on we suose non-colanar f g: Then for each t 2 R there is an ane transformation : i 7! i (t) for i = ; : : : ; 3. When we use matrix notation and write the coordinate vectors as columns, then we can set u (2) (t): i 7! i(t) = a(t) + A(t) i with a() = o; A() = I 3 ; where I 3 denotes the 33 unit matrix. The coordinates of a(t) and the entries of A(t) are olynomials of degree n. In the new notation equation (3) reads (3) h i (t) q j (t)i 2 (i q j ) 2 = o(t n ) for all i 2 f; : : : ; 3g and j 2 f; 2g: We subtract the equation for i = and obtain [ (t) h i (t)]t (t) + i (t) 2q j (t)i ( i ) T ( i + 2q j ) = o(t n ): The dierence of these equation for j = ; 2 gives 2 [ (t) i (t)]t [q (t) 2 q (t)] + 2( i ) T ( q ) = o(t n ); and due to (2) we get 2( i ) T A(t) T [q (t) 2 q (t)] + 2( i ) T ( q ) = o(t n ) for i 2 f; 2; 3g: The suosed linear indeendence of the dierence vectors f ; 2 ; 3 g imlies A(t) T [q (t) 2 q (t)] (q 2 q ) = o(t n ): This means that for each t 2 R there is a second ane transformation (4) b(t): q j (t) 7! ba(t) + A(t)T q j (t) = q j + o(t n ) for j = ; 2 with ba() = o. For t suciently near to the image oints f (t); : : : ; 3 (t)g are non-colanar, too. Hence there exists the inverse (5) b(t) : q j 7! b(t) + A(t) T q j = q j + o(t n ) with b := A T ba We substitute in (3) the matrix reresentations (2) of and (4) of b and obtain (6) a T a + 2a T (A i q j) + T i A T A i 2 T i A T q j + q T j q j ba T ba 2ba T (A T q j i ) q T j AA T q j + 2 T i A T q j T i i = o(t n ): 5

6 Searating the terms with i from those with q j(t) leads to T i (A T A I 3 ) i + 2(a T A + ba T ) i + a T a = = q T j (AA T I 3 )q j + 2(a T + ba T A T )q j + ba T ba + o(t n ): As this equation must hold for all i 2 f; : : : ; 3g and j 2 f; 2g, both sides of the equation must be constant for each t 2 R. This results in two equations which are quadratic in the coordinates of i and q j, resectively. With (5) we obtain { after subtracting bt a = a T b from both sides { (7) T i (A T A I 3 ) i + 2(a T b T )A i + (a T b T )a = (t); i = ; : : : ; 3; and for j = ; 2 q T j (t)(aa T I 3 )q j(t) + 2(a T b T AA T )q j(t) + b T AA T b a T b = (t) + o(t n ) with a rational function (t). For the sake of brevity we have ceased to indicate that the matrix A as well as the vectors a and b are functions of t. Now we aly b to q j (t) in the second quadratic equation. Due to (5) we get nally (8) q T j (I 3 A A T )q j + 2(a T b T )A T q j + (a T b T )b = (t) + o(t n ) for j = ; 2. onversely, the two equations (7) and (8) imly (6) and hence (3). Until now we have only dealt with the biartite sub-framework O of the octahedron. The edges i i+ of the equator (subscrits modulo 4) imose the additional condition h i (t) i+ (t)i 2 (i i+ ) 2 = o(t n ) for i = ; : : : ; 3: We substitute the reresentation (2) of and obtain Due to (7) this is equivalent to ( i i+ ) T (A T A I 3 )( i i+ ) = o(t n ): (9) T i (A T A I 3 ) i+ + (a T b T )A( i + i+ ) + (a T b T )a (t) = o(t n ): Due to (7) and (8) we dene two bilinear functions (2) f(t; x; y) := x T (A T A I 3 )y + (a T b T )A(x + y) + (a T b T )a (t); g(t; x; y) := x T (I 3 A A T )y + (a T b T )A T (x + y) + (a T b T )b (t): Then we can combine the equations (7), (8) and (9) in Theorem 2: The octahedron O with non-colanar f g is exible of order n if and only if in a neighborhood of t = there are functions a(t), b(t), A(t), and (t) of class n such that the vertices ; q obey the equations f(t; i ; i ) = ; f(t; i ; i+ ) = o(t n ); g(t; q j ; q j ) = o(t n ) for all i 2 f; : : : ; 3g and j 2 f; 2g with f and g according to (2). 6

7 Similar to the roof of Lemma 2 in [4] we could show that for each t suciently near to the equations f(t ; x; x) = and g(t ; x; x) = reresent two confocal surfaces of second order. The condition f(t ; i ; i+ ) = exresses conjugate osition of two oints i and i+ on the rst of these two surfaces. This is equivalent to the fact that the connecting line is a generator of this surface. Hence, if for any t the equations f(t ; i ; i ) = f(t ; i ; i+ ) = g(t ; q j ; q j ) = hold for all i = ; : : : ; 3 and j = ; 2, then we have found a second-order surface assing through the sides of the skew quadrangle such that there is a confocal surface assing through q and (comare [] or [2],. 42, Satz ). Whenever such a air of surfaces is found, the theorem of Ivory gives a new octahedron ( (t ); : : : (t )) which is incongruent but isometric to the initial octahedron ( (); : : : ()). The conditions (7), (9) and (8) show that in the case of higher-order exibility there is a multile zero at t = for the determination of airs of confocal second-order surfaces with the above mentioned roerty. Since this roblem is algebraic of degree 8, a multile zero of order > 8 at t = imlies already a one-arametric set of solutions, i.e., a continuously exible octahedron. 4 haracterizing 2 nd - and 3 rd -order exibility The equations in Theorem 2 are necessary and sucient for an octahedron O being innitesmally exible of order n. In order to obtain geometric characterizations for n = ; 2; : : : we comare the coecients of t i, i = ; 2; : : :, in these equations. For this urose we set u the Taylor exansions as A(t) = I 3 + A t + A 2 t 2 + : : : ; a(t) = a t + a 2 t 2 + : : : ; (t) = t + 2 t 2 + : : : ; b(t) = b t + b 2 t 2 + : : : : The inverse matrix A (t) can be exanded in the form with A (t) = I 3 + B t + B 2 t 2 + : : : B = A B 3 = A 3 + A 2 A + A A 2 A 3 B 2 = A 2 + A 2 B k = A k A k B : : : A B k : This imlies for the exion () according to (2) and (5) (2) i;r = a i + A i i ; q j;r = b j + B T j q j for r = ; : : : ; n : with The coecients of t in (7), (9) and (8) yield f ( i ; i ) = f ( i ; i+ ) = f (q j ; q j ) = for i = ; : : : ; 3 and j = ; 2 (22) f (x; y) := x T (A + A T )y + (at bt )(x + y) : : f (x; x) = is the second-order surface mentioned in statement (iv) of Theorem (Fig. ). 7

8 For the velocity vectors i; = a + A i and q j; = b A T q j according to (2) we can suose a symmetric matrix A and b = a. This is ossible since otherwise we suerimose a motion which aoints to each oint x the instantaneous velocity vector v(x) := 2 (a + b ) 2 (A A T )x with the skew symmetric matrix (A A T ). Then we obtain the vectors (23) i; = 2 (a b ) + 2 (A + A T ) i ; q j; = 2 (a b ) 2 (A + A T )q j ; which still solve (5). These vectors are orthogonal to (see Fig. ) as according to (22) the equation of the olar lane of oint r with resect to reads (24) f (x; r) = x T h (A + A T )r + (a b ) i + (a T bt )r = : The coecients of t 2 in (7), (9) and (8) lead to the following equations: f 2 ( i ; i ) = f 2 ( i ; i+ ) = for i = ; : : : ; 3 with (25) f 2 (x; y) := x T (A T 2 + A T A + A 2 )y + + h (a T bt )A + (a T 2 bt 2 )i (x + y) + (a T bt )a 2 ; and g 2 (q j ; q j ) =, j = ; 2, with (26) g 2 (x; y) := x T (A 2 A A T + AT 2 A2 AT 2 )y + + h (a T b T )A T + (a T 2 b T 2 ) i (x + y) + (a T b T )b 2 : The dierence (27) h 2 (x; y) := f 2 (x; y) g 2 (x; y) = = x T (A +A T ) 2 y + (a T b T )(A +A T )(x+y) + (a T b T )(a b ) = = h x T (A +A T ) + (a T b T ) i h (A +A T )y + (a b ) i deends only on the rst derivatives a, b, A, but not. : f 2 (x; x) = and q : g 2 (x; x) = are two surfaces of second order, one assing through the sides of : : : 3, the other through q and. The surface : h 2 (x; x) = belongs to the encil [ q] sanned by and q. According to the last line in (27) the equation h 2 (x; y) = is equivalent to the statement that the oints x and y have orthogonal olar lanes with resect to (note (24)). Hence : h 2 (x; x) = is olar to the absolute conic with resect to, rovided is regular. In Table all ossible cases for are listed. Here the equations (22) of are given in standard form. The corresonding equations (27) of and of one articular surface selected from the encil [ ] can be found in Table 2. These secications rove, that for a nontrivial ex the olynomial h 2 (x; x) is never roortional to f (x; x), f 2 (x; x) or g 2 (x; x). Since for regular the (imaginary) surface is olar to the absolute conic, the encil [ ] must be olar to the linear system of surfaces confocal to. The cylinder in the rst two cases of Table 2 is olar to one focal conic of with resect to. 8

9 tye of reduced equ. of A = A T a = b one-sheet hyerb. hy. araboloid x 2 a 2 + y2 b 2 z2 c 2 = x 2 a 2 y2 b 2 z = intersecting lanes k 2 x 2 z 2 = =a 2 =b 2 =c 2 =a 2 =b 2 k 2 A A A 2 A 2 A A Table : Dierent cases of tye of h 2 (x; x) surface hyerboloid hy. araboloid 4x 2 a 4 4x 2 a 4 + 4y2 b 4 + 4z2 a 2 +c 2 x 2 + b2 +c 2 y 2 = : : : ell. cylinder c 4 a 4 b 4 + 4y2 b + a 2 +b 2 x 2 z + b2 4 a 4 4 = : : : arab. cylinder inters. lanes 4k 4 x 2 + 4z 2 z 2 = : : : two-fold lane of symmetry For a 2 nd -order ex of O obeying (22) it is necessary that there is a surface assing through the sides of the skew quadrangle : : : 3 and a surface q through q and such that the encil [ q] sanned by and q contains. With and each surface in the encil [ ] asses through the equator : : : 3. In the same way each q 2 [ q] contains q and. And any 2 [ ] n fg and 2 [ q q] n fg san a encil which contains any 2 [ ] since all these surfaces are contained in the twodimensional linear system sanned by, and (see Fig. 4 where the linear system is reresented as a rojective lane). Table 2: Dierent cases of and Ψ Ψ q Ψ Ψ q Φ Ψ Ψ Figure 2: The linear system of secondorder surfaces sanned by, and onversely, if a air ( ; ) of second-order surfaces with q assing through the equator and q 2 is given such that the encil [ ] contains any q q 2 [; ] n fg, then the octahedron is innitesimally exible of order 2. This results from the following arguments: We can determine 2 [ ] and [ q] such that the dierence of their equations gives h 2 (x; x) = in (27). Then from the equations of, and q one gets A 2 + A T 2 and 9

10 a 2 b 2. Also for the acceleration vectors 2 i;2 and 2q j;2 of the vertices we can assume a symmetric matrix A 2 and a 2 = b 2 in (2) i;2 = a 2 + A 2 i and q j;2 = b 2 + B T 2 q j = b 2 (A T 2 AT 2 )q j ; because otherwise according to (9) we could add at each vertex x the vector a(x) := 2 (a 2 + b 2 ) 2 (A 2 A T 2 )x and obtain the new solution of the system (6) (28) i;2 = 2 (a 2 b 2 ) + 2 (A 2+A T 2 ) i ; q j;2 = 2 (a 2 b 2 ) 2 (A 2+A T 2 2AT 2 )q j : We demonstrate this method at the following Examle: Let be a one-sheet hyerboloid according to Table, i.e., : f (x; x) = x 2 + y2 2 z2 = =) : h 2 (x; x) = x 2 + y2 4 + z2 = : We secify as a air of tangent lanes of and q as a hyerboloid: : y 2 (z + ) 2 = and q : 2x y2 4 6(z + ) 2 = : Then the encil [ q] contains the associated ellitic cylinder : 2x y2 = from Table 2. Now we relace by 2 [ ] and set q = q such that the dierence of the equations f 2 (x; x) = of and g 2 (x; x) = of q gives exactly the equation h 2 (x; x) = of according to (27). This yields f 2 (x; x) = 4x 2 26y z z + 28 and g 2 (x; x) = 8x 2 27y z z + 28 : The comarison with (22), (25) and (26) gives A = A T = 2 5 A ; A 2 + A T = B A ; a T = bt = (; ; ); at 2 bt 2 = (; ; 24); = 2 ; 2 = 28 : The coordinates of the four vertices dening the quadrangle \ = \ and those of the oles q selected from \ q = \ q are listed in Table 3 as well as the corresonding velocity vectors (23) and acceleration vectors (28) obeying the linear systems (5) and (6). We summarize:

11 vertex velocity vector half acceleration vector T = (; 2; ) T ; = (; ; ) T ;2 = (; 5 4 ; 5 2 ) T = ( 2; ; ) T ; = ( 2; ; ) T ;2 = ( 5 2 2; ; 3 2 ) T 2 = (; 2; ) T 2; = (; ; ) T 2;2 = (; 5 4 ; 5 2 ) T = ( 3 2; ; ) T = ( 3; 2; ; ) T = ( 5 3;2 2 2; ; 3 q T = ( 2 5; ; ) 2 qt ; = ( 2 5; ; ) 2 qt ;2 = ( 7 4 5; ; 23 q T 2 = (c; 2; c) q T 2; = ( c; 2 2; c) q T 2;2 = ( 7 2 c; ) 4 ) 2; c) c := obeying 8c2 24c 3 = : Table 3: Examle of a 2 nd -order exible octahedron Theorem 3: A rst-order innitesimally exible octahedron O with vertices ; : : : 2 and a non-colanar equator : : : 3 is innitesimally exible of order two if and only if there are second-order surfaces 6= through the sides of the equator and q 6= through the oles q such that the encil sanned by and q shares a surface with the encil sanned by and the associated : h 2 (x; x) = as listed in Table 2. orollary : A second-order exible octahedron O can be built from ve arbitrary vertices ; : : : ; q, rovided f g are not colanar. There is a free choice for the last vertex on a sace-curve of order four assing through q. Proof: The equator : : : 3 and the ole q dene the surface uniquely. Then we secify any other second-order surface through the equator. There is one surface 2 [ ] q assing through q. In order to meet the conditions of Theorem 3, it is sucient to secify ole on the curve of intersection between and q. Remark: Second-order exibility of any framework means that to each vertex we can assign a curvature center in a way which is comatible with the given edges (see [3]). In this sense there is also a more kinematic characterization for 2 nd -order exibility of octahedra based on the relative motion between oosite faces 4 as dislayed in Fig. 4: An octahedron O is exible of order two if and only if there is a satial motion such that the colanar lines q r ; r ; q can serve as curvature axes for the trajectories of the moving oints 2,, r 2, resectively. The relation between movings oints and their instantaneous curvature axes under a satial motion is e.g. treated in [3],. 69. The coecients of t 3 in (7) and (8) give rise to the equations f 3 ( i ; i ) =, i = ; : : : ; 3, 4 These motions lay also a role in robotics: They are unexected innitesimal or even nite self-motions at \singular ostures" of articular arallel maniulators (see [7] or [8]).

12 r q 2 r 2 Figure 4: The relative motion between oosite faces q r and 2 r 2 of a 2 nd -order exible octahedron O: The curvature circles of the moving vertices 2 ; r 2 have colanar axes q r, r, q, resectively. and g 3 (q j ; q j ) =, j = ; 2, with bilinear functions (29) f 3 (x; y) := x T (A T 3 + A T 2 A + A T A 2 + A 3 )y + h (a T b T )A 2 + (a T 2 b T 2 )A + +(a T 3 bt 3 )i (x+y) + h (a T bt )a 2 + (a T 2 bt 2 )a i 3 g 3 (x; y) := x T (A 3 A 2 A A A 2 + A 3 A 2 A T + A 2 A T A A T 2 + +A A T 2 + A T 3 AT AT 2 AT 2 AT + AT 3)y + h (3) (a T bt )( AT + 2 AT 2) (a T 2 bt 2 )AT + (at 3 bt 3 )i (x+y) + h (a T bt )b 2 + (a T 2 bt)b 2 i 3 Again, these dene surfaces of second order, one assing through the sides of : : : 3, the other through the oles q. The dierence of these functions (3) h 3 (x; y) := x T h (A2 A 2 + A T 2 )(A +A T ) + (A +A T )(A 2 A T 2 + A T 2 ) i y + + h (a T b T )(A 2 A T 2 + A T 2 ) + (a T 2 b T 2 )(A +A T ) i (x+y)+ +2(a T b T )(a 2 b 2 ) = deends only on the rst and second derivatives of a(t), b(t) and A(t) at t =, hence on the bilinear functions f, f 2 and g 2 according to (22), (25) and (26). A geometric interretation for these conditions in analogy to Theorem 3 is based on the surfaces : h 3 (x; x) =, : f 3 (x; x) = and q : g 3 (x; x) =. This yields Theorem 4: A second-order exible octahedron O with non-colanar equator : : : 3 and oles q according to Theorem 3 is innitesimally exible of order three if and only if 2

13 there are surfaces 6= through the sides of : : : 3 and q 6= through q such that the encil sanned by and q shares a surface with the encil sanned by and : h 3 (x; x) =, associated to, and q. The quadric is dened by, and q. But the geometric meaning of this deendence has not been gured out yet. orollary 2: A third-order exible octahedron O can be built from ve arbitrary vertices ; : : : ; q, rovided f g are not colanar. The last vertex is a oint of intersection between the three quadrics, q, q assing through q. Proof: In addition to the choice used in the roof of orollary we set = and secify q 2 [ ] as the surface assing through q. Suose O is of Bricard's tye or 2. Then the quadrangle must be symmetric with resect to a lane or line. For tye 3 the quadric must be a hyerboloid of revolution (see e.g. [2]). According to orollary 2 there are third-order exible octahedra which do not obey any of these necessary conditions. This reveals that for octahedra innitesimal exibility of order 3 does not imly continous exibility. Examle: For the data given above in Table 3 we get : h 3 (x; x) = 24x 2 53y 2 4z 2 96z = ; q : 24x2 + 85y z z 32 = : The only real solutions for are 2 5; ; 2. The solution q2 6= q gives a Bricard octahedron of tyes and 2, simultaneously. The examle resented in Table 3 with a dierent choice of is exactly of 2 nd -order exibility. There are analogous conditions for innitesimal exibility of order 4 and higher. In the sense of Fig. 4 these conditions exress rojective deendencies of articular quadrics in the rojective 9-sace of quadrics in E 3. Acknowledgement This research is artially suorted by the INTAS-RFBR-97 grant 778. References [] G.T. Bennet: Deformable Octahedra. Proc. London math. soc., Sec. Series, 39{ 343 (92). [2] W. Blaschke: Uber ane Geometrie XXVI: Wackelige Achtache. Math. Z. 6, 85{93 (92). 3

14 [3] O. Bottema, B. Roth: Theoretical Kinematics. North-Holland Publishing omany, Amsterdam 979. [4] R. Bricard: Memoire sur la theorie de l'octaedre articule. J. math. ur. al., Liouville 3, 3-48 (897). [5] R. Bricard: Lecon de cinematique II. Paris 927. [6] R. onnelly, H. Servatius: Higher-order rigidity { What is the roer denition? Discrete omut. Geom., no. 2, 93{2 (994). [7] A. Karger, M. Husty: On Self-Motions of a lass of Parallel Maniulators. In J. Lenarcic, V. Parenti-astelli (eds.): Recent Advances in Robot Kinematics, Kluwer Acad. Publ., 996,. 339{348. [8] A. Karger, M. Husty: Singularities and Self-motions of Stewart-Gough Platforms. In J. Angeles, E. Zakhariev (eds.): omutational methods in Kinematics, NATO Advanced Study Institute, Varna, Bulgaria, June 997, vol. II, 279{288. [9] I.Kh. Sabitov: Local Theory of Bendings of Surfaces. In Yu.D. Burago, V.A. Zalgaller (eds.): Geometry III, Theory of Surfaces. Encycl. of Math. Sciences, vol. 48, Sringer-Verlag 992,. 79{25. [] J. Graver, B. Servatius, H. Servatius: ombinatorial rigidity. Graduate Studies in Mathematics, vol. 2, American Mathematical Society, Providence 993. [] H. Stachel: Bemerkungen uber zwei raumliche Trilaterationsrobleme. Z. Angew. Math. Mech. 62, 329{34 (982). [2] H. Stachel: Zur Einzigkeit der Bricardschen Oktaeder. J. Geom. 28, 4{56 (987). [3] H. Stachel: Innitesimal Flexibility of Higher Order for a Planar Parallel Maniulator. Institut fur Geometrie, TU Wien, Technical Reort 64 (999). [4] H. Stachel: Higher-Order Flexibility of a Biartite Planar Framework. In A. Kecskemethy, M. Schneider,. Woernle (eds.): Advances in Multibody Systems and Mechatronics. Inst. f. Mechanik und Getriebelehre, TU Graz, Duisburg 999 (ISBN ),. 345{357. [5] W. Wunderlich: Starre, kiende, wackelige und bewegliche Achtache. Elem. Math. 2, 25{32 (965). [6] W. Wunderlich: Fast bewegliche Oktaeder mit zwei Symmetrieebenen. Rad Jugosl. Akad. Zagreb 428, Mat. Znan. 6, 29{35 (987). 4