Moving Frames Classical contributions: G. Darboux, É. Cotton, É. Cartan Modern contributions: P. Griffiths, M. Green, G. Jensen I did not quite unders

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1 Moving Frames Peter J. Olver University of Minnesota http : ==www:math:umn:edu= ο olver Beijing, March, 2001 ch 1

2 Moving Frames Classical contributions: G. Darboux, É. Cotton, É. Cartan Modern contributions: P. Griffiths, M. Green, G. Jensen I did not quite understand how he [Cartan] does this in general, though in the examples he gives the procedure is clear." Nevertheless, I must admit I found the book, like most of Cartan's papers, hard reading." Cartan on groups and differential geometry" Bull. Amer. Math. Soc. 44 (1938) Hermann Weyl ch 2

3 ffl Fels, M., Olver, P.J., References Part I, Acta Appl. Math. 51 (1998) Part II, Acta Appl. Math. 55 (1999) ffl Olver, P.J., Selecta Math. 6 (2000) ffl Calabi, Olver, Shakiban, Tannenbaum, Haker, Int. J. Computer Vision 26 (1998) ffl Mar Beffa, G., Olver, P.J., Commun. Anal. Geom. 7 (1999) ffl Olver, P.J., Classical Invariant Theory, Cambridge Univ. Press, 1999 ffl Berchenko, I.A., Olver, P.J. J. Symb. Comp. 29 (2000) ffl Olver, P.J., Found. Comput. Math. 1 (2001) 3 67 ffl Olver, P.J., Geometric foundations of numerical algorithms and symmetry", Appl. Alg. Engin. Commun. Comput., to appear ffl Kogan, I.A., Olver, P.J., Invariant Euler-Lagrange equations and the invariant variational bicomplex", preprint, ch 3

4 Applications of Moving Frames ffl Differential geometry ffl Equivalence ffl Symmetry ffl Differential invariants ffl Rigidity ffl Joint Invariants and Semi-Differential Invariants ffl Invariant differential forms and tensors ffl Identities and syzygies ffl Classical invariant theory ffl Computer vision ffi object recognition ffi symmetry detection ffl Invariant numerical methods ffl Poisson geometry & solitons ffl Lie pseudogroups ch 4

5 The Basic Equivalence Problem M smooth m-dimensional manifold. G transformation group acting on M ffl finite-dimensional Lie group ffl infinite-dimensional Lie pseudo-group Equivalence: Determine when two n-dimensional submanifolds are congruent: N and N ρ M N = g N for g 2 G Symmetry: Self-equivalence or self-congruence: N = g N ch 5

6 Classical Geometry Equivalence Problem: Determine whether or not two given submanifolds N and N are congruent under a group transformation: N = g N. Symmetry Problem: Given a submanifold N, find all its symmetries (belonging to the group). ffl Euclidean group G = SE(n) or E(n) ) isometries of Euclidean space ) translations, rotations (& reflections) z 7! R z + a 8 >< >: R 2 SO(n) or O(n) a 2 R n z 2 R n ffl Equi-affine group: G = SA(n) R 2 SL(n) area-preserving ffl Affine group: R 2 GL(n) ffl Projective group: acting on RP n 1 G = A(n) G = PSL(n) =) Applications in computer vision ch 6

7 Classical Invariant Theory Binary form: Q(x) = nx k=0 ψ! n a k k x k Equivalence of polynomials (binary forms): Q(x) = (flx + ffi) n Q ψ! ffx + fi flx + ffi ψ ff g = fl fi ffi! 2 GL(2) ) multiplier representation of GL(2) ) modular forms Transformation group: g : (x; u) 7! ψ ffx + fi flx + ffi ;! u (flx + ffi) n Equivalence of functions () equivalence of graphs N Q = f (x; u) = (x; Q(x)) g ρ C 2 ch 7

8 Moving Frames Definition. A moving frame is a G-equivariant map ρ : M! G Equivariance: ρ(g z) = 8 < : g ρ(z) ρ(z) g 1 left moving frame right moving frame ρ left (z) = ρ right (z) 1 ch 8

9 Theorem. A moving frame exists in a neighborhood of a point z 2 M if and only if G acts freely and regularly near z. Necessity: Let z 2 M. Let ρ : M! G be a left moving frame. Freeness: If g 2 G z, so g z = z, then by left equivariance: ρ(z) = ρ(g z) = g ρ(z): Therefore g = e, and hence G z = feg for all z 2 M. Regularity: Suppose z n = g n z! z as n! 1 By continuity, ρ(z n ) = ρ(g n z) = g n ρ(z)! ρ(z) Hence g n! e in G. Sufficiency: By construction normalization". Q.E.D. ch 9

10 Isotropy subgroup for z 2 M: Isotropy G z = f g j g z = z g ffl free the only group element g 2 G which fixes one point z 2 M is the identity: G z = feg for all z 2 M. ffl locally free the orbits all have the same dimension as G: G z is a discrete subgroup of G. ffl regular all orbits have the same dimension and intersect sufficiently small coordinate charts only once ( 6ß irrational flow on the torus) ffl effective the only group element g 2 G which fixes every point z 2 M is the identity: g z = z for all z 2 M iff g = e: G M = G z = feg z2m ch 10

11 Geometrical Construction Normalization = choice of cross-section to the group orbits K z O z k g =ρ(z ) K cross-section to the group orbits O z orbit through z 2 M k 2 K O z unique point in the intersection ffl k is the canonical form of z ffl the (nonconstant) coordinates of k are the fundamental invariants g 2 G unique group element mapping k to z ρ(z) = g left moving frame ρ(h z) = h ρ(z) k = ρ 1 (z) z = ρ right (z) z =) freeness ch 11

12 Construction of Moving Frames Coordinate cross-section r = dim G» m = dim M K = f z 1 = c 1 ; ::: ;z r = c r g left w(g; z) = g 1 z right w(g; z) = g z Choose r = dim G components to normalize: w 1 (g; z) = c 1 ::: w r (g; z) = c r The solution is a (local) moving frame. g = ρ(z) =) Implicit Function Theorem ch 12

13 The Fundamental Invariants Substituting the moving frame formulae g = ρ(z) into the unnormalized components of w(g; z) produces the fundamental invariants: I 1 (z) = w r+1 (ρ(z);z) ::: I m r (z) = w m (ρ(z);z) =) These are the coordinates of the canonical form k 2 K. Theorem. Every invariant I(z) can be (locally) uniquely written as a function of the fundamental invariants: I(z) = H(I 1 (z);:::;i m r (z)) ch 13

14 Invariantization Definition. The invariantization of a function F : M! R with respect to a right moving frame ρ is the the invariant function I = (F ) defined by I(z) = F (ρ(z) z). [ F (z 1 ;:::;z m )] = F (c 1 ;;:::c r ;I 1 (z);:::;i m r (z)) Invariantization amounts to restricting F to the cross-section I j K = F j K and then requiring that I = (F ) be constant along the orbits. In particular, if I(z) is an invariant, then (I) = I. Invariantization defines a canonical projection : functions 7! invariants ch 14

15 The Rotation Group G = SO(2) acting on R 2 z = (x; u) 7! g z = ( x cos u sin ; x sin + u cos ) =) Free on M = R 2 nf0g Left moving frame: w(g; z) = g 1 z = (y; v) y = x cos + u sin v = x sin + u cos Cross-section K = f u = 0; x > 0 g Normalization equation Left moving frame: v = x sin + u cos = 0 = tan 1 u x =) = ρ(x; u) 2 SO(2) Fundamental invariant r = (x) = p x 2 + u 2 Invariantization [ F (x; u)]= F (r; 0) ch 15

16 Prolongation Most interesting group actions (Euclidean, affine, projective, etc.) are not free! An effective action can usually be made free by: ffl Prolonging to derivatives (jet space) G (n) : J n (M;p)! J n (M;p) =) differential invariants ffl Prolonging to Cartesian product actions G n : M M! M M =) joint invariants ffl Prolonging to multi-space" G (n) : M (n)! M (n) =) joint or semi-differential invariants =) invariant numerical approximations ch 16

17 Jet Space ffl Although in use since the time of Lie and Darboux, jet space was first formally defined by Ehresmann in ffl Jet space is the proper setting for the geometry of partial differential equations. M smooth m-dimensional manifold 1» p» m 1 J n = J n (M;p) (extended) jet bundle =) Defined as the space of equivalence classes of p- dimensional submanifolds under the equivalence relation of n th order contact at a single point. =) Can be identified as the space of n th order Taylor polynomials for submanifolds given as graphs u = f(x) ch 17

18 Local Coordinates on Jet Space J n = J n (M;p) n th extended jet bundle for p-dimensional submanifolds N ρ M Local coordinates: Assume N = fu = f(x)g is a graph (section). x = (x 1 ;:::;x p ) independent variables u = (u 1 ;:::;u q ) dependent variables p + q = m = dim M z (n) = (x; u (n) ) = ( ::: x i ::: u ff J ::: ) u ff J J uff 0» #J» n induced jet coordinates ffl No bundle structure assumed on M. ffl Projective completion of J n E when E! X is a bundle. ch 18

19 Prolongation of Group Actions G transformation group acting on M =) G maps submanifolds to submanifolds and preserves the order of contact G (n) prolonged action of G on the jet space J n The prolonged group formulae w (n) = (y; v (n) ) = g (n) z (n) are obtained by implicit differentiation: dy i = D y j = px j=1 px i=1 Pj i (g; z(1) ) dx j =) Q = P T Q i j (g; z(1) ) D x i vj ff = D y j 1 D y j k (v ff ) Differential invariant I : J n! R I(g (n) z (n) ) = I(z (n) ) =) curvatures ch 19

20 Freeness Theorem. If G acts (locally) effectively on M, then G acts (locally) freely on a dense open subset V n ρ J n for n fl 0. Definition. N ρ M is regular at order n if j n N ρ V n. Corollary. Any regular submanifold admits a (local) moving frame. Theorem. A submanifold is totally singular, j n N ρ J n n V n for all n, if and only if its symmetry group does not act freely on N. G N = f g j g N ρ N g ch 20

21 Moving Frames on Jet Space w (n) = (y; v (n) ) = 8 >< >: g (n) z (n) right (g (n) ) 1 z (n) left Choose r = dim G jet coordinates z 1 ;:::;z r x i or u ff J Coordinate cross-section K ρ J n z 1 = c 1 ::: z r = c r Corresponding lifted differential invariants: w 1 ;:::;w r y i or v ff J Normalization Equations Solution: w 1 (g; x; u (n) ) = c 1 ::: w r (g; x; u (n) ) = c r g = ρ (n) (z (n) ) = ρ (n) (x; u (n) ) =) moving frame ch 21

22 The Fundamental Differential Invariants I (n) (z (n) ) = w (n) (ρ (n) (z (n) );z (n) ) H i (x; u (n) ) = y i (ρ (n) (x; u (n) );x;u) I ff K (x; u(k) ) = v ff K (ρ(n) (x; u (n) );x;u (k) ) Phantom differential invariants w 1 = c 1 ::: w r = c r =) normalizations Theorem. Every n th order differential invariant can be locally uniquely written as a function of the non-phantom fundamental differential invariants in I (n). ch 22

23 Contact-invariant coframe dy i 7!! i = Invariant Differentiation px j=1 Invariant differential operators: D y j 7! D j = Duality: df = px i=1 px i=1 Pj i (ρ(n) (z (n) );z (n) ) dx i =) arc length element Q i j (ρ(n) (z (n) );z (n) ) D x i =) arc length derivative D i F! i Theorem. The higher order differential invariants are obtained by invariant differentiation with respect to D 1 ;:::;D p. ch 23

24 Euclidean Curves G = SE(2) Assume the curve is (locally) a graph: C = fu = f(x)g Prolong to J 3 via implicit differentiation y = cos (x a) + sin (u b) v = sin (x a) + cos (u b) v y = sin + u x cos cos + u x sin v yy = u xx (cos + u x sin ) 3 v yyy = (cos + u x sin )u xxx 3u2 xx sin (cos + u x sin ) 5. Normalization r = dim G = 3 9 = ; w = R 1 (z b) y = 0; v = 0; v y = 0 Left moving frame ρ : J 1! SE(2) a = x; b = u; = tan 1 u x ch 24

25 Differential invariants v yy 7!» = v yyy 7! d» ds v yyyy 7! d2» ds 2 3»3 = u xx (1 + u 2 x )3=2 = (1 + u2 x )u xxx 3u x u2 xx (1 + u 2 x )3 Invariant one-form arc length dy = (cos + u x sin ) dx 7! ds = q 1+u 2 x dx Invariant differential operator d dy = 1 cos + u x sin d dx 7! d ds = 1 q 1+u 2 x d dx Theorem. All differential invariants are functions of the derivatives of curvature with respect to arc length:»; d» ds ; d 2» ds ; 2 ch 25

26 Euclidean Curves e 2 e 1 x Moving frame R = 1 q 1+u 2 x ψ 1 ρ : (x; u; u x ) 7! (R; a) 2 SE(2)! ψ ux x = (e u x 1 1 ; e 2 ) a = u! Frenet frame e 1 = dx ds = ψ xs y s! e 2 = e? 1 ψ! = ys x s Frenet equations = Maurer Cartan equations: dx ds = e 1 de 1 ds =» e 2 de 2 ds =» e 1 ch 26

27 The Replacement Theorem Any differential invariant has the form I = F (x; u (n) ) = F (y; w (n) ) = F (I (n) ) =) T.Y. Thomas» = v yy (1 + v 2 y )2 = u xx (1 + u 2 x )2 (x) = (u) = (u x ) = 0 (u xx ) =» ch 27

28 Equi-affine Curves G = SA(2) z 7! Az+ b A 2 SL(2); b 2 R 2 Prolong to J 3 via implicit differentiation dy = (ffi u x fi) dx D y = 1 ffi u x fi D x y = ffi(x a) fi(u b) v = fl(x a)+ff(u b) 9 = ; w = A 1 (z b) v y = fl ffu x ffi fiu x v yy = v yyy = (ffi fiu x )u xxx +3fiu2 xx (ffi fiu x ) 5 u xx (ffi fiu x ) 3 v yyyy = u xxxx (ffi fiu x )2 +10u xx u xxx fi(ffi fiu x )+15u 3 xx fi2 (ff + fiu x ) 7. Nondegeneracy u xx =0 =) Straight lines are totally singular (three-dimensional equi-affine symmetry group) Normalization r = dim G =5 y = 0; v = 0; v y = 0; v yy = 1; v yyy = 0: ch 28

29 Left Moving frame ρ : J 3! SA(2) A = ψ = q 3 u xx q u x 3 dz ds u xx u 1=3 xx 1 3 u 5=3 xx u xxx 1 3 u 5=3 xx u xxx! d 2 z ds 2 1 A b = z = ψ! x u e 2 e 1 x ch 29

30 Frenet frame e 1 = dz ds e 2 = d2 z ds 2 Frenet equations = Maurer Cartan equations: dz ds = e 1 de 1 ds = e 2 de 2 ds =» e 1 Equi-affine arc length dy 7! ds = 3 qu xx dx = 3 q _z ^ z dt Invariant differential operator D y 7! d ds = qu 1 D x = xx 3 1 q D t _z ^ z 3 Equi-affine curvature v 4y 7!» = 5u xx u xxxx 3u2 xxx 9u 8=3 xx = z s ^ z ss v 5y 7! d» ds v 6y 7! d2» ds 2 5»2 ch 30

31 Equivalence & Signature Cartan's main idea: The equivalence and symmetry properties of submanifolds will be found by restricting the differential invariants to the submanifold J(x) = I(j n Nj x ). Equivalent submanifolds should have the same invariants. However, unless an invariant J(x) is constant, it carries little information by itself, since the equivalence map will typically drastically change the dependence of the invariant on the parameter x. =) Constant curvature submanifolds However, a functional dependency or syzygy among the invariants is intrinsic: J k (x) = Φ(J 1 (x);:::;j k 1 (x)) ch 31

32 The Signature Map Equivalence and symmetry properties of submanifolds are governed by the functional dependencies syzygies" among the differential invariants. J k (x) = Φ(J 1 (x);:::;j k 1 (x)) The syzygies are encoded by the signature map ± : N! S of the submanifold N, which is parametrized by the fundamental differential invariants: ±(x) = (J 1 (x);:::;j m (x)) = ( I 1 j N; ::: ;I m j N ) The image S = Im ± is the signature subset (or submanifold) of N. ch 32

33 Geometrically, the signature S ρ K is the image of j n N in the cross-section K ρ J n, where n fl 0 is sufficiently large. ± : N! j n N! S ρ K Theorem. Two submanifolds are equivalent N = g N if and only if their signatures are identical S = S ch 33

34 Signature Curves Definition. The signature curve S ρ R 2 of a curve C ρ R 2 is parametrized by the first two differential invariants» and» s S = ( ψ» ; d» ds! ) ρ R 2 Theorem. Two curves C and C are equivalent C = g C if and only if their signature curves are identical S = S =) object recognition ch 34

35 Symmetry Signature map ± : N! S Theorem. Let S denote the signature of the submanifold N. Then the dimension of its symmetry group G N = f g j g N ρ N g equals dim G N = dim N dim S Corollary. For a regular submanifold N ρ M, 0» dim G N» dim N =) Only totally singular submanifolds can have larger symmetry groups! ch 35

36 Maximally Symmetric Submanifolds Theorem. The following are equivalent: ffl The submanifold N has a p-dimensional symmetry group ffl The signature S degenerates to a point dim S = 0 ffl The submanifold has all constant differential invariants ffl N = H fz 0 g is the orbit of a p-dimensional subgroup H ρ G =) In Euclidean geometry, these are the circles, straight lines, spheres & planes. =) In equi-affine plane geometry, these are the conic sections. ch 36

37 Discrete Symmetries Definition. The index of a submanifold N equals the number of points in C which map to a generic point of its signature S: N = min n #± 1 fwg fi fi fi w 2 S o =) Self intersections Theorem. The cardinality of the symmetry group of N equals its index N. =) Approximate symmetries ch 37

38 Classical Invariant Theory (ψ! fi ff fi fifififi ) M = R 2 nfu = 0g G = GL(2) = ffffi fifl 6= 0 fl ffi (x; u) 7! ψ ffx + fi flx + ffi ;! u (flx + ffi) n n 6= 0; 1 ff = flx+ ffi = ffffi fifl Prolongation: y = ffx + fi flx + ffi v = ff n u v y = ffu x nflu ff n 1 v yy = ff2 u xx 2(n 1)flffu x + n(n 1)fl 2 u 2 ff n 2 v yyy = Normalization: y = 0 v = 1 v y = 0 v yy = 1 n(n 1) ch 38

39 Moving frame: ff = u (1 n)=np H fl = 1 n u(1 n)=n fi = xu (1 n)=np H ffi = u 1=n 1 n xu(1 n)=n H = n(n 1)uu xx (n 1) 2 u 2 x Hessian Nonsingular form: H 6= 0 Note: H 0 if and only if Q(x) = (ax + b) n =) Totally singular forms Differential invariants: J v yyy 7! n 2 (n 1) ß» v yyyy 7! K +3(n 2) n 3 (n 1) ß d» ds Absolute rational covariants: J 2 = T 2 H 3 K = U H 2 H = 1 2 (Q; Q)(2) = n(n 1)QQ 00 (n 1) 2 Q 02 ο Q xx Q yy Q 2 xy T = (Q; H) (1) =(2n 4)Q 0 H nqh 0 ο Q x H y Q y H x U = (Q; T ) (1) =(3n 6)Q 0 T nqt 0 ο Q x T y Q y T x deg Q = n deg H = 2n 4 deg T = 3n 6 deg U = 4n 8 ch 39

40 Signatures of Binary Forms Signature curve of a nonsingular binary form Q(x): S Q = ( (J(x) 2 ;K(x)) = ψ T (x) 2 H(x) 3 ; U(x) H(x) 2!) Nonsingular: H(x) 6= 0 and (J 0 (x);k 0 (x)) 6= 0. Signature map ±: N Q! S Q ±(x) = (J(x) 2 ;K(x)) Theorem. Two nonsingular binary forms are equivalent if and only if their signature curves are identical. ch 40

41 Maximally Symmetric Binary Forms Theorem. If u = Q(x) is a polynomial, then the following are equivalent: ffl Q(x) admits a one-parameter symmetry group ffl T 2 is a constant multiple of H 3 ffl Q(x) ' x k is complex-equivalent to a monomial ffl the signature curve degenerates to a single point ffl all the (absolute) differential invariants of Q are constant ffl the graph of Q coincides with the orbit of a one-parameter subgroup =) diagonalizable ch 41

42 Symmetries of Binary Forms Theorem. The symmetry group of a nonzero binary form Q(x) 6 0 of degree n is: ffl A two-parameter group if and only if H 0 if and only if Q is equivalent to a constant. =) totally singular ffl A one-parameter group if and only if H 6 0 and T 2 is a constant multiple of H 3 if and only if Q is complexequivalent to a monomial x k, with k 6= 0;n. =) maximally symmetric ffl In all other cases, a finite group whose cardinality equals the index Q = min n #± 1 fwg fi fi fi w 2 S o of the signature curve, and is bounded by Q» 8 < : 6n 12 U = ch 2 4n 8 otherwise ch 42

43 Let G act on M. Joint Invariants A k-point joint invariant is an invariant of the k-fold Cartesian product action on M M I(g z 1 ;:::;g z k ) = I(z 1 ;:::;z k ) A k-point joint differential invariant is an invariant of the prolonged action G (n) on a k-fold Cartesian product of jet space J n J n I(g (n) z (n) 1 ;:::;g (n) z (n) ) = I(z(n) 1 ;:::;z (n) k k ) =) Joint differential invariants are known as semi-differential invariants" in the computer vision literature, and are proposed as noise resistant" alternatives for object recognition. ch 43

44 Joint Euclidean Invariants SE(2) acts on M = R 2 R 2 : z i = (x i ;u i ) w i = (y i ;v i ) = g 1 z i i =0; 1; 2;::: y i = cos (x i a) + sin (u i b) v i = sin (x i a) + cos (u i b) Normalization (cross-section) Left moving frame y 0 = 0 v 0 = 0 y 1 > 0 v 1 =0 ρ : M! SE(2) ψ! a = x 0 b = u 0 = tan 1 u1 u 0 x 1 x 0 Joint invariants: y i 7! (z i z 0 ) (z 1 z 0 ) k z 1 z 0 k v i 7! (z i z 0 ) ^ (z 1 z 0 ) k z 1 z 0 k ch 44

45 Theorem. Every joint Euclidean invariant is a function of the interpoint distances k z i z j k and, in the orientation preserving case, a single signed area A(z 0 ;z 1 ;z 2 ) ch 45

46 Joint Invariant Signatures If the invariants depend on k points on a p-dimensional submanifold, then you need at least ` > kp distinct invariants I 1 ;:::;I` in order to construct a syzygy: The total number of syzygies is Φ(I 1 ;:::;I`) 0 ` kp Typically, the number of joint invariants is ` = km r = (#points)(dim M) dim G Therefore, to find a joint invariant signature, that involes no differentiation, we need at least k points on our submanifold. r m p +1 ch 46

47 Joint Euclidean Signature For the Euclidean group G = SE(2) acting on curves C ρ R 2 (or R 3 ) we need at least four points Joint invariants: z 0 ;z 1 ;z 2 ;z 3 2 C a = k z 1 z 0 k; b = k z 2 z 0 k; c = k z 3 z 0 k; d = k z 2 z 1 k; e = k z 3 z 1 k; f = k z 3 z 2 k: =) six functions of four variables Joint Signature: ±: C 4! S ρ R 6 dim S = 4 =) two syzygies Φ 1 (a; b; c; d; e; f) = 0 Φ 2 (a; b; c; d; e; f) = 0 Universal Cayley Menger syzygy: fi fi fi fi 2a det fi 2 a 2 + b 2 d 2 a 2 + c 2 e 2 fi fi fi a fi 2 + b 2 d 2 2b 2 b 2 + c 2 fi f 2 fi fi fi a 2 + c 2 e 2 b 2 + c 2 f 2 2c 2 fi =0 () C ρ R 2 ch 47

48 z 0 c z 3 a e b f z 1 d z 2 Four-Point Euclidean Joint Signature ch 48

49 Euclidean Joint Differential Invariants Planar Curves ffl One point ffl Two point ) curvature» = Π z ^ z ΠΠ k z Π k 3 ) distances k z 1 z 0 k ) tangent angles ffi k = <) (z 1 z 0 ; Π z k ) ch 49

50 Equi Affine Joint Differential Invariants Planar Curves ffl One point ) affine curvature» = (z t ^ z tttt )+4(z tt ^ z ttt ) 3(z t ^ z tt ) 5=3 5(z t ^ z ttt )2 9(z t ^ z tt ) 8=3 = z s ^ z ss ffl Two point ) tangent triangle area ratio Π z 0 ^ z ΠΠ 0 [(z 1 z 0 ) ^ z Π 0 ] = [ Π 0 Π Π 0 ] 3 [ Π ] 3 ffl Three point ) triangle area 1 (z z ) ^ (z z ) = 1 [ ] ch 50

51 Projective Joint Differential Invariants Planar Curves ffl One point ffl Two point ) projective curvature» = ::: ) tangent triangle area ratio ffl Three point ffl Four point ) tangent triangle ratio [ Π ] 3 [ 1 Π 1 ΠΠ ] [ Π ] 3 [ 0 Π 0 ΠΠ ] [ 0 2 Π 0 ][0 1 Π 1 ][1 2 Π 2 ] [ 0 1 Π 0 ][1 2 Π 1 ][0 2 Π 2 ] : ) area cross ratio [ ][0 3 4 ] [ ][0 2 4 ] ch 51

52 Transformation Groups and Jets (x 1 ;:::;x p ) independent variables (u 1 ;:::;u q ) dependent variables z (n) = (x; u (n) ) 2 J n n th order jet space u ff J derivative coordinates on J n G transformation group G (n) prolonged action on J n v 2 g Lie algebra v (n) 2 g (n) Prolonged inf. gens. The Prolongation Formula v (n) = px i=1 ' ff J = D J Qff + ο i i + nx px i=1 ο i u ff J;i ff;j ' ff J (x; ff J Characteristic Q ff (x; u (1) ) = ' ff px i=1 i ch 52

53 Rotation group SO(2) (x; u) 7! (x cos u sin ; x sin + u cos ) Transformed function Second prolongation v = μ f(y): y = x cos f(x) sin ; v = x sin + f(x) cos ; (x; u; u x ;u xx ) 7! (x cos u sin ; x sin + u cos ; Infinitesimal generator Second prolongation sin + u x cos cos u x sin ; v v (2) +(1+u2 x ) Q = x + uu u xx (cos u x sin ) x +3u x xx ' x = D x Q + οu xx = D x (x + uu x ) uu xx = 1+u 2 x ' xx = D 2 x Q + οu xxx = D2 x (x + uu x ) uu xxx = 3u x u xx ch 53

54 Differential invariant: Infinitesimal criterion: I(g (n) (x; u (n) )) = I(x; u (n) ) v (n) (I) = 0 for all v (n) 2 g (n) =) Solve the first order linear partial differential equation by the method of characteristics. =) Moving frames avoids integration! Note: If I 1 ;:::;I k are differential invariants, so is Φ(I 1 ;:::;I k ). =) Classify differential invariants up to functional independence. =) Local results on open subsets of jet space. ch 54

55 Theorem. Any transformation group admits a finite system of fundamental differential invariants J 1 ;:::;J` and p invariant differential operators D 1 ;:::;D p such that every differential invariant is a function of the differentiated invariants: I = Φ( ::: D K J ν ::: ) ch 55

56 Classification Problem. How many fundamental differential invariants J 1 ;:::;J` are required? =) For curves (p = 1), we have ` = q. Syzygy Problem. Determine the algebraic relations Φ( ::: D K J ν ::: ) = 0 among the differentiated invariants. Commutation Formulae. The order of invariant differentiation matters [ D i ; D j ] =??? =) Only an issue when p > 1. ch 56

57 The Fundamental Differential Invariants I (n) (z (n) ) = ρ (n) (z (n) ) 1 z (n) H i (x; u (n) ) = y i (ρ (n) (x; u (n) );x;u) I ff K (x; u(k) ) = v ff K (ρ(n) (x; u (n) );x;u (k) ) Recurrence Formulae: D j H i = ffi i j + M i j D j I ff K = Iff K;j + M ff K;j M i j ;Mff K;j correction terms Commutation Formulae: [D i ; D j ] = px i=1 A k ij D k ffl The correction terms can be computed directly from the infinitesimal generators! ch 57

58 Generating Invariants Theorem. A generating system of differential invariants consists of ffl all non-phantom differential invariants H i and I ff coming from the un-normalized zero th order lifted invariants y i, v ff, and ffl all non-phantom differential invariants of the form IJ;i ff where IJ ff is a phantom differential invariant. order» order ρ + 1 In other words, every other differential invariant can, locally, be written as a function of the generating invariants and their invariant derivatives, D K H i, D K I ff J;i. =) Not necessarily a minimal set! ch 58

59 Syzygies A syzygy is a functional relation among differentiated invariants: H( ::: D J I ν ::: ) 0 Derivatives of syzygies are syzygies =) find a minimal basis Remark: There are no syzygies among the normalized differential invariants I (n) except for the phantom syzygies" I ν = c ν corresponding to the normalizations. ch 59

60 Classification of Syzygies Theorem. All syzygies among the differentiated invariants are differential consequences of the following three fundamental types: D j H i = ffi i j + M i j H i non-phantom D J I ff K = c ν + M ff K;J I ff K generating I ff J;K = w ν = c ν phantom D J I ff LK D K Iff LJ = M ff LK;J M ff LJ;K I ff LK, Iff LJ generating, K J =? =) Not necessarily a minimal system! ch 60

61 Right Regularization If G acts on M, then the lifted action (h; z) 7! (h g 1 ;g z) on the trivial right principal bundle B = G M is always regular and free! The functions w : B! M given by w(g; z) = g z provide a complete system of global invariants for the lifted action. ch 61

62 Example. G = SO(2) M = R 2 B = SO(2) R 2 solid torus (x; u; ffi) 7! (x cos u sin ; x sin + u cos ; ffi + mod 2 ß) ch 62

63 Jet Regularization B n = J n G J n J n w = w (n) = g (n) z (n) ff (n) (z (n) ) = (z (n) ;ρ (n) (z (n) )) I (n) (z (n) ) = w (n) ffi ff (n) (z (n) ) = ρ (n) (z (n) ) z (n) Invariantization (F ) = (ff (n) ) Λ ffi (w (n) ) Λ F = F ffi I (n) ch 63

64 General Philosophy of Lifting All invariant objects on B n = J n G are well-behaved and easily understood. =) lifted invariants We use the G-equivariant moving frame section ff (n) : J n! B ff(z (n) ) = (ρ(z (n) );z (n) ) to pull back lifted invariants to construct ordinary invariants on J n. For example, ff Λ w (n) = w (n) ffi ff = I (n) gives the fundamental differential invariants. Similarly for lifted invariant differential forms, differential operators, tensors, etc. =) The key complication is that the pull-back process does not commute with differentiation! ch 64

65 Infinite jet space Inverse limit Local coordinates The Variational Bicomplex M = J 0 ψ J 1 ψ J 2 ψ J 1 = lim n!1 Jn z (1) = (x; u (1) ) = ( ::: x i ::: u ff J ::: ) Coframe basis for the cotangent space T Λ J 1 : Horizontal one-forms dx 1 ;:::;dx p Contact (vertical) one-forms ff J = duff J p X i=1 u ff J;i dxi Intrinsic definition of contact form j j 1 N = 0 () = X A ff J ff J ch 65

66 Vertical and Horizontal Differentials Bigrading of the differential forms on J 1 Differential Ω Λ = M r;s Ωr;s d = d H + d V d H F = px i=1 d V F = X ff;j d H : Ω r;s! Ω r+1;s d V : Ω r;s! Ω r;s+1 (D i F ) dx i ff J ff J variation =) Vinogradov, Tsujishita, I. Anderson ch 66

67 The Simplest Example. M = R 2 x; u 2 R Horizontal form Contact (vertical) forms Differential Total derivative dx = du u x dx x = du x u xx dx xx = du xx u xxx dx. x du xx du xx + = (D x F ) @u x xx xx + = d H F + d V F D x u x u xx u xxx + ch 67

68 Lifted Variational Tricomplex B 1 = J 1 G ffl Lifted horizontal forms d J y i i = 1;:::;p ffl Lifted invariant contact forms ff J = d J vff J p X i=1 v ff J;i d J yi ffl Right-invariant Maurer Cartan forms μ = dg g 1 =) μ 1 ;:::;μ r r = dim G Differential forms on B 1 Ω Λ = M Differential r;s;t bω r;s;t d = d H + d V + d G d H : d V : d G : b Ω r;s;t b Ω r;s;t b Ω r;s;t! b Ω r+1;s;t! b Ω r;s+1;t! b Ω r;s;t+1 ch 68

69 Invariantization : Functions! Invariants Forms! Invariant Forms Functions: (F ) = ff Λ ffi w Λ (F ) = F ffi I (1) Differential Forms: (Ω) = ff Λ (ß J (w Λ Ω)): ß J Jet projection T Λ B 1 = T Λ (J 1 G) ' T Λ J 1 Φ T Λ G ch 69

70 Invariant Variational Complex Fundamental differential invariants H i (x; u (n) ) = (x i ) IK ff (x; u(l) ) = (u ff K ) Invariant horizontal one-forms $ i = (dx i ) =! i + i! i contact-invariant forms i contact corrections" Invariant contact forms # ff K = ( ff J ) Differential forms Ω Λ = M r;s bω r;s Differential d = d H + d V + d W d H : Ω b r;s! Ω b r+1;s d V : Ω b r;s! Ω b r;s+1 d W : Ω b r;s! Ω b r 1;s+2 ch 70

71 The Key Formula d (Ω) = (dω) + px k=1 ν» ^ [v» (Ω)] v 1 ;:::;v r basis for g ν» = ff Λ μ» = fl» + "»» = 1;:::;r fl» 2 b Ω 1;0 "» 2 b Ω 0;1 pull back of the dual basis Maurer Cartan forms via the moving frame section ff Λ : J 1! B 1??? All recurrence formulae, syzygies, commutation formulae, etc. are found by applying the key formula for various forms and functions Ω ch 71

72 Euclidean Curves Lifted invariants y = w Λ (x) = x cos ffi u sin ffi + a v = w Λ (u) =x cos ffi + u sin ffi + b v y = w Λ (u x )= sin ffi + u x cos ffi cos ffi u x sin ffi v yy = w Λ (u xx ) = u xx (cos ffi u x sin ffi) 3 v yyy = w Λ (u xx ) = (cos ffi u x sin ffi )u xxx 3u2 xx sin ffi (cos ffi u x sin ffi ) 5 dy = (cos ffi u x sin ffi) dx (sin ffi) + da vdffi d J y = ß J (dy) = (cos ffi u x sin ffi) dx (sin ffi) D y = 1 cos ffi u x sin ffi D x = du u x dx Normalization Right moving frame y =0 v = 0 v y = 0 ρ : J 1! SE(2) ffi = tan 1 u x a = x + uu x q 1+u 2 x b = q xu x u 1+u 2 x ch 72

73 Fundamental normalized differential invariants 9 (x) = H =0 >= (u) = I 0 = 0 phantom diff. invs. >; (u x ) = I 1 = 0 (u xx ) = I 2 =» (u xxx ) = I 3 =» s (u xxxx ) = I 4 =» ss +3» 3 Invariant horizontal one-form (dx) = ff Λ ( d J y) = $ =! + q = 1+u 2 x dx + u q x 1+u 2 x Invariant contact forms ( ) = # = q 1+u 2 x ( x ) = # 1 = (1 + u2 x ) x u x u xx (1 + u 2 x )2 ch 73

74 Prolonged infinitesimal generators v 1 x v 2 u v 3 = u@ x + x@ u +(1+u 2 x ux +3u x u uxx + d H I = D s I $ Horizontal recurrence formula d H (F ) = (d H F )+ (v 1 (F )) fl 1 + (v 2 (F )) fl 2 + (v 3 (F )) fl 3 Use phantom invariants 0 = d H H = (d H x)+ X (v» (x)) fl» = $ + fl 1 ; 0 = d H I 0 = (d H u)+ X (v» (u)) fl» = fl 2 ; 0 = d H I 1 = (d H u x )+ X (v» (u x )) fl» =»$+ fl 3 ; to solve for fl 1 = $ fl 2 = 0 fl 3 =»$ ch 74

75 fl 1 = $ fl 2 = 0 fl 3 =»$ Recurrence formulae» s $ = d H» = d H (I 2 ) = (d H u xx )+ (v 3 (u xx )) fl 3 = (u xxx dx) (3u x u xx ))»$ = I 3 $» ss $ = d H (I 3 )= (d H u xxx )+ (v 3 (u xxx ) fl 3 = (u xxxx dx) (4u x u xxx +3u 2 xx )»$ = I 4 3I 3 2 $» = I 2 I 2 =»» s = I 3 I 3 =» s» ss = I 4 3I 3 2 I 4 =» ss +3» 3» sss = I 5 19I 2 2 I 3 I 4 =» sss +19» 2» s ch 75

76 Vertical recurrence formula d V (F ) = ( d V F )+ (v 1 (F )) " 1 + (v 2 (F )) " 2 + (v 3 (F )) " 3 Use phantom invariants 0= d V H = " 1 0= d V I 0 = # + " 2 0= d V I 1 = # 1 + " 3 to solve for " 1 = 0 " 2 = # = ( ) " 3 = # 1 = ( 1 ) Recurrence formulae d V I 2 = d V» = ( 2 )+ (v 3 (u xx )) " 3 = # 2 = (D 2 +» 2 ) #; d H #: D# = # 1 D# 1 = # 2» 2 # d V $ =»#^ $ ch 76

77 Example (x 1 ;x 2 ;u) 2 M = R 3 G = GL(2) (x 1 ;x 2 ;u) 7! (ffx 1 + fix 2 ;flx 1 + ffix 2 ; u) = ffffi fifl =) Classical invariant theory Prolongation (lifted differential invariants): y 1 = 1 (ffix 1 fix 2 ) y 2 = 1 ( flx 1 + ffx 2 ) v = 1 u v 1 = ffu 1 + flu 2 v 2 = fiu 1 + ffiu 2 Normalization v 11 = ff2 u 11 +2ffflu 12 + fl 2 u 22 v 12 = fffiu 11 +(ffffi + fifl)u 12 + flffiu 22 v 22 = fi2 u 11 +2fiffiu 12 + ffi 2 u 22 y 1 = 1 y 2 = 0 v 1 = 1 v 2 = 0 Nondegeneracy = 0 ch 77

78 First order moving frame ψ ff fl fi ffi! ψ! x 1 u = 2 x 2 u 1 Normalized differential invariants J 1 = 1 J 2 =0 u I = x 1 u 1 + x 2 u 2 I 1 = 1 I 2 = 0 I 11 = (x1 ) 2 u 11 +2x 1 x 2 u 12 +(x 2 ) 2 u 22 x 1 u 1 + x 2 u 2 I 12 = x1 u 2 u 11 +(x 1 u 1 x 2 u 2 )u 12 + x 2 u 1 u 22 x 1 u 1 + x 2 u 2 I 22 = (u 2 )2 u 11 2u 1 u 2 u 12 +(u 1 ) 2 u 22 x 1 u 1 + x 2 u 2 Phantom differential invariants Generating differential invariants I 1 I 2 I I 11 I 12 I 22 Invariant differential operators D 1 = x 1 D 1 + x 2 D 2 D 2 = u 2 D 1 + u 1 D 2 scaling process Jacobian process ch 78

79 Recurrence formulae D 1 J 1 = ffi 1 1 1=0 D 2 J 1 = ffi 1 2 0=0 D 1 J 2 = ffi 2 1 0=0 D 2 J 2 = ffi 2 2 1=0 D 1 I = I 1 I(1 + I 11 )= I(1 + I 11 ) D 2 I = I 2 II 12 = I I 12 D 1 I 1 = I 11 I 11 = 0 D 2 I 1 = I 12 I 12 = 0 D 1 I 2 = I 12 I 12 = 0 D 2 I 2 = I 22 I 22 = 0 D 1 I 11 = I 111 +(1 I 11 )I 11 D 2 I 11 = I 112 +(2 I 11 )I 12 D 1 I 12 = I 112 I 11 I 12 D 2 I 12 = I 122 +(1 I 11 )I 22 D 1 I 22 = I 122 +(I 11 1)I 22 2I 2 12 D 2 I 22 = I 222 I 12 I 22 Syzygies =) Use I to generate I 11 and I 12 D 1 I 12 D 2 I 11 = 2I 12 D 1 I 22 D 2 I 12 = 2(I 11 1)I 22 2I 2 12 (D 1 ) 2 I 22 (D 2 ) 2 I 11 = = 2I 22 D 1 I 11 +(5I 12 2)D 1 I 12 +(3I 11 5)D 1 I 22 (2I 11 5)(I 11 1)I 12 +4(I 11 1)I 2 12 Commutation formulae [D 1 ; D 2 ]= I 12 D 1 +(I 11 1)D 2 ch 79

80 Invariant Variational Problems I[u] = Z L(x; u (n) ) dx = Z P ( ::: D K I ff ::: )! I 1 ;:::;I` D K I ff fundamental differential invariants differentiated invariants! =! 1 ^ ^! p contact-invariant volume form Invariant Euler-Lagrange equations E(L) = F ( ::: D K I ff ::: ) = 0 Problem. Construct F directly from P. =) P. Griffiths, I. Anderson ch 80

81 Example. Planar Euclidean group G = SE(2) Invariant variational problem Z P (»;» s ;» ss ; ::: ) ds Euler-Lagrange equations E(L) = F (»;» s ;» ss ; ::: ) = 0 The Elastica (Euler): I[u] = Z 1 2»2 ds = Z u 2 xx dx (1 + u 2 x )5=2 Euler-Lagrange equation E(L) =» ss + 1 2»3 = 0 =) elliptic functions ch 81

82 Z P (»;» s ;» ss ; ::: ) ds Invariantized Euler operator E = 1X n=0 ( n D = d ds Invariantized Hamiltonian operator H(P ) = X i>j» i j ( i P Invariant Euler-Lagrange formula E(L) = (D 2 +» 2 ) E(P )+» H(P ): Elastica P = 1 2»2 E(P ) =» H(P ) = P = 1 2»2 E(L) =» ss + 1 2»3 = 0 ch 82

83 Euler-Lagrange Equations Integration by Parts: ß : Ω p;1! F 1 = Ω p;1 =d H Ω p 1;1 =) Source forms Variational derivative or Euler operator: ffi = ß ffi d V : Ω p;0! F 1 Variational Problems! Source Forms Hamiltonian ffi : = Ldx! H(L) = mx ff=1 X i>j 0 qx ff=1 u ff i j ( x ff i E ff (L) ff ^ dx L ch 83

84 The Simplest Example. M = R 2 x; u 2 R Lagrangian form = L(x; u (n) ) dx Vertical derivative d = d V @u x xx + xx Integration by parts d H (A ) = (D x A) dx ^ A x ^ dx! = [(D x A) + A x ] ^ dx ^ dx 2 Ω 1;1 Variational derivative ffi x + D 2 x = E(L) ^ dx 2 F ^ xx ch 84

85 Plane Curves Invariant Lagrangian Z P (»;»s;:::) $» fundamental differential invariant (curvature) $ =! + fully invariant horizontal form! = ds contact-invariant arc length Invariant integration by parts d V (P $) = E(P ) d V» ^ $ H(P ) d V $ Vertical differentiation formulae d V» = A(#) d V $ = B(#) ^ $ A Eulerian operator B Hamiltonian operator =) The explicit formulae follow from our fundamental recurrence formula, based on the infinitesimal generators of the action. Invariant Euler-Lagrange equation A Λ E(P ) B Λ H(P ) = 0 ch 85

86 General Framework Fundamental differential invariants Invariant horizontal coframe I 1 ;:::;I` $ 1 ;:::;$ p Dual invariant differential operators D 1 ;:::;D p Invariant volume form $ = $ 1 ^ ^$ p Differentiated invariants I ff ;K = DK J ff = D k1 D kn J ff =) order is important! ch 86

87 Eulerian operator d V I ff = qx fi=1 A ff fi (#fi ) A = ( A ff fi ) =) m q matrix of invariant differential operators Hamiltonian operator complex d V $ j = qx fi=1 B j i;fi (#fi ) ^ $ i B j i = ( B j i;fi ) =) p 2 row vectors of invariant differential operators $ (i) = ( 1) i 1 $ 1 ^ ^$ i 1 ^ $ i+1 ^ ^$ p Twist invariants Twisted adjoint d H $ (i) = Z i $ D y i = (D i + Z i ) ch 87

88 Invariant variational problem Z P (I (n) )$ Invariant Eulerian E ff (P ) = X K D ff ;K Invariant Hamiltonian tensor Hj i(p ) = P ffii j + qx ff=1 X J;K I ff ;J;j D ff ;J;i;K ; Invariant Euler-Lagrange equations A y E(P ) px i;j=1 (Bi j ) y Hj i (P ) = 0: ch 88

89 Euclidean Surfaces S ρ M = R 3 coordinates z = (x; y; u) Group: G = E(3) z 7! Rz+ a; R 2 O(3) Normalization coordinate cross-section x = y = u = u x = u y = u xy = 0: Left moving frame a = z R = ( t 1 t 2 n ) ffl t 1 ; t 2 2 TS ffl n Frenet frame unit normal ch 89

90 Fundamental differential invariants» 1 = (u xx )» 2 = (u yy ) =) principal curvatures Frenet coframe $ 1 = (dx 1 ) =! $ 2 = (dx 2 ) =! Invariant differential operators D 1 D 2 =) Frenet differentiation Fundamental Syzygy: Use the recurrence formula to compare (u xxyy ) with» 1 ;22 = D2 2 (u xx )» 2 ;11 = D2 1 (u yy )» 1 ;22»2 ;11 +»1 ;1»2 ;1 +»1 ;2»2 ;2 2(»2 ;1 )2 2(» 1 ;2 )2» 1» 2» 1» 2 (» 1» 2 ) =0 =) Codazzi equations ch 90

91 Twisted adjoints D y 1 = (D 1 + Z 1 ) Z 1 =»2 ;1» 1» 2 D y 2 = (D 2 + Z 2 ) Z 2 =»1 ;2» 2» 1 Gauss curvature Codazzi equations: K =» 1» 2 = D y 1 (Z 1 )+D y 2 (Z 2 ) = (D 1 + Z 1 )Z 1 (D 2 + Z 2 )Z 2 K is an invariant divergence =) Gauss Bonnet Theorem! ch 91

92 Invariant contact form # = ( ) = (du u x dx u y dy) Invariant vertical derivatives d V» 1 = ( xx )=( D Z D 2 2 +(»1 ) 2 ) # d V» 2 = ( yy ) =(D Z D 1 1 +(»2 ) 2 ) # Eulerian operator A = ψ D Z D! 2 2 +(»1 ) 2 D Z D 1 1 +(»2 ) 2 d V $ 1 =» 1 # ^ $ 1 1» 1» 2( D D 1 2 Z D 2 1 )# ^ $2 d V $ 2 = Hamiltonian operator complex 1» 1» 2( D 2 D 1 Z 1 D 2 )# ^ $1 +» 2 # ^ $ 2 B 1 1 =»1 ; B 2 2 =» 2 ; B 1 2 = 1» 1» 2( D 1 D 2 Z 2 D 1 )= B2 1 ch 92

93 Euclidean-invariant variational problem Z Z P (» (n) )! 1 ^! 2 = P (» (n) ) da Euler-Lagrange equations Special case: P (» 1 ;» 2 ) E(L) = A y E(P ) B y H(P ) =0; E(L) =[(D y 1 ) 2 D y 2 Z 2 +(» 1 ) 1 + Minimal surfaces: P = 1 +[(D y 2 ) 2 D y 1 Z 1 +(» 2 ) 2 +(»1 +» 2 ) P» 1 +» 2 =2H =0 Minimizing mean curvature: P = H = 1 2 (»1 +» 2 ) h i (» 1 ) 2 +(» 2 ) 2 +» 1 +» 2 = 2H 2 + H K =0: 1 2 Willmore surfaces: P = 1 2 (»1 ) (»2 ) 2 (» 1 +» 2 )+ 1 2 (»1 +» 2 )(» 1» 2 ) 2 =2 H +4(H 2 K)H = 0 Laplace Beltrami operator =(D 1 + Z 1 )D 1 +(D 2 + Z 2 )D 2 = D y 1 D 1 D y 2 D 2 ch 93

94 Multi Space Although in use since the time of Lie and Darboux, jet space was first formally defined by Ehresmann in Jet space is the proper setting for the geometry of partial differential equations. In this talk, I will propose a setting, named multispace, for the geometry of numerical approximations to derivatives and differential equations. =) Multi-space is the context for geometric integration. ch 94

95 Invariant Numerical Approximations Key remark: Every (finite difference) numerical approximation to the derivatives of a function require evaluating the function at several points z i = (x i ;u i ) = (x i ;f(x i )). ch 95

96 In other words, we seek to approximate the n th order jet of a submanifold N ρ M by a function F (z 0 ;:::;z n ) defined on the (n + 1)-fold Cartesian product space M (n+1) = M M, or, more correctly, on the off-diagonal" part M Π(n+1) = f z i 6= z j for all i 6= j g =) distinct (n + 1)-tuples of points. ch 96

97 Thus, multi-space should contain both the jet space and the off-diagonal Cartesian product space as submanifolds: M Π(n+1) # 9 >= ρ M (n) J n (M;p) >; Functions F : M (n)! R are given by F (z 0 ;:::;z n ) on M Π(n+1) and extend smoothly to J n as the points coalesce. In this manner, F j M Π(n+1) provides a finite difference approximation to the differential function F j J n. ch 97

98 Construction of M (n) Definition. An (n + 1)-pointed manifold M = (z 0 ;:::;z n ; M) M smooth manifold z 0 ;:::;z n 2 M not necessarily distinct Given M, let #i = # n j fi fi fi zj = z i o denote the number of points which coincide with the i th one. ch 98

99 Multi-contact for Curves Definition. Two (n + 1)-pointed curves C = (z 0 ;:::;z n ; C); f C = (~z 0 ;:::;~z n ; e C); have n th order multi-contact if and only if z i = ~z i ; and j #i 1 Cj zi = j e #i 1 Cj zi ; for each i = 0;:::;n. #i = # n j fi o fi fi zj = z i Definition. The n th order multi-space M (n) is the set of equivalence classes of (n + 1)-pointed curves in M under the equivalence relation of n th order multi-contact. ch 99

100 The Fundamental Theorem Theorem. If M is a smooth m-dimensional manifold, then its n th order multi-space M (n) is a smooth manifold of dimension (n + 1)m, which contains the off-diagonal part M Π(n+1) of the Cartesian product space as an open, dense submanifold, and the n th order jet space J n as a smooth submanifold. M Π(n+1) 9 J k 1 Π Π J k ν >= ρ M (n) J n (M;p) >; ch 100

101 Example. Let M = R m (i) M (1) is the space of two-pointed lines M (1) ' f (z 0 ;z 1 ; L) j z 0 ;z 1 2 L line g : =) Blow-up construction in algebraic geometry (ii) M (2) is the space of three-pointed circles, i.e., M (2) ' f (z 0 ;z 1 ;z 2 ;C) j z 0 ;z 1 ;z 2 2 C circle g : Straight lines are included as circles of infinite radius, but points are not included (even though they could be viewed as circles of zero radius). (iii) M (3)???? =) Grassmann bundles. Topology local and global. ch 101

102 Finite Differences Local coordinates on J n are provided by the coefficients of Taylor polynomials =) derivatives Local coordinates on M (n) are provided by the coefficients of interpolating polynomials. =) finite differences Given (z 0 ;:::;z n ) 2 M Π(n+1), define the classical divided differences by the standard recursive rule [ z 0 z 1 :::z k 1 z k ]= [ z 0 z 1 z 2 :::z k 2 z k ] [ z 0 z 1 z 2 :::z k 2 z k 1 ] x k x k 1 [ z j ]=u j =) Well-defined provided no two points lie on the same vertical line. =) Symmetric functions of z i. ch 102

103 Definition. Given an (n + 1)-pointed graph C = (z 0 ;:::;z n ; C), its divided differences are defined by [ z j ] C = f(x j ) [ z [ z 0 z 1 :::z k 1 z k ] C = lim 0 z 1 z 2 :::z k 2 z ] C [ z 0 z 1 z 2 :::z k 2 z k 1 ] C z!zk x x k 1 =) When taking the limit, the point z = (x; f(x)) must lie on the graph C, and take limiting values x! x k and f(x)! f(x k ). Theorem. Two (n + 1)-pointed graphs C; f C have n th order multi-contact if and only if they have the same divided differences: [ z 0 z 1 :::z k ] C = [ z 0 z 1 :::z k ]e C ; k = 0;:::;n: ch 103

104 Local coordinates on M (n) They consist of the independent variables along with all the divided differences u (0) = u 0 = [ z 0 ] C u (1) =[z 0 z 1 ] C x 0 ;:::;x n u (2) = 2[z 0 z 1 z 2 ] C ::: u (n) = n![z 0 z 1 :::z n ] C prescribed by (n + 1)-pointed graphs C = (z 0 ;:::;z n ; C) The n! factor is included so that u (n) agrees with the usual derivative coordinate when restricted to J n. ch 104

105 Numerical Approximations (x; u (n) ) differential function : J n! R System of differential equations: 1(x; u (n) ) = = k(x; u (n) ) = 0: Definition. An (n + 1)-point numerical approximation of order k to a differential function : J n! R is a k th order extension F : M (n)! R of to multi-space, based on the inclusion J n ρ M (n). F (x 0 ;:::;x n ;u (0) ;:::;u (n) )! F (x;:::;x;u (0) ;:::;u (n) ) = (x; u (n) ) ch 105

106 Invariant Numerical Approximations G Lie group acting on M Basic Idea: Every invariant finite difference approximation to a differential invariant must expressible in terms of the joint invariants of the transformation group. Differential Invariant Joint Invariant I(g (n) z (n) ) = I(z (n) ) J(g z 0 ;:::;g z k ) = J(z 0 ;:::;z k ) Semi-differential invariant = Joint differential invariant =) Approximate differential invariants by joint invariants ch 106

107 Euclidean Invariants Joint Euclidean invariant: Euclidean curvature: Euclidean arc length: d(z;w) = k z w k» = u xx (1 + u 2 x )3=2 ds = q 1+u 2 x dx Higher order differential invariants:» s = d» ds» ss = d2» ds 2 ::: Euclidean invariant differential equation: F (»;» s ;» ss ;:::) = 0 ch 107

108 Three point approximation Heron's formula e»(a; B; C) = 4 abc = 4 q s(s a)(s b)(s c) abc s = a + b + c 2 semi-perimeter Expansion: e» =» + 1 d» (b a) 3 ds (b2 ab + a 2 ) d2» ds (b3 ab 2 + a 2 b a 3 ) d3» ds (b a)(3b2 +5ab +3a 2 )» d» 2 ds + : ch 108

109 Higher order invariants» s = d» ds Invariant finite difference approximation: e» s (P i 2 ;P i 1 ;P i ;P i+1 )= e»(p i 1 ;P i ;P i+1 ) e»(p i 2 ;P i 1 ;P i ) d(p i ;P i 1 ) Unbiased centered difference: e» s (P i 2 ;P i 1 ;P i ;P i+1 ;P i+2 )= e»(p i ;P i+1 ;P i+2 ) e»(p i 2 ;P i 1 ;P i ) d(p i+1 ;P i 1 ) Better approximation (M. Boutin): e» s (P i 2 ;P i 1 ;P i ;P i+1 ) =3 e»(p i 1 ;P i ;P i+1 ) e»(p i 2 ;P i 1 ;P i ) d i 2 +2d i 1 +2d i + d i+1 d j = d(p j ;P j+1 ) ch 109

110 Affine Joint Invariants x! Ax + b det A = 1 Area is the fundamental joint affine invariant [ijk ] = (P i P j ) ^ (P i P k ) = det fi fi fi fi fi fi fi x i y i 1 x j y j 1 x k y k 1 = Area of parallelogram = 2 Area of triangle (P i ;P j ;P k ) fi fi fi fi fi fi fi Syzygies: [ijl ] + [jkl] = [ijk ] + [ikl ] [ijk ] [ilm] [ijl ] [ikm] + [ijm] [ikl ] = 0 ch 110

111 Affine curvature Affine Differential Invariants» = 3u xx u xxxx 5u2 xxx 9(u xx ) 8=3 Affine arc length ds = 3 q uxx dx Higher order affine invariants:» s = d» ds» ss = d2» ds 2 ::: ch 111

112 Conic Sections Ax 2 +2Bxy + Cy 2 +2Dx +2Ey + F = 0 Affine curvature: Ellipse:» = S T 2=3 S = AC B 2 = det fi fi A B T = det fi fi fi fi fi fi fi A B D B C E D E F fi fi fi fi fi fi fi fi fi fi» = (ß=A) 2=3 B C fi fi fi fi fi Affine arc length of ellipse: Z Q P A = ß T = Area S3=2 s 1=3 T CT ds = S arcsin 1=2 S 2 = 2ST 2=3 A(P; Q) ψ x +!fi CD BE fififififi Q S P ch 112

113 Q A(P; Q) : P Triangular approximation: (O; P; Q) : O Q P Total affine arc length: q q 3 T L = 2 3 A = 2ß p S ch 113

114 Conic through five points P 0 ;:::;P 4 : [013][024][x12][x34] = [012][034][x13][x24] Affine curvature and arc length:» = S T 2=3 x = (x; y) ds = Area (O; P 1 ;P 3 ) = 1 2 [O; P 1 ;P 3 ] = N 2S 4T = Y 0»i<j<k»4 [ijk] 4S = [013] 2 [024] 2 ([124] [123]) [012] 2 [034] 2 ([134] + [123]) 2 2[012][034][013][024]([123][234] + [124][134]) 4N = [123][134] f[023] 2 [014] 2 ([124] [123]) + + [012] 2 [034] 2 ([134] + [123]) + + [012][023][014][034]([134] [123])g Theorem. P 0 ;P 1 ;P 2 ;P 3 ;P 4 points on the convex curve C. ch 114

115 » affine curvature of C at P 2 e» = e»(p 0 ;P 1 ;P 2 ;P 3 ;P 4 ) affine curvature of conic L i = Z Pi P 2 ds affine arc length of conic Expansion: e» =» ψ 4X i=0! 0 d» L i ds + 30 X 0»i»j»4 L i L j 1 A d2» ds 2 + ch 115

116 Multi-Invariants G Lie group which acts smoothly on M =) G preserves the multi-contact equivalence relation G (n) n th multi-prolongation to M (n) =) On J n ρ M (n) it coincides with the usual jet space prolongation =) On M Π(n+1) ρ M (n) it coincides with the (n + 1)-fold Cartesian product action. K : M (n)! R multi-invariant K(g (n) z (n) ) = K(z (n) ) =) K j J n differential invariant =) K j M Π(n+1) joint invariant =) K j J k 1 Π ΠJ k ν joint diff. invariant The theory of multi-invariants is the theory of invariant numerical approximations! ch 116

117 Moving frames provide a systematic algorithm for constructing multi-invariants! A moving frame on multi-space is called a multi-frame. ρ : M (n)! G ch 117

118 Example. G = R 2 nr (x; u) 7! ( 1 x + a; u + b) Multi-prolonged action: compute the divided differences of the basic lifted invariants We find y k = 1 x k + a; v k = u k + b: v (1) = [ w 0 w 1 ] = v 1 v 0 y 1 y 0 = 2 u 1 u 0 x 1 x 0 = 2 [ z 0 z 1 ] = 2 u (1) ; v (n) = n+1 u (n) : Moving frame cross-section Solve for the group parameters y 0 = 0 v 0 = 0 v (1) = 1 a = p u (1) x 0 b = u 0 p u (1) = 1 p u (1) =) multi-frame ρ : M (n)! G. ch 118

119 Multi-invariants: y k : H k = (x k x 0 ) p u (1) = (x k x 0 ) s u1 u 0 x 1 x 0 u k : K k = u k u 0 p u (1) u (n) : K (n) = = (u k u 0 ) s x1 x 0 u 1 u 0 u (n) (u (1) ) = n![z 0 z 1 :::z n ] (n+1)=2 [ z 0 z 1 ] (n+1)=2 Coalescent limit K (0) = K 0 = 0 K (1) =1 K (n)! I (n) = u (n) (u (1) ) (n+1)=2 =) K (n) is a first order invariant numerical approximation to the differential invariant I (n). =) Higher order invariant numerical approximations are obtained by invariantization of higher order divided difference approximations. F ( ::: x k ::: u (n) ::: )! F ( ::: H k ::: K (n) ::: ) ch 119

120 Invariantization To construct an invariant numerical scheme for any similarity-invariant ordinary differential equation F (x; u; u (1) ;u (2) ;:::u (n) ) = 0; we merely invariantize the defining differential function, leading to the general similarity invariant numerical approximation F (0; 0; 1;K (2) ;:::;K (n) ) = 0: =) Nonsingular! ch 120

121 Example. Euclidean group SE(2) y = x cos u sin + a v = x sin + u cos + b Multi-prolonged action on M (1) : y 0 = x 0 cos u 0 sin + a v 0 = x 0 sin + u 0 cos + b y 1 = x 1 cos u 1 sin + a v (1) = sin + u(1) cos cos u (1) sin Cross-section y 0 = v 0 = v (1) = 0 Right moving frame a = x 0 cos + u 0 sin = x + 0 u(1) u q 0 1+(u (1) ) 2 tan = u (1) : b = x 0 sin u 0 cos = x 0 u(1) u q 0 1+(u (1) ) 2 ch 121

122 Euclidean multi-invariants (y k ;v k )! I k = (H k ;K k ) H k = (x k x 0 )+u(1) (u k u 0 ) q =(x k x 0 ) 1+[z 0 z 1 ][z 0 z k q ] 1+(u (1) ) 2 1+[z0 z 1 ] 2 K k = (u k u ) 0 u(1) (x k x 0 ) q =(x k x 0 ) [ z 0 z k ] [ z 0 z 1 q ] 1+(u (1) ) 2 1+[z0 z 1 ] 2 Difference quotients I (1) = [ I 0 I 1 ]=0 [ I 0 I k ]= K k K 0 H k H 0 = K k H k = (x k x 1 )[ z 0 z 1 z k ] 1+[z 0 z k ][z 0 z 1 ] I (2) = 2[I 0 I 1 I 2 ] =2 [ I 0 I 2 ] [ I 0 I 1 ] H 2 H 1 = = 2[z 0 z 1 z 2 ] q 1+[z 0 z 1 ] 2 (1+[z 0 z 1 ][z 1 z 2 ])(1+[z 0 z 1 ][z 0 z 2 ]) u (2)q 1+(u (1) ) 2 h ih i 1+(u (1) ) u(1) u (2) (x 2 x 0 ) 1+(u (1) ) u(1) u (2) (x 2 x 1 ) Invariant numerical approximation to the Euclidean curvature: u (2) lim I (2) =» = z 1 ;z 2!z 0 (1+(u (1) ) 2 ) 3=2 Euclidean invariant approximation for» s = (u xxx ): I (3) = 6[I 0 I 1 I 2 I 3 ] = 6 [ I 0 I 1 I 3 ] [ I 0 I 1 I 2 ] H 3 H 2 ch 122

123 Higher Dimensional Submanifolds T (n) Mj z n th order tangent space Proposition. Two p-dimensional submanifolds N; f N have n th order contact at a common point z 2 N f N if and only if T (n) Nj z = T (n)f Nj z ψ! p + n =) Requires coalescing points to approximate n th order n derivatives ch 123

124 Surfaces p = 2 n ψ p + n n! ch 124

125 Definition. A subspace V ρ T (n) Mj z is called admissible if for every vector v 2 V T (k) Mj z, 1» k» n, there exists a submanifold N ρ M such that v 2 T (k) Nj z ρ V. Definition. Two submanifolds N; N f have r th order subcontact at a common point if and only if for some n, there exists an admissible common r- dimensional subspace S ρ T (n) Nj z T (n)f Nj z ρ T (n) Mj z ch 125

126 Example. Surfaces: order S; e S ρ M Conditions 0 z 2 S e S common point 1 tangent curves: TCj z = T e Cj z >< >: 8 >< >: tangent surfaces: osculating curves: TSj z = T e Sj z T (2) Cj z = T (2) e Cj z TSj z = T e Sj z and T (2) Cj z = T (2) e Cj z T (3) Cj z = T (3) e Cj z. 8. T (2) Sj z = T (2) e Sj z 5 >< TSj z = T e Sj z ; T (3) Cj z = T (3) e Cj z ; T (2) C 0 j z = T (2) e C 0 j z >: TSj z = T e Sj z ; T (4) Cj z = T (4) e Cj z T (5) Cj z = T (5) e Cj z ch 126