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Mo ving F rames
Moving Frames
http
Peter J. Olver
University of Minnesota
: ==www:math:umn:edu= olver
Beijing, March, 2001
ch 1
Moving Frames
Classical contributions:
Cotton, E.
Cartan
G. Darboux, E.
Modern contributions:
P. GriÆths, 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."
| Hermann Weyl
\Cartan on groups and dierential geometry"
44 (1938) 598{601
Bull. Amer. Math. Soc.
ch 2
References
Fels, M., Olver, P.J.,
Part I, Acta Appl. Math. 51 (1998) 161{213
Part II, Acta Appl. Math. 55 (1999) 127{208
Olver, P.J., Selecta Math. 6 (2000) 41{77
Calabi, Olver, Shakiban, Tannenbaum, Haker, Int. J.
Computer Vision 26 (1998) 107{135
Mar{Bea, G., Olver, P.J.,
Commun. Anal. Geom. 7 (1999) 807{839
Olver, P.J., Classical Invariant Theory ,
Cambridge Univ. Press, 1999
Berchenko, I.A., Olver, P.J.
J. Symb. Comp. 29 (2000) 485{514
Olver, P.J., Found. Comput. Math. 1 (2001) 3{67
Olver, P.J., \Geometric foundations of numerical algorithms
and symmetry", Appl. Alg. Engin. Commun. Comput., to
appear
Kogan, I.A., Olver, P.J., \Invariant Euler-Lagrange equations and the invariant variational bicomplex", preprint,
2001
http://www.math.umn.edu/olver
ch 3
Applications of Moving Frames
Dierential geometry
Equivalence
Symmetry
Dierential invariants
Rigidity
Joint Invariants and Semi-Dierential Invariants
Invariant dierential forms and tensors
Identities and syzygies
Classical invariant theory
Computer vision
Æ object recognition
Æ symmetry detection
Invariant numerical methods
Poisson geometry & solitons
Lie pseudogroups
ch 4
The Basic Equivalence Problem
M
| smooth m-dimensional manifold.
G | transformation group acting on M
nite-dimensional Lie group
innite-dimensional Lie pseudo-group
Equivalence:
Determine when two n-dimensional submanifolds
N and N M
are congruent :
N =gN
for g 2 G
Symmetry:
Self-equivalence or self-congruence :
N =gN
ch 5
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 , nd all its
symmetries (belonging to the group).
| G = SE(n) or E(n)
) isometries of Euclidean space
) translations, rotations (& reections)
R 2 SO(n) or O(n)
z 7 !Rz+a
a 2 Rn
Euclidean group
8
>
>
>
>
<
>
>
>
>
:
z 2 Rn
G = SA(n)
Equi-aÆne group :
R 2 SL(n) | area-preserving
AÆne group :
G = A(n)
R 2 GL(n)
Projective group :
G = PSL(n)
acting on RP n
1
=) Applications in computer vision
ch 6
Classical Invariant Theory
Binary form:
Q(x) =
n
!
n
ak xk
k
X
k=0
Equivalence of polynomials (binary forms):
x + Q(x) = (x + Æ)n Q
x + Æ
!
g=
Æ
!
2 GL(2)
) multiplier representation of GL(2)
) modular forms
Transformation group:
g : (x; u) 7 !
u
x + ;
x + Æ (x + Æ)n
!
Equivalence of functions () equivalence of graphs
NQ = f (x; u) = (x; Q(x)) g C
2
ch 7
Moving Frames
Denition.
A moving frame is a G-equivariant map
: M !G
Equivariance:
8
<
(g z ) = :
g (z )
(z ) g
1
left moving frame
right moving frame
left(z ) = right(z )
1
ch 8
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 : M ! G be a left moving frame.
Theorem.
Freeness :
If g 2 Gz , so g z = z, then by left equivariance:
(z ) = (g z ) = g (z ):
Therefore g = e, and hence Gz = feg for all z 2 M .
Regularity :
Suppose
zn = gn z ! z
as n ! 1
By continuity,
(zn) = (gn z ) = gn (z ) ! (z )
Hence gn ! e in G.
SuÆciency :
By construction | \normalization".
Q.E.D.
ch 9
Isotropy
Isotropy subgroup for z 2 M :
Gz = f g j g z = z g
free | the only group element g 2 G which xes one point
z
2 M is the identity:
Gz = feg for all z 2 M .
locally free | the orbits all have the same dimension as G:
is a discrete subgroup of G.
regular | all orbits have the same dimension and intersect
suÆciently small coordinate charts only once
( 6 irrational ow on the torus)
eective | the only group element g 2 G which xes every
point z 2 M is the identity: g z = z for all z 2 M i
g = e:
\ Gz = feg
GM =
Gz
z2M
ch 10
Geometrical Construction
Normalization = choice of cross-section to the group orbits
K
Oz
z
g = ρ(z )
k
| cross-section to the group orbits
Oz | orbit through z 2 M
k 2 K \ Oz | unique point in the intersection
k is the canonical form of z
the (nonconstant) coordinates of k are the fundamental
invariants
g 2 G | unique group element mapping k to z
=) freeness
(z ) = g left moving frame (h z ) = h (z )
k = (z ) z = right(z ) z
K
1
ch 11
Construction of Moving Frames
r = dim G
m = dim M
Coordinate cross-section
K = fz = c ;
1
1
left
w(g; z ) = g z
: : : ; zr = cr g
right
w(g; z ) = g z
1
Choose r = dim G components to normalize :
w (g; z ) = c
:::
wr (g; z ) = cr
1
The solution
is
1
g = (z )
a (local) moving frame.
=) Implicit Function Theorem
ch 12
The Fundamental Invariants
Substituting the moving frame formulae
g = (z )
into the unnormalized components of w(g; z) produces the
fundamental invariants:
I1(z ) = wr+1((z ); z ) : : : Im r (z ) = wm((z ); z )
=) These are the coordinates of the canonical form k 2 K .
Every invariant I (z) can be (locally) uniquely
written as a function of the
fundamental invariants:
I (z ) = H (I (z ); : : : ; Im r (z ))
Theorem.
1
ch 13
Invariantization
The invariantization of a function F : M !
R with respect to a right moving frame is the the
invariant function I = (F ) dened by I (z) = F ((z) z).
Denition.
[ F (z1; : : : ; zm)]
= F (c ; ; : : : cr; I (z); : : : ; Im r (z))
1
1
Invariantization amounts to restricting F to the cross-section
I jK = F jK
and then requiring that I = (F ) be constant along the
orbits.
In particular, if I (z) is an invariant, then (I ) = I .
Invariantization denes a canonical projection
:
functions 7 ! invariants
ch 14
The Rotation Group
G = SO(2)
acting on R
z = (x; u) 7 ! g z = ( x cos u sin ; x sin + u cos )
=) Free on M = R n f0g
2
2
Left moving frame:
w(g; z ) = g
y = x cos + u sin Cross-section
1
z = (y; v)
v = x sin + u cos K = f u = 0; x > 0 g
Normalization equation
v = x sin + u cos = 0
Left moving frame:
u
=) = (x; u) 2 SO(2)
= tan
x
Fundamental invariant
p
r = (x) = x + u
Invariantization
[ F (x; u)] = F (r; 0)
1
2
2
ch 15
Prolongation
Most interesting group actions (Euclidean, aÆne, projective,
etc.) are not free!
An eective action can usually be made free by:
Prolonging to derivatives (jet space)
G n : Jn(M; p) ! Jn(M; p)
=) dierential invariants
( )
Prolonging to Cartesian product actions
Gn : M M ! M M
=) joint invariants
Prolonging to \multi-space"
G(n) : M (n)
!M n
( )
=) joint or semi-dierential invariants
=) invariant numerical approximations
ch 16
Jet Space
Although in use since the time of Lie and
Darboux, jet space was rst formally dened by Ehresmann in 1950.
Jet space is the proper setting for the geometry of partial
dierential equations.
| smooth m-dimensional manifold
1pm 1
Jn = Jn(M; p) | (extended) jet bundle
=) Dened as the space of equivalence classes of pdimensional submanifolds under the equivalence relation of n order contact at a single point.
=) Can be identied as the space of n order Taylor polynomials for submanifolds given as graphs u = f (x)
M
th
th
ch 17
Local Coordinates on Jet Space
Jn = Jn(M; p) | n extended jet bundle for
p-dimensional submanifolds N M
th
Local coordinates:
Assume N = f u = f (x) g is a graph (section).
x = (x ; : : : ; xp)
| independent variables
u = (u ; : : : ; uq )
| dependent variables
p + q = m = dim M
1
1
z (n) = (x; u(n)) = ( : : : xi : : : uJ : : : )
uJ = @J u
0 #J n
| induced jet coordinates
No bundle structure assumed on M .
Projective completion of JnE when E ! X is a bundle.
ch 18
Prolongation of Group Actions
| 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 Jn
G
( )
The prolonged group formulae
w n = (y; v n ) = g n z n
are obtained by implicit dierentiation:
p
i
dy =
Pji(g; z ) dxj
j
=) Q = P
p
Dy j =
Qij (g; z ) Dxi
( )
X
( )
( )
( )
(1)
=1
X
T
(1)
i=1
vJ = Dyj1
Dierential invariant
Dyjk (v)
I : Jn ! R
I (g(n) z (n)) = I (z (n))
=) curvatures
ch 19
Freeness
If G acts (locally) eectively on M , then G acts
(locally) freely on a dense open subset V n Jn for n 0.
Denition. N M is regular at order n if jnN V n.
Corollary. Any regular submanifold admits a (local) moving
frame.
Theorem.
A submanifold is totally singular, jnN Jn n V n
for all n, if and only if its symmetry group
GN = f g j g N N g
does not act freely on N .
Theorem.
ch 20
Moving Frames on Jet Space
8
>
<
w(n) = (y; v(n)) = >
:
g(n) z (n)
(g n ) z n
( )
1
( )
Choose r = dim G jet coordinates
xi or uJ
z1; : : : ; zr
Coordinate cross-section K Jn
z = c : : : z r = cr
Corresponding lifted dierential invariants:
1
right
left
1
w1; : : : ; wr
yi or vJ
Normalization Equations
w1(g; x; u(n)) = c1 : : : wr (g; x; u(n)) = cr
Solution:
g = (n)(z (n)) = (n)(x; u(n))
=) moving frame
ch 21
The Fundamental Dierential Invariants
I (n)(z (n)) = w(n)((n)(z (n)); z (n))
H i(x; u(n)) = yi((n)(x; u(n)); x; u)
((n)(x; u(n)); x; u(k))
IK (x; u(k)) = vK
Phantom dierential invariants
w = c : : : wr = cr
1
1
=) normalizations
Every n order dierential invariant can be
locally uniquely written as a function of the non-phantom
fundamental dierential invariants in I n .
Theorem.
th
( )
ch 22
Invariant Dierentiation
Contact-invariant coframe
dyi
7!
!i
=
p
X
j =1
Pji((n)(z (n)); z (n)) dxi
Invariant dierential operators:
p
Dyj 7 ! Dj =
X
i=1
Duality:
dF
=
p
X
i=1
=) arc length element
Qij ((n)(z (n)); z (n)) Dxi
=) arc length derivative
Di F ! i
The higher order dierential invariants are
obtained by invariant dierentiation with respect to
D ; : : : ; Dp.
Theorem.
1
ch 23
G = SE(2)
Euclidean Curves
Assume the curve is (locally) a graph:
C = fu = f (x)g
Prolong to J via implicit dierentiation
3
y = cos (x a) + sin (u b)
9
=
v=
sin (x a) + cos (u b)
sin + ux cos vy =
cos + ux sin uxx
vyy =
(cos + u sin )
x
;
w=R
1
(z
b)
3
(cos + ux sin )uxxx 3uxx sin vyyy =
(cos + ux sin )
..
2
5
Normalization
r = dim G = 3
y = 0;
v = 0;
vy = 0
Left moving frame : J ! SE(2)
a = x;
b = u;
= tan
1
1
ux
ch 24
Dierential invariants
vyy
= (1 +uuxx) =
x
= (1 + ux(1)u+xxxu ) 3uxuxx
7!
vyyy
7!
vyyyy
7!
d
ds
d2
ds2
2 3 2
2
2 3
x
3 = 3
Invariant one-form | arc length
dy = (cos + ux sin ) dx
Invariant dierential operator
d
dy
2
= cos +1u sin x
d
dx
7 ! ds = 1 + ux dx
q
2
d
7 ! dsd = 1 dx
1 + ux
q
2
All dierential invariants are functions of the
derivatives of curvature with respect to arc length:
Theorem.
;
d
;
ds
d2
;
ds2
ch 25
Euclidean Curves
e2
e1
x
Moving frame : (x; u; ux) 7 ! (R; a) 2 SE(2)
1
1 ux = (e ; e ) a = x
R=
u
1 + ux ux 1
!
q
1
2
Frenet frame
dx
e1 = = xy s
ds
s
!
2
e2 = e?1 =
Frenet equations = Maurer{Cartan equations:
de
de
dx
=
e
=
e
=
ds
ds
ds
1
1
2
!
2
ys
xs
!
e1
ch 26
The Replacement Theorem
Any dierential invariant has the form
I = F (x; u(n)) = F (y; w(n)) = F (I (n))
=) T.Y. Thomas
=
vyy
uxx
=
(1 + vy2)2 (1 + u2x)2
(x) = (u) = (ux) = 0
(uxx) = ch 27
Equi-aÆne Curves
z7
! Az + b
G = SA(2)
A 2 SL(2);
Prolong to J via implicit dierentiation
dy = (Æ ux ) dx
b 2 R2
3
y = Æ(x a) (u b)
v=
vy =
(x a) + (u b)
ux
Æ ux
(Æ
vyyy =
vyyyy =
ux)uxxx + 3u2xx
(Æ ux)5
Dy =
1
Æ ux
9
=
;
vyy =
Dx
w=A
1
(z
b)
uxx
(Æ ux)3
uxxxx(Æ ux)2 + 10uxxuxxx (Æ ux) + 15u3xx 2
( + ux)7
..
Nondegeneracy
uxx = 0
=) Straight lines are totally singular
(three-dimensional equi-aÆne symmetry group)
Normalization r = dim G = 5
y = 0; v = 0; vy = 0; vyy = 1; vyyy = 0:
ch 28
Left Moving frame
0
[email protected]
=
q
: J3
uxx
ux 3 uxx uxx
3
q
! SA(2)
1
3
1=3
5=3
uxx uxxx
1
u 5=3uxxx
3 xx
dz
ds
dz
ds2
2
!
1
A
b=z=
x
u
!
e2
e1
x
ch 29
Frenet frame
d2 z
e2 = 2
ds
dz
e1 =
ds
Frenet equations = Maurer{Cartan equations:
de
dz
=
e
=e
ds
ds
1
Equi-aÆne arc length
dy
1
de2
ds
2
= e
1
7 ! ds = uxx dx = 3 z_ ^ z dt
q
q
3
Invariant dierential operator
Dy
7 ! dsd = 3 1 Dx = 3 1 Dt
z_ ^ z
uxx
q
q
Equi-aÆne curvature
v4y
7 ! = 5uxxuxxxx= 3uxxx = zs ^ zss
9uxx
v5y
7 ! d
ds
2
8 3
v6y
7 ! dds 5
2
2
2
ch 30
Equivalence & Signature
Cartan's main idea :
The equivalence and symmetry
properties of submanifolds will be found by restricting the
dierential invariants to the submanifold J (x) = I ( jnN jx ).
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:
Jk (x) = (J (x); : : : ; Jk (x))
1
1
ch 31
The Signature Map
Equivalence and symmetry properties of submanifolds are
governed by the functional
dependencies | \syzygies" | among the
dierential invariants.
Jk (x) = (J1(x); : : : ; Jk
1
(x))
The syzygies are encoded by the signature map
: N ! S
of the submanifold N , which is parametrized by the fundamental dierential invariants:
(x) = (J (x); : : : ; Jm(x))
= ( I j N; : : : ; Im j N )
The image S = Im is the signature subset (or submanifold)
of N .
1
1
ch 32
Geometrically, the signature
SK
is the image of jnN in the cross-section K Jn, where n 0
is suÆciently large.
: N
Theorem.
! jnN
! SK
Two submanifolds are equivalent
N
=gN
if and only if their signatures are identical
S=S
ch 33
Signature Curves
The signature curve S R of a curve C R is
parametrized by the rst two dierential invariants and
Denition.
2
2
s
S=
(
;
d
ds
! )
R
2
Two curves C and C are equivalent
C =gC
if and only if their signature curves are identical
S=S
Theorem.
=) object recognition
ch 34
Symmetry
Signature map
:N ! S
Let S denote the signature of the submanifold
N . Then the dimension of its symmetry group GN =
f g j g N N g equals
dim GN = dim N dim S
Theorem.
Corollary.
For a regular submanifold N M ,
0 dim GN dim N
=) Only totally singular submanifolds can have larger symmetry groups!
ch 35
Maximally Symmetric Submanifolds
The following are equivalent:
The submanifold N has a p-dimensional
symmetry group
The signature S degenerates to a point
dim S = 0
Theorem.
The submanifold has all constant dierential invariants
N = H f z g is the orbit of a p-dimensional subgroup
H G
0
=) In Euclidean geometry, these are the circles, straight
lines, spheres & planes.
=) In equi-aÆne plane geometry, these are the conic sections.
ch 36
Discrete Symmetries
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 # fwg w 2 S
Denition.
n
1
o
=) Self{intersections
The cardinality of the symmetry group of N
equals its index N .
Theorem.
=) Approximate symmetries
ch 37
Classical Invariant Theory
M
= R n fu = 0g
2
(x; u) 7 !
G = GL(2) =
(
Æ
u
x + ;
x + Æ (x + Æ)n
=x+Æ
! Æ 6= 0
!
= Æ
)
n 6= 0; 1
Prolongation:
x + x + Æ
v = nu
u nu
vy = x n 1
2uxx 2(n
vyy =
vyyy = y=
Normalization:
y=0
v=1
1)ux + n(n 1) u
n
2
2
vy = 0
2
vyy =
n(n
1
1)
ch 38
Moving frame:
p
n
=n
H
=u
= n u n =n
(1
1
H = n(n
= xu
)
(1
Æ = u1=n
)
! n (nJ 1) 2
2
= (Q; H )
U = (Q; T )
(1)
(1)
| Hessian
Q(x) = (ax + b)n
=) Totally singular forms
vyyyy 7
Absolute rational covariants:
T
J =
H
T
(1 n)=n
xu
n
1
2 2
Dierential invariants:
H = 12 (Q; Q)(2)
n)=npH
1)uuxx (n 1) ux
Nonsingular form: H 6= 0
Note : H 0 if and only if
vyyy 7
(1
2
3
n 2)
d
! Kn+(3(
n 1)
ds
K=
3
U
H2
= n(n 1)QQ00 (n 1) Q0 QxxQyy Qxy
= (2n 4)Q0H nQH 0
QxHy Qy Hx
= (3n 6)Q0T nQT 0
QxTy Qy Tx
2
2
2
deg Q = n deg H = 2n 4 deg T = 3n 6 deg U = 4n 8
ch 39
Signatures of Binary Forms
Signature curve
of a nonsingular binary form Q(x):
(
SQ = (J (x) ; K (x)) =
2
Nonsingular :
T (x)2 U (x)
;
H (x)3 H (x)2
!)
H (x) 6= 0 and (J 0(x); K 0(x)) 6= 0.
Signature map
: NQ ! SQ
(x) = (J (x) ; K (x))
2
Two nonsingular binary forms are equivalent if
and only if their signature curves are identical.
Theorem.
ch 40
Maximally Symmetric Binary Forms
If u = Q(x) is a polynomial, then the following
are equivalent:
Q(x) admits a one-parameter symmetry group
Theorem.
T is a constant multiple of H
2
3
Q(x) ' xk is complex-equivalent to a monomial
the signature curve degenerates to a single point
all the (absolute) dierential invariants of Q are constant
the graph of Q coincides with the orbit of a
one-parameter subgroup
=) diagonalizable
ch 41
Symmetries of Binary Forms
The symmetry group of a nonzero binary form
Q(x) 6 0 of degree n is:
Theorem.
A two-parameter group if and only if H 0 if and only if
Q is equivalent to a constant.
=) totally singular
A one-parameter group if and only if H 6 0 and T is
2
a constant multiple of H if and only if Q is complexequivalent to a monomial xk, with k 6= 0; n.
3
=) maximally symmetric
In all other cases, a nite group whose cardinality equals
the index
Q = min
# fw g w 2 S
of the signature curve, and is bounded by
6n 12
U = cH
Q 4n 8
otherwise
8
<
n
1
o
2
:
ch 42
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 z1; : : : ; g zk )
= I (z ; : : : ; zk )
1
A k-point joint dierential invariant is an invariant of
the prolonged action G n on a k-fold Cartesian product of jet
space
Jn Jn
( )
I (g(n) z1(n); : : : ; g(n) zk(n))
= I (z n ; : : : ; zkn )
( )
1
( )
=) Joint dierential invariants are known as \semi-dierential invariants"
in the computer vision literature, and are proposed as \noise
resistant" alternatives for object recognition.
ch 43
Joint Euclidean Invariants
SE(2) acts on M = R R :
zi = (xi; ui )
wi = (yi; vi ) = g zi
yi = cos (xi a) + sin (ui b)
vi = sin (xi a) + cos (ui b)
2
2
1
Normalization (cross-section)
y =0
0
Left moving frame
: M
a = x0
Joint invariants:
(z
y 7 ! i
i
v0 = 0
y1 > 0
i = 0; 1; 2; : : :
v1 = 0
! SE(2)
b = u0
z0) (z1 z0 )
k z1 z0 k
= tan
vi 7
1
u1 u0
x1 x0
!
! (zi kzz ) ^ (zz k z )
0
1
1
0
0
ch 44
Every joint Euclidean invariant is a function of
the interpoint distances k zi zj k and, in the orientation
preserving case, a single signed area A(z ; z ; z )
Theorem.
0
1
2
ch 45
Joint Invariant Signatures
If the invariants depend on k points on a p-dimensional
submanifold, then you need at least
` > kp
distinct invariants I ; : : : ; I` in order to construct a syzygy:
(I ; : : : ; I`) 0
The total number of syzygies is
1
1
` kp
Typically, the number of joint invariants is
` = k m r = (#points)(dim M ) dim G
Therefore, to nd a joint invariant signature, that involes
no dierentiation, we need at least
r
+1
k
m p
points on our submanifold.
ch 46
Joint Euclidean Signature
For the Euclidean group G = SE(2) acting on curves C R (or R )
we need at least four points
2
3
z0 ; z1 ; z2 ; z3 2 C
Joint invariants:
a = kz
d = kz
1
z 0 k;
2
z 1 k;
b = k z 2 z 0 k;
c = k z 3 z 0 k;
e = k z 3 z 1 k;
= k z z k:
=) six functions of four variables
f
3
2
Joint Signature:
: C ! S R
dim S = 4 =) two syzygies
(a; b; c; d; e; f ) = 0
(a; b; c; d; e; f ) = 0
Universal Cayley{Menger syzygy:
2a
a +b d a +c e
2b
b +c f =0
det a + b d
a +c e b +c f
2c
4
6
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
() C R
2
ch 47
z0
c
z3
a
f
b
e
z1
d
z2
Four-Point Euclidean Joint Signature
ch 48
Euclidean Joint Dierential Invariants
| Planar Curves
One{point
) curvature
=
z^z
k z k3
Two{point
) distances k z z k
) tangent angles k = < (z
1
0
)
z0; z k )
1
ch 49
Equi{AÆne Joint Dierential Invariants
| Planar Curves
One{point
) aÆne curvature
(z ^ z ) + 4(ztt ^ zttt) 5(zt ^ zttt)
= t tttt
3(zt ^ ztt) =
9(zt ^ ztt) =
= zs ^ zss
2
5 3
8 3
Two{point
) tangent triangle area ratio
[
]
z ^z
0
0
[ (z z ) ^ z ] = [ 0 1 0 ]
0
0
1
0
0
3
3
Three{point
) triangle area
1
2
(z
1
z0) ^ (z2 z0) = 12 [ 0 1 2 ]
ch 50
Projective Joint Dierential Invariants
| Planar Curves
One{point
) projective curvature
= :::
Two{point
) tangent triangle area ratio
[0 1 0] [1 1]
[0 1 1] [0 0]
3
3
Three{point
) tangent triangle ratio
[0 2 0][0 1 1][1 2 2] :
[0 1 0][1 2 1][0 2 2]
Four{point
) area cross{ratio
[0 1 2][0 3 4]
[0 1 3][0 2 4]
ch 51
Transformation Groups and Jets
(x ; : : : ; xp) | independent variables
(u ; : : : ; uq) | dependent variables
z n = (x; u n ) 2 Jn | n order jet space
uJ | derivative coordinates on Jn
1
1
( )
( )
th
G | transformation group
G(n) | prolonged action on Jn
v2g |
v(n) 2 g(n)
Lie algebra
| Prolonged inf. gens.
The Prolongation Formula
p
@
n
v = i(x; u) i +
@x
( )
'J
n
X
X
i=1
;J
= DJ +
Q
Characteristic
Q
p
X
i=1
@
@uJ
i uJ;i
(x; u ) =
(1)
'J(x; u(j ))
'
p
X
i=1
@xi
@u
i
ch 52
Rotation group | SO(2)
(x; u) 7 ! (x cos u sin ; x sin + u cos )
Transformed function v = f(y):
y = x cos f (x)sin ;
v = x sin + f (x)cos ;
Second prolongation
(x; u; ux; uxx) 7 ! (x cos Innitesimal generator
u sin ; x sin + u cos ;
sin + ux cos ;
cos ux sin (cos v= u
Second prolongation
uxx
ux sin )3
!
@
@
+
x
@x
@u
@
@
@
@
+
x + (1 + u2x)
+
3
uxuxx
@x
@u
@ux
@uxx
Q = x + uux
'x = DxQ + uxx = Dx(x + uux) uuxx = 1 + u2x
v(2) = u
'xx = Dx2 Q + uxxx = Dx2 (x + uux) uuxxx = 3uxuxx
ch 53
Dierential invariant:
I (g(n) (x; u(n))) = I (x; u(n))
Innitesimal criterion:
v(n)(I ) = 0
for all
v(n) 2 g(n)
=) Solve the rst order linear partial dierential equation by
the method of characteristics.
=) Moving frames avoids integration!
Note :
If I ; : : : ; Ik are dierential invariants, so is (I ; : : : ; Ik).
1
1
=) Classify dierential invariants up to functional independence.
=) Local results on open subsets of jet space.
ch 54
Any transformation group admits a nite
system of fundamental dierential invariants
Theorem.
J1; : : : ; J`
and p invariant dierential operators
D ; : : : ; Dp
1
such that every dierential invariant is a function of the
dierentiated invariants:
I = ( : : : DK J : : : )
ch 55
Classication Problem.
How many fundamental dierential invariants J ; : : : ; J` are
required?
1
=) For curves (p = 1), we have ` = q.
Syzygy Problem.
Determine the algebraic relations
( : : : DK J : : : ) = 0
among the dierentiated invariants.
Commutation Formulae.
The order of invariant dierentiation matters
[ Di ; Dj ] = ???
=) Only an issue when p > 1.
ch 56
The Fundamental Dierential Invariants
I (n)(z (n)) = (n)(z (n))
1
z n
( )
H i(x; u(n)) = yi((n)(x; u(n)); x; u)
((n)(x; u(n)); x; u(k))
IK (x; u(k)) = vK
Recurrence Formulae:
Dj H i = Æji + Mji
+ M
Dj IK = IK;j
K;j
Mji; MK;j
| correction terms
Commutation Formulae:
[Di; Dj ] =
p
X
i=1
Akij Dk
The correction terms can be computed directly from the
innitesimal generators!
ch 57
Generating Invariants
A generating system of dierential invariants
consists of
all non-phantom dierential invariants H i and I coming
from the un-normalized zero order lifted invariants yi,
v, and
all non-phantom dierential invariants of the form IJ;i
where IJ is a phantom dierential invariant.
order order + 1
Theorem.
th
In other words, every other dierential invariant can, locally,
be written as a function of the generating invariants and
.
their invariant derivatives, DK H i, DK IJ;i
=) Not necessarily a minimal set!
ch 58
Syzygies
A syzygy is a functional relation among
dierentiated invariants:
H ( : : : DJ I : : : )
0
Derivatives of syzygies are syzygies
=) nd a minimal basis
Remark :
There are no syzygies among the normalized dierential invariants I n except for the \phantom syzygies"
I = c
corresponding to the normalizations.
( )
ch 59
Classication of Syzygies
All syzygies among the dierentiated invariants
are dierential consequences of the following three fundamental types:
Theorem.
Dj H i = Æji + Mji
DJ IK = c + MK;J
| H i non-phantom
| IK generating
= w = c phantom
| IJ;K
= M
DJ ILK
DK ILJ
LK;J MLJ;K
, I generating, K \ J = ?
| ILK
LJ
=) Not necessarily a minimal system!
ch 60
Right Regularization
If G acts on M , then the lifted action
(h; z) 7 ! (h g
on the trivial right principal bundle
B = GM
is always regular and free!
1
; g z)
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
Example. G = SO(2)
B = SO(2) R
(x; u; ) 7 !
(x cos M
2
=R
2
solid torus
u sin ; x sin + u cos ; + mod2 )
ch 62
Jet Regularization
B n = Jn G
Jn
Jn
w = w(n) = g(n) z (n)
(n)(z (n)) = (z (n); (n)(z (n)))
I (n)(z (n)) = w(n) Æ (n)(z (n)) = (n)(z (n)) z (n)
Invariantization
(F ) = ( n ) Æ (w n ) F = F Æ I n
( )
( )
( )
ch 63
General Philosophy of Lifting
All invariant objects on Bn = Jn G
are well-behaved and easily understood.
=) lifted invariants
We use the G-equivariant moving frame section
n : Jn ! B
(z n ) = ((z n ); z n )
to pull back lifted invariants to construct ordinary invariants
on Jn.
For example,
w n = w n Æ = I n
gives the fundamental dierential invariants.
( )
( )
( )
( )
( )
( )
( )
Similarly for lifted invariant dierential forms, dierential
operators, tensors, etc.
=) The key complication is that the pull-back process does
not commute with dierentiation!
ch 64
The Variational Bicomplex
Innite jet space
M = J0
Inverse limit
J
J
1
2
n
J1 = nlim
!1 J
Local coordinates
z 1 = (x; u 1 ) = ( : : :
(
)
(
)
xi : : : uJ : : : )
Coframe | basis for the cotangent space T J1:
Horizontal one-forms
dx1; : : : ; dxp
Contact (vertical) one-forms
p
J = duJ
X
i=1
uJ;i dxi
Intrinsic denition of contact form
j j1 N = 0
() =
X
AJ J
ch 65
Vertical and Horizontal Dierentials
Bigrading of the dierential forms on J1
r;s
= M
r;s
Dierential
d = dH
dH
dH F
=
dV F
=
p
dV
X
i=1
: r;s
: r;s
(DiF ) dxi
@F J
@u
J
;J
X
=)
+ dV
! r
! r;s
+1
;s
+1
| total derivatives
| variation
Vinogradov, Tsujishita, I. Anderson
ch 66
The Simplest Example.
Horizontal form
M
=R
2
x; u 2 R
dx
Contact (vertical) forms
= du ux dx
x = dux uxx dx
xx = duxx uxxx dx
..
Dierential
@F
@F
@F
@F
dF =
dx +
du +
dux +
@x
@u
@u
@u
x
xx
duxx + @F
@F
= (DxF ) dx + @F
+
x +
+ @u
@ux
@uxx xx
= dH F + dV F
Total derivative
@F
@F
@F
ux +
uxx +
uxxx + DxF =
@u
@u
@u
x
xx
ch 67
Lifted Variational Tricomplex
B 1 = J1 G
Lifted horizontal forms
i = 1; : : : ; p
dJ yi
Lifted invariant contact forms
=
J
p
dJ vJ
X
i=1
d yi
vJ;i
J
Right-invariant Maurer{Cartan forms
= dg g
=) ; : : : ; r
1
r = dim G
1
Dierential forms on B1
M r;s;t
= r;s;t
Dierential
d = dH + dV + dG
b
: r;s;t
dV : r;s;t
dG : r;s;t
dH
b
b
b
! r ;s;t
! r;s ;t
! r;s;t
b +1
b
b
+1
+1
ch 68
Invariantization
:
Functions
Forms
! Invariants
! Invariant Forms
Functions:
(F )
= Æ w (F ) =
F Æ I (1)
Dierential Forms:
(
) = (J (w )):
J
| Jet projection
T B1
= T (J1 G) ' T J1 T G
ch 69
Invariant Variational Complex
Fundamental dierential invariants
H i(x; u n ) = (xi)
IK (x; u l ) = (uK )
( )
( )
Invariant horizontal one-forms
$i = (dxi) = !i + i
!i | contact-invariant forms
i | contact \corrections"
Invariant contact forms
#K = (J)
r;s
= M
r;s
d = dH + dV + dW
dH : r;s
! r ;s
dV : r;s
! r;s
dW : r;s
! r ;s
Dierential forms
Dierential
b
b
b +1
b
b
+1
b
1 +2
b
ch 70
The Key Formula
d (
) = (d
) +
v1; : : : ; vr
p
X
k=1
^ [v(
)]
| basis for g
= = + "
= 1; : : : ; r
2 b 1;0
" 2 b 0;1
| pull back of the dual basis Maurer{Cartan forms via
the moving frame section
: J1 ! B 1
???
All recurrence formulae, syzygies,
commutation formulae, etc. are found by applying
the key formula for various forms and functions ch 71
Euclidean Curves
Lifted invariants
y = w (x) = x cos u sin + a
v = w (u) = x cos + u sin + b
sin + ux cos vy = w (ux) =
cos ux sin uxx
vyy = w (uxx) =
(cos u sin )
vyyy = w (uxx) =
(cos x
3
ux sin )uxxx 3u2xx sin (cos ux sin )5
dy = (cos ux sin ) dx
dJ y = J (dy) = (cos Dy =
cos Normalization
ux sin y=0
Right moving frame
=
1
tan
: J1
1
ux
(sin ) + da v d
ux sin ) dx (sin ) = du ux dx
Dx
v=0
vy = 0
! SE(2)
a=
x + uux
1 + u2x
q
xu u
b= q x 2
1 + ux
ch 72
Fundamental normalized dierential invariants
(x) = H = 0
(u) = I = 0
phantom di. invs.
(ux) = I = 0
(uxx) = I = (uxxx) = I = s
(uxxxx) = I = ss + 3
9
>
>
>
>
=
0
>
>
>
>
;
1
2
3
3
4
Invariant horizontal one-form
(dx) = ( dJ y) = $ =
+
!
= 1 + ux dx +
q
2
ux
1 + ux
q
2
Invariant contact forms
() = # =
1 + ux
(1 + ux) x uxuxx
(x) = # =
(1 + ux)
q
2
2
1
2 2
ch 73
Prolonged innitesimal generators
v = @x
v = @u
v = u @x + x @u + (1 + ux) @ux + 3uxuxx @uxx + 1
2
2
3
dH I = DsI $
Horizontal recurrence formula
dH (F ) = (dH F ) + (v (F )) + (v (F )) + (v (F )) Use phantom invariants
0 = dH H = (dH x) + (v(x)) = $ + ;
0 = dH I = (dH u) + (v(u)) = ;
0 = dH I = (dH ux) + (v(ux)) = $ + ;
to solve for
= $
=0
= $
1
1
2
2
X
2
0
X
3
1
2
3
1
X
1
3
3
ch 74
1 = $
2 = 0
3 = $
Recurrence formulae
s$ = dH = dH (I ) = (dH uxx) + (v (uxx)) = (uxxx dx) (3uxuxx)) $ = I $
ss$ = dH (I ) = (dH uxxx) + (v (uxxx) = (uxxxx dx) (4uxuxxx + 3uxx) $ = I 3I
2
3
3
3
3
3
3
2
= I2
s = I3
ss = I4
sss = I5
4
3
2
$
I2 = 3I
19I
I3 = s
I4 = ss + 33
3
2
I
2
2 3
I4 = sss + 192 s
ch 75
Vertical recurrence formula
dV (F ) = ( dV F ) + (v (F )) " + (v (F )) " + (v (F )) "
Use phantom invariants
0 = dV H = "
0 = dV I = # + "
0 = dV I = # + "
to solve for
" =0
" = # = ()
" = # = ( )
Recurrence formulae
dV I = dV = ( ) + (v (uxx)) " = # = (D + ) #;
1
1
2
2
3
3
1
2
0
1
1
2
2
dH #:
3
1
3
2
3
D# = #
3
1
2
D# = #
1
1
2
1
2
2
2 #
dV $ = # ^ $
ch 76
Example
(x ; x ; u) 2 M = R
G = GL(2)
(x ; x ; u) 7 ! (x + x ; x + Æx ; u)
= Æ =) Classical invariant theory
1
2
3
1
2
1
2
1
2
Prolongation (lifted dierential invariants):
y = (Æx x )
y = ( x + x )
v= u
u + u
u + Æu
v =
v =
u + 2u + u
v =
1
1
1
2
2
1
1
1
2
2
1
1
1
2
2
11
2
12
2
22
u11 + (Æ + )u12 + Æu22
v12 =
2 u11 + 2Æu12 + Æ2u22
v22 =
11
Normalization
Nondegeneracy
y1 = 1
y2 = 0
x1
v1 = 1
@u
2 @u
+
x
@x1
@x2
v2 = 0
6= 0
ch 77
First order moving frame
Æ
!
x1
x2
=
Normalized dierential invariants
J =1
J =0
u
I=
x u +x u
1
u2
u1
!
2
1
2
1
I1 = 1
2
I2 = 0
(x ) u + 2x x u + (x ) u
I =
x u +x u
x u u + (x u x u )u + x u u
I =
x u +x u
(u ) u 2u u u + (u ) u
I =
x u +x u
Phantom dierential invariants
1 2
11
1
22
1
2
1
1
11
1
12
2
1
2 11
12
2
1 2
11
1
1
I1
2 2
2
2
1
2
1
2 12
2
22
2
12
2
1 22
2
1
2
22
2
I2
Generating dierential invariants
I
I11
Invariant dierential operators
D =x D +x D
D = u D +u D
1
2
1
2
1
2
1
2
1
2
I12
I22
| scaling process
| Jacobian process
ch 78
Recurrence formulae
D J =Æ 1=0
D J =Æ 0=0
D J =Æ 0=0
D J =Æ 1=0
D I = I I (1 + I ) = I (1 + I ) D I = I I I = I I
D I =I I =0
D I =I I =0
D I =I I =0
D I =I I =0
D I = I + (1 I )I
D I = I + (2 I )I
D I =I
I I
D I = I + (1 I )I
D I = I + (I 1)I 2I
D I =I
I I
=) Use I to generate I and I
Syzygies
D I D I = 2I
D I D I = 2(I 1)I 2I
(D ) I (D ) I =
= 2I D I + (5I 2)D I + (3I 5)D I
(2I 5)(I 1)I + 4(I 1)I
Commutation formulae
[D ; D ] = I D + (I 1)D
1
1
1
1
1
2
2
2
1
2
1
1
11
11
1
1
2
2
2
2
2
2
12
1 1
11
11
2 1
12
12
1 2
12
12
2 2
22
22
1 11
111
1 12
112
1 22
122
11
11
11 12
11
2
12
22
12
2 11
112
11
12
2 12
122
11
22
2 22
222
12 22
11
1
2
1 12
2 11
1 22
2 12
2
2
11
22
22
12
11
1 11
11
1
2
2
12
22
12
1 12
11
12
1
12
11
12
11
11
1 22
2
12
2
ch 79
Invariant Variational Problems
Z
Z
I [ u ] = L(x; u ) dx = P ( : : : DK I : : : ) !
I1; : : : ; I`
DK I n
( )
| fundamental dierential invariants
| dierentiated invariants
! = !1 ^ ^ !p | contact-invariant volume form
Invariant Euler-Lagrange equations
E(L) = F ( : : : DK I ::: ) = 0
Problem.
Construct F directly from P .
=) P. GriÆths, I. Anderson
ch 80
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
ds =
2
Euler-Lagrange equation
E(L) = ss +
Z
1
2
u2xx dx
(1 + u2x)5=2
3 = 0
=) elliptic functions
ch 81
Z
P (; s; ss; : : : ) ds
Invariantized Euler operator
E=
1
X
n=0
( D)n @@
D = dsd
n
Invariantized Hamiltonian operator
H(P ) =
X
i>j
i j
@P
( D)j @
i
P
Invariant Euler-Lagrange formula
E(L) = (D + ) E (P ) + H(P ):
2
2
Elastica
P
=
1
2
2
E (P ) = H(P ) = P =
1
2
2
E(L) = ss + 21 3 = 0
ch 82
Euler-Lagrange Equations
Integration by Parts:
: p; ! F = p;
1
1
1
= dH p 1;1
Variational derivative or Euler operator:
Æ = Æ dV : p;
! F
0
1
Variational Problems
Æ : = L dx
Hamiltonian
H(L) =
m
X
=) Source forms
X
=1 i>j 0
! Source Forms
!
q
X
=1
E(L) ^ dx
@L
ui j ( Dx)j @u
i
L
ch 83
The Simplest Example.
Lagrangian form
M
=R
2
x; u 2 R
= L(x; u(n)) dx
Vertical derivative
d = dV @L
@L
+
x +
= @L
@u
@u
@u
x
xx
xx + !
^ dx 2 ;
11
Integration by parts
dH (A ) = (DxA) dx ^ A x ^ dx
= [(DxA) + A x ] ^ dx
Variational derivative
@L
@L
@L
Dx
+
Dx2
Æ =
@u
@ux
@uxx
= E(L) ^ dx 2 F 1
!
^ dx
ch 84
Invariant Lagrangian
Plane Curves
Z
P (; s; : : :) $
| fundamental dierential invariant (curvature)
$ = ! + | fully invariant horizontal form
! = ds | contact-invariant arc length
Invariant integration by parts
dV (P $) = E (P ) dV ^ $ H(P ) dV $
Vertical dierentiation formulae
dV = A(#)
A | Eulerian operator
dV $ = B(#) ^ $
B | Hamiltonian operator
=) The explicit formulae follow from our fundamental recurrence
formula, based on the innitesimal generators of the action.
Invariant Euler-Lagrange equation
AE (P ) BH(P ) = 0
ch 85
General Framework
Fundamental dierential invariants
I 1; : : : ; I `
Invariant horizontal coframe
$1; : : : ; $p
Dual invariant dierential operators
D ; : : : ; Dp
1
Invariant volume form
$ = $1 ^ ^ $p
Dierentiated invariants
= DK J = D D J I;K
k1
kn
=) order is important!
ch 86
Eulerian operator
dV
=)
=
I
q
X
=1
A(# )
A = ( A )
m q matrix of invariant dierential operators
Hamiltonian operator complex
dV
=)
$j
=
q
X
=1
j (# ) ^ $i
Bi;
j )
Bij = ( Bi;
p2 row vectors of invariant dierential operators
$(i) = ( 1)i 1 $1 ^ ^ $i 1 ^ $i+1 ^ ^ $p
Twist invariants
Twisted adjoint
dH $(i) = Zi $
Diy = (Di + Zi)
ch 87
Invariant variational problem
Z
P (I (n)) $
Invariant Eulerian
E(P ) =
X
Invariant Hamiltonian tensor
q
i
i
Hj (P ) = P Æj +
K
X
=1
DKy @[email protected]
;K
D y @P ;
I;J;j
K @I ;J;i;K
J;K
X
Invariant Euler-Lagrange equations
A y E (P )
p
X
i;j =1
(Bij ) y Hji (P ) = 0:
ch 88
Euclidean Surfaces
SM
Group:
=R
3
coordinates
z = (x; y; u)
G = E(3)
z7
! R z + a;
R 2 O(3)
Normalization | coordinate cross-section
x = y = u = ux = uy = uxy = 0:
Left moving frame
a=z
R = ( t1 t2 n
)
t ; t 2 T S | Frenet frame
n
| unit normal
1
2
ch 89
Fundamental dierential invariants
= (uxx)
= (uyy )
=) principal curvatures
1
2
Frenet coframe
$ = (dx ) = ! + Invariant dierential operators
1
1
1
1
D
1
$2 = (dx2) = !2 + 2
D
2
=) Frenet dierentiation
Fundamental Syzygy:
Use the recurrence formula to compare
; = D (uxx)
(uxxyy )
with
; = D (uyy )
1
;22
2
;11
1;1 2;1 + 1;22;2 2(2;1)2
+
1 2
1
22
2
2
2
11
2
1
2(; )
1 2
2
12 (1 2) = 0
=) Codazzi equations
ch 90
Twisted adjoints
Dy =
1
(D + Z )
1
Z1 =
1 2
1;2
Z2 = 2 1
1
D y = (D + Z )
2
2
2;1
2
Gauss curvature | Codazzi equations:
K = = D y(Z ) + D y(Z )
= (D + Z )Z (D + Z )Z
1 2
1
1
K
2
1
is an invariant divergence
1
2
1
2
2
2
=) Gauss{Bonnet Theorem!
ch 91
Invariant contact form
# = () = (du ux dx uy dy)
Invariant vertical derivatives
dV = (xx) = ( D
dV = (yy ) = ( D
Eulerian operator
+Z
A= D
D +Z
1
2
1
2
2
2
2
1
2
2
dV $1 = 1 # ^ $1
2
1
1
+ Z D + ( ) ) #
+ Z D + ( ) ) #
2
2
1
1
D + ( )
D + ( )
1 2
2
1 2
2 2
!
2 2
1
1 (D D
1
2
2
Z2 D1 )# ^ $2
1 ( D D Z D )# ^ $ + # ^ $
Hamiltonian operator complex
B = ; B = 1 (D D Z D ) = B
B = ;
dV $2 =
1
1
2
2
1
2
1
2
1
2
2
1
1
1
2
1
2
1
2
2
2
1
2
2
1
ch 92
Euclidean-invariant variational problem
P ( n ) ! ^ ! = P ( n ) dA
Euler-Lagrange equations
E(L) = A y E (P ) B y H(P ) = 0;
Special case: P ( ; )
Z
( )
1
1
Z
2
( )
2
@P
D y Z + ( ) ] @
+
E(L) = [ (D1y)2
2
1 2
2
1
@P
+ [ (D y) D y Z + ( ) ] @
+ ( + ) P
2
2
Minimal surfaces:
1
2 2
1
1
2
2
=1
P
1 + 2 = 2H = 0
Minimizing mean curvature: P = H = ( + )
( ) + ( ) + + = 2H + H K = 0:
Willmore surfaces: P = ( ) + ( )
( + ) + ( + )( ) = 2H + 4(H K )H = 0
1
2
1
2
h
1 2
2 2
1
2
1
2
1
2
1
1
1 2
2
1
2
1
2
i
1
2
2
2 2
2 2
2
Laplace{Beltrami operator
= (D + Z )D + (D + Z )D = D y D D y D
1
1
1
2
2
2
1
1
2
2
ch 93
Multi{Space
Although in use since the time of Lie and Darboux,
jet space was rst formally dened by Ehresmann in 1950.
Jet space is the proper setting for the geometry
of partial dierential equations.
In this talk, I will propose a setting, named multispace , for the geometry of
numerical approximations to
derivatives and dierential equations.
=)
Multi-space is the context for geometric integration.
ch 94
Invariant Numerical Approximations
: Every (nite dierence) numerical
approximation to the derivatives of a function
require evaluating the function at several points
zi = (xi ; ui ) = (xi ; f (xi )).
Key remark
ch 95
In other words, we seek to approximate the nth
order jet of a submanifold N M by a function
F (z0 ; : : : ; zn ) dened on the (n + 1)-fold Cartesian
product space M (n+1) = M M , or, more
correctly, on the \o-diagonal" part
M
(n+1) = f z =
zj for all i 6= j g
i 6
=)
distinct
(n + 1)-tuples of points.
ch 96
Thus, multi-space should contain both the jet
space and the o-diagonal Cartesian product space
as submanifolds:
(n+1)
M
#
Jn(M; p)
Functions
F
: M (n)
9
>
>
>
>
>
=
>
>
>
>
>
;
!R
M
(n)
are given by
(n+1)
(0
)
on
M
and extend smoothly to Jn as the points coalesce. In
this manner, F j M (n+1)
provides a nite dierence approximation to the
dierential function F j Jn.
F z ; : : : ; zn
ch 97
Construction of M (n)
Denition. An (n + 1)-pointed manifold
M = (z0; : : : ; zn; M )
M
0
| smooth manifold
z ; : : : ; zn
2
M
| not necessarily distinct
Given M, let
#i = #
n
j z
j
= zi
o
denote the number of points which coincide with the
th one.
i
ch 98
Multi-contact for Curves
Denition. Two (n + 1)-pointed curves
C = (z0; : : : ; zn; C ); C = (~z 0; : : : ; ~z n; C );
have nth order multi-contact if and only if
f
zi
= z~i;
and
e
j#i 1C jzi = j#i 1Ce jzi ;
for each i = 0; : : : ; n.
#i = #
n
j z
j
= zi
o
Denition. The nth order multi-space
M
(n) is the
set of equivalence classes of (n + 1)-pointed
curves in M under the equivalence relation of
th order multi-contact.
n
ch 99
The Fundamental Theorem
Theorem. If M is a smooth m-dimensional
manifold, then its nth order multi-space M (n) is a
smooth manifold of dimension (n + 1)m, which contains the o-diagonal part M (n+1) of the Cartesian
product space as an open, dense submanifold, and
the nth order jet space Jn as a smooth submanifold.
(n+1)
M
Jk1 Jk
Jn(M; p)
9
>
>
>
>
>
>
=
>
>
>
>
>
>
;
M
(n)
ch 100
Let M = R m
(i ) M is the space of two-pointed lines
M ' f (z ; z ; L) j z ; z 2 L | line g :
=) Blow-up construction in algebraic geometry
Example.
(1)
(1)
(ii )
0
1
0
1
M (2) is the space of three-pointed circles, i.e.,
M (2) ' f (z0; z1; z2; C ) j z0; z1; z2 2 C
| circle g :
Straight lines are included as circles of innite radius, but
points are not included (even though they could be viewed
as circles of zero radius).
=) Grassmann bundles.
(iii ) M
????
(3)
Topology | local and global.
ch 101
Finite Dierences
Local coordinates on Jn are provided by the coeÆcients of
Taylor polynomials
=) derivatives
Local coordinates on M n are provided by the coeÆcients of
interpolating polynomials.
=) nite dierences
( )
Given (z ; : : : ; zn) 2 M n , dene the classical divided
dierences by the standard recursive rule
( +1)
0
[z z
0
1 : : : zk 1 zk ] =
[z z z
[ zj ] = uj
0 1 2
: : : zk 2zk ] [ z0 z1z2 : : : zk 2 zk
xk xk 1
1
]
=) Well-dened provided no two points lie on the same
vertical line.
=) Symmetric functions of zi.
ch 102
Given an (n + 1)-pointed graph C =
(z ; : : : ; zn; C ), its divided dierences are dened by
Denition.
0
[z z
0
1 : : : zk
[ zj ]C = f (xj )
[z z z
z ] = lim
1
k C
z!zk
0 1 2
: : : zk 2 z ]C [ z0 z1 z2 : : : zk 2zk
x xk 1
1
]C
=) When taking the limit, the point z = (x; f (x)) must lie
on the graph C , and take limiting values x ! xk and
f (x) ! f (xk ).
Two (n + 1)-pointed graphs C; C have n
order multi-contact if and only if they have the same divided
dierences:
[ z z : : : zk ]C = [ z z : : : zk ]C ; k = 0; : : : ; n:
Theorem.
0 1
f
0 1
th
e
ch 103
Local coordinates on M (n)
They consist of the independent variables along with all
the divided dierences
x0 ; : : : ; xn
u(0) = u0 = [ z0 ]C
u(1) = [ z0 z1 ]C
u(2) = 2 [ z0 z1z2 ]C : : : u(n) = n! [ z0 z1 : : : zn ]C
prescribed by (n + 1)-pointed graphs
C = (z ; : : : ; zn; C )
0
The n! factor is included so that u n agrees with the
usual derivative coordinate when restricted to Jn.
( )
ch 104
Numerical Approximations
(x; u n ) | dierential function
: Jn ! R
( )
System of dierential equations:
(x; u n ) = = k(x; u n ) = 0:
1
( )
( )
An (n + 1)-point numerical
approximation of order k to a dierential function
: Jn ! R is a k order extension F : M n ! R of
to multi-space, based on the inclusion Jn M n .
Denition.
th
( )
( )
F (x0; : : : ; xn; u(0); : : : ; u(n))
! F (x; : : : ; x; u ; : : : ; u n ) = (x; u n )
(0)
( )
( )
ch 105
Invariant Numerical Approximations
G
| Lie group acting on M
Basic Idea:
Every invariant nite dierence approximation to a
dierential invariant must expressible in terms of the joint
invariants of the transformation group.
Dierential Invariant
I (g(n) z (n)) = I (z (n))
Joint Invariant
J (g z0; : : : ; g zk ) = J (z0; : : : ; zk )
Semi-dierential invariant =
Joint dierential invariant
=) Approximate dierential invariants by joint invariants
ch 106
Euclidean Invariants
Joint Euclidean invariant:
d(z; w) = k z w k
Euclidean curvature:
uxx
=
(1 + u ) =
2 3 2
x
Euclidean arc length:
1 + ux dx
Higher order dierential invariants:
d
d
s =
ss =
ds
ds
Euclidean{invariant dierential equation:
F (; s; ss; : : :) = 0
ds =
q
2
2
2
:::
ch 107
Three point approximation
Heron's formula
e (A; B; C ) = 4
s=
= 4 s(s
abc
a)(s b)(s c)
abc
q
a+b+c
| semi-perimeter
2
Expansion:
1
d 1
d
= + (b a) + (b ab + a )
+
3
ds 12
ds
+ 601 (b ab + a b a ) dds +
1 (b a)(3b + 5ab + 3a ) d + :
+ 120
ds
2
e
3
2
2
2
2
2
3
2
3
3
2
2
ch 108
Higher order invariants
s =
d
ds
Invariant nite dierence approximation:
(Pi ; Pi; Pi ) (Pi ; Pi ; Pi )
s(Pi ; Pi ; Pi ; Pi ) =
d(Pi ; Pi )
Unbiased centered dierence:
(Pi; Pi ; Pi ) (Pi ; Pi ; Pi)
s(Pi ; Pi ; Pi; Pi ; Pi ) =
d(Pi ; Pi )
Better approximation (M. Boutin):
(Pi ; Pi; Pi ) (Pi ; Pi ; Pi)
s(Pi ; Pi ; Pi; Pi ) = 3
di + 2di + 2di + di
dj = d(Pj ; Pj )
e
e
2
1
1
+1
2
1
+1
2
1
+1
+1
+2
1
e
+2
+1
e
e
2
1
e
e
e
+1
1
e
+1
2
1
2
1
1
2
1
+1
+1
ch 109
AÆne Joint Invariants
x ! Ax + b
det A = 1
Area is the fundamental joint aÆne invariant
[ ijk ] = (Pi
= det
Pj ) ^ (Pi Pk )
x
y
1
i
i
x
y
1
j
j
x
y
1
k k
= Area of parallelogram
= 2 Area of triangle (Pi; Pj ; Pk)
Syzygies:
[ ij l ] + [ j kl ] = [ ij k ] + [ ikl ]
[ ij k ] [ ilm ]
[ ij l ] [ ikm ] + [ ij m ] [ ikl ] = 0
ch 110
AÆne Dierential Invariants
AÆne curvature
3u u
5u
= xx xxxx = xxx
9(uxx)
2
8 3
AÆne arc length
ds = 3 uxx dx
q
Higher order aÆne invariants:
d
d
ss =
s =
ds
ds
2
2
:::
ch 111
Conic Sections
Ax2 + 2Bxy + Cy2 + 2Dx + 2Ey + F
AÆne curvature:
=
S = AC
T
= det
Ellipse:
S
T 2=3
B 2 = det
=0
A B B C
A B D B C E D E F
= (=A)2=3
A=
AÆne arc length of ellipse:
Q
Z
P
T =
S 1=2
T
S 3=2
s
= Area
Q
!
CT
CD BE
x
+
S2
S
P
= 2ST 2=3A(P; Q)
ds =
1 3
arcsin
ch 112
Q
A(P; Q) :
P
Triangular approximation:
Q
(O; P; Q) :
P
O
Total aÆne arc length:
L=2 A=
q
3
q
2 pTS
3
ch 113
Conic through ve points P ; : : : ; P :
[013][024][x12][x34] = [012][034][x13][x24]
0
4
x = (x; y)
AÆne curvature and arc length:
S
= =
T
2 3
ds = Area (O; P1; P3) = 12 [O; P1; P3] =
4T =
Y
0
i<j<k4
N
2S
[ijk]
4S = [013] [024] ([124] [123]) +
+ [012] [034] ([134] + [123])
2[012][034][013][024]([123][234] + [124][134])
2
2
4N =
2
2
2
2
[123][134] f[023] [014] ([124] [123]) +
+ [012] [034] ([134] + [123]) +
+ [012][023][014][034]([134] [123])g
2
2
2
2
Theorem. P0; P1; P2; P3; P4 | points on the convex curve C .
ch 114
| aÆne curvature of C
at P
2
e = e (P0; P1; P2; P3; P4)
Li =
Z
Pi
P2
ds
Expansion:
1
=+
5
e
| aÆne curvature of conic
| aÆne arc length of conic
2
d 1 @
d
Li
+
LiLj A 2 + ds 30 0ij 4
ds
i=0
4
X
!
0
1
X
ch 115
G
Multi-Invariants
| Lie group which acts smoothly on M
=)
G
preserves the multi-contact equivalence relation
| n multi-prolongation to M n
=) On Jn M n it coincides with the usual
jet space prolongation
=) On M n M n it coincides with the
(n + 1)-fold Cartesian product action.
G(n)
th
( )
( )
( +1)
K : M (n) ! R
=)
=)
=)
|
( )
multi-invariant
K (g(n) z (n)) = K (z (n))
K j Jn | dierential invariant
K j M (n+1) | joint invariant
K j Jk1 Jk | joint di. invariant
The theory of multi-invariants is the theory of invariant
numerical approximations!
ch 116
Moving frames provide a
systematic algorithm for
constructing multi-invariants!
A moving frame on multi-space
: M n
!
is called a multi-frame .
( )
G
ch 117
Example. G = R 2 n R
(x; u) 7 ! ( x + a; u + b)
Multi-prolonged action: compute the divided dierences of the
basic lifted invariants
yk = xk + a;
vk = uk + b:
We nd
v v
v = [w w ] =
y y
1
1
(1)
0
=
2
1
1
0
1
0
u1 u0
=
2 [ z0z1 ] = 2 u(1);
x1 x0
v(n) = n+1 u(n):
Moving frame cross-section
y =0
v =0
Solve for the group parameters
p
u
b= p
a=
u x
0
(1)
v(1) = 1
0
0
=)
1
= p (1)
(1)
u
u
multi-frame : M (n) ! G.
0
ch 118
Multi-invariants:
p
yk :
Hk = (xk
uk :
uk u 0
Kk = p
= (uk
u(1)
u(n)
(n)
K = (1) (n+1)=2
(u )
u(n) :
K (0) = K0 = 0
Coalescent limit
K
x0 ) u
(1)
n
( )
= (xk
x0 )
s
u1 u0
x1 x0
x1 x0
u1 u0
: : zn ]
= n[!z[ zz0z]1(n:+1)
=2
0 1
u0)
K (1) = 1
u(n)
! I = (u ) n
n
( )
s
=
(1) ( +1) 2
=) K n is a rst order invariant numerical approximation
to the dierential invariant I n .
=) Higher order invariant numerical approximations are
obtained by invariantization of higher order divided dierence
approximations.
F ( : : : xk : : : u n : : : )
! F ( : : : Hk : : : K n : : : )
( )
( )
( )
( )
ch 119
Invariantization
To construct an invariant numerical scheme for any
similarity-invariant ordinary dierential equation
F (x; u; u ; u ; : : : u n ) = 0;
we merely invariantize the dening dierential function, leading to the general similarity{invariant numerical approximation
F (0; 0; 1; K ; : : : ; K n ) = 0:
(1)
(2)
(2)
( )
( )
=) Nonsingular!
ch 120
Euclidean group SE(2)
y = x cos u sin + a
v = x sin + u cos + b
Multi-prolonged action on M :
Example.
(1)
y0 = x0 cos u0 sin + a
v0 = x0 sin + u0 cos + b
y1 = x1 cos u1 sin + a
v(1) =
Cross-section
b=
(1)
(1)
y0 = v0 = v(1) = 0
Right moving frame
a=
sin + u cos cos u sin x0 cos + u0 sin =
x0 + u(1) u0
1 + (u(1))2
q
x u(1) u0
x0 sin u0 cos = q0
1 + (u(1))2
tan =
u(1) :
ch 121
Euclidean multi-invariants
(yk ; vk ) ! Ik = (Hk ; Kk )
(x x ) + u (uk u ) = (x x ) 1 + [ z z ][ z zk ]
Hk = k
k
1 + [z z ]
1 + (u )
(u u ) u (xk x ) = (x x ) [ z zk ] [ z z ]
Kk = k
k
1 + [z z ]
1 + (u )
Dierence quotients
Kk (xk x )[ z z zk ]
[ I Ik ] = KHk K
=
=
Hk 1 + [ z zk ][ z z ]
k H
I = [I I ] = 0
[I I ] [I I ]
I = 2[ I I I ] = 2
0
q
0
q
(1)
0
(1)
0
0 1
(1)
0
0
0 1
0
1
0
2
0 1
q
0
(1) 2
0
0 1
q
0
(1) 2
2
0 1
0
0 1
0 1
(2)
0 1 2
0 2
0 1
H2 H1
= (1 + [ z 2[z z][zz zz ]] )(11++[ z[ zzz ] ][ z z ])
q
0 1 2
0 1
=
0 1
1 2
0 1
2
0 2
q
u(2) 1 + (u(1) )2
h
ih
1 (1) (2)
1 + (u(1) )2 + 2
u u (x2 x0 ) 1 + (u(1) )2 + 12 u(1) u(2) (x2
x1 )
i
Invariant numerical approximation to the Euclidean curvature:
u
lim
I
=
=
z1 ;z2!z0
(1 + (u ) ) =
Euclidean{invariant approximation for s = (uxxx):
[I I I ] [I I I ]
I = 6 [I I I I ] = 6
H H
(2)
(2)
(1) 2 3 2
(3)
0 1 2 3
0 1 3
0 1 2
3
2
ch 122
Higher Dimensional Submanifolds
T
(n)M jz | nth order tangent space
Proposition.
f
Two p-dimensional submanifolds N; N
have nth order
contact
f
at a common point z 2 N \ N
if and only if
T
=) Requires
(n)N j = T (n)N
f
jz
z
!
p
+n
n
coalescing points to approxi-
mate nth order derivatives
ch 123
Surfaces
n
=2
p
p
+
!
n
n
0
1
1
3
2
6
3
10
..
..
ch 124
Denition. A subspace V T (n)M jz is called
if for every vector
\ T (k)M jz , 1 k n,
admissible
v2V
there exists a submanifold N
v 2 T (k)N jz V .
M
such that
Denition. Two submanifolds N; N have rth order
f
at a common point if and only if for
some n, there exists an admissible common rdimensional subspace
subcontact
S
() j \ () j () j
T
n
N z
T
n f
N z
T
n
M z
ch 125
Surfaces:
Example.
S; Se M
order
0
1
2
3
..
5
Conditions
| common point
tangent curves: T C jz = T C jz
tangent surfaces: T S jz = T S jz
osculating curves: T C jz = T C jz
T S jz = T S jz and T C jz = T C jz
T C jz = T C jz
..
T S jz = T S jz
T S jz = T S jz ; T C jz = T C jz ;
T C 0jz = T C 0jz
T S jz = T S jz ; T C jz = T C jz
T C jz = T C jz
z 2 S \ Se
e
8
>
<
e
>
:
(2)
8
>
<
>
:
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
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>
>
>
>
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>
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:
(2)
e
(3)
(3) e
(2)
(2) e
(3)
e
(2) e
(2) e
(3) e
(2)
(4)
e
(5)
(2) e
(4) e
(5) e
ch 126
Multi-space and Multi-variate Interpolation
Denition.
The n
th
Let M be a smooth manifold.
order multi-space M n is the set of all n-point
( )
interpolant data
Z = (z0; : : : ; zn
1
; V ; : : : ; Vn );
0
1
consisting of
(a) an ordered set of n points z ; : : : ; zn 2 M .
#i = # j zj = zi
(b) an ordered collection of admissible subspaces Vi T n M jzi
such that
Vi = Vj if zi = zj
dim Vi = #i 1
In particular, if #i = 1, and so zi only appears once in Z, then
Vi = f0g is trivial.
0
n
1
o
( )
8
<
:
Multivariate Hermite Interpolation
Denition. An interpolant to Z is a submanifold N M
such that zi 2 N and Vi T n N jzi .
( )
ch 127
The multispace M n is a manifold of dimension (n + 1)m. It contains
M n as an open, dense submanifold
all Jk(M; p) that have dimension (n + 1)m as submanifolds
various o-diagonal copies of multi-jet spaces Ji1 (M; p) Jik (M; p) for i + + ik = n k as submanifolds.
=) smooth or analytic
Conjecture.
( )
1
DiÆculties




Multi-variate interpolation theory.
Multi-variate divided dierences.
Coordinates at coalescent points.
Topological structure | local and global
ch 128
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