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Moving Framesin Applications

Peter J. Olver

University of Minnesota

http://www.math.umn.edu/! olver

Varna, June, 2012

Moving Frames

Classical contributions:M. Bartels (!1800), J. Serret, J. Frenet, G. Darboux,

E. Cotton, Elie Cartan

Modern developments: (1970’s)

S.S. Chern, M. Green, P. Gri!ths, G. Jensen, . . .

The equivariant approach: (1997 – )PJO, M. Fels, G. Marı–Be"a, I. Kogan, J. Cheh,

J. Pohjanpelto, P. Kim, M. Boutin, D. Lewis, E. Mansfield,E. Hubert, O. Morozov, R. McLenaghan, R. Smirnov, J. Yue,A. Nikitin, J. Patera, F. Valiquette, R. Thompson, . . .

“I did not quite understand how he [Cartan] does thisin general, though in the examples he gives theprocedure is clear.”

“Nevertheless, I must admit I found the book, likemost of Cartan’s papers, hard reading.”

— Hermann Weyl

“Cartan on groups and di"erential geometry”Bull. Amer. Math. Soc. 44 (1938) 598–601

Applications of Moving Frames

• Di"erential geometry

• Equivalence

• Symmetry

• Di"erential invariants

• Rigidity

• Joint invariants and semi-di"erential invariants

• Invariant di"erential forms and tensors

• Identities and syzygies

• Classical invariant theory

• Computer vision — object recognition & symmetrydetection

• Invariant numerical methods

• Invariant variational problems

• Invariant submanifold flows

• Poisson geometry & solitons

• Killing tensors in relativity

• Invariants of Lie algebras in quantum mechanics

• Lie pseudo-groups

The Basic Equivalence Problem

M — smooth m-dimensional manifold.

G — transformation group acting on M

• finite-dimensional Lie group

• infinite-dimensional Lie pseudo-group

Equivalence:Determine when two p-dimensional submanifolds

N and N " M

are congruent :

N = g · N for g # G

Symmetry:Find all symmetries,

i.e., self-equivalences or self-congruences :

N = g · N

Classical Geometry — F. Klein

• Euclidean group: G =

!"

#

SE(m) = SO(m) ! Rm

E(m) = O(m) ! Rm

z $%& A · z + b A # SO(m) or O(m), b # Rm, z # R

m

' isometries: rotations, translations , (reflections)

• Equi-a!ne group: G = SA(m) = SL(m) ! Rm

A # SL(m) — volume-preserving

• A!ne group: G = A(m) = GL(m) ! Rm

A # GL(m)

• Projective group: G = PSL(m + 1)acting on Rm " RPm

=' Applications in computer vision

Tennis, Anyone?

Classical Invariant Theory

Binary form:

Q(x) =n$

k=0

%n

k

&

ak xk

Equivalence of polynomials (binary forms):

Q(x) = (!x + ")n Q

%#x + $

!x + "

&

g =

%# $! "

&

# GL(2)

• multiplier representation of GL(2)• modular forms

Q(x) = (!x + ")n Q

%#x + $

!x + "

&

Transformation group:

g : (x, u) $%&%#x + $

!x + ",

u

(!x + ")n

&

Equivalence of functions (' equivalence of graphs

#Q = { (x, u) = (x, Q(x)) } " C2

Moving Frames

Definition.

A moving frame is a G-equivariant map

% : M %& G

Equivariance:

%(g·z) =

'g · %(z) left moving frame

%(z) · g!1 right moving frame

%left(z) = %right(z)!1

The Main Result

Theorem. A moving frame exists ina neighborhood of a point z # M if andonly if G acts freely and regularly near z.

Isotropy & Freeness

Isotropy subgroup: Gz = { g | g · z = z } for z # M

• free — the only group element g # G which fixes one pointz # M is the identity

=' Gz = {e} for all z # M

• locally free — the orbits all have the same dimension as G=' Gz " G is discrete for all z # M

• regular — the orbits form a regular foliation)* irrational flow on the torus

• e"ective — the only group element which fixes every point inM is the identity: g · z = z for all z # M i" g = e:

G+M =\

z"MGz = {e}

Geometric Construction

z

Oz

Normalization = choice of cross-section to the group orbits


z

Oz

K

k



z

Oz

K

k

g = %left(z)



z

Oz

K

k

g = %right(z)


Algebraic Construction

r = dim G , m = dim M

Coordinate cross-section

K = { z1 = c1, . . . , zr = cr }

left right

w(g, z) = g!1 · z w(g, z) = g · z

g = (g1, . . . , gr) — group parameters

z = (z1, . . . , zm) — coordinates on M

Choose r = dim G components to normalize:

w1(g, z)= c1 . . . wr(g, z)= cr

Solve for the group parameters g = (g1, . . . , gr)

=' Implicit Function Theorem

The solutiong = %(z)

is a (local) moving frame.

The Fundamental Invariants

Substituting the moving frame formulae

g = %(z)

into the unnormalized components of w(g, z) produces thefundamental invariants

I1(z) = wr+1(%(z), z) . . . Im!r(z) = wm(%(z), z)

=' These are the coordinates of the canonical form k # K.

Completeness of Invariants

Theorem. Every invariant I(z) canbe (locally) uniquely written as afunction of the fundamental invariants:

I(z) = H(I1(z), . . . , Im!r(z))

Invariantization

Definition. The invariantization of a functionF : M & R with respect to a right moving frameg = %(z) is the the invariant function I = &(F )defined by

I(z) = F (%(z) · z).

&(z1) = c1, . . . &(zr) = cr, &(zr+1) = I1(z), . . . &(zm) = Im!r(z).

cross-section variables fundamental invariants“phantom invariants”

& [ F (z1, . . . , zm) ] = F (c1, . . . , cr, I1(z), . . . , Im!r(z))

Invariantization amounts to restricting F to the cross-section

I |K = F |Kand then requiring that I = &(F ) be constantalong the orbits.

In particular, if I(z) is an invariant, then &(I) = I.

Invariantization defines a canonical projection

& : functions $%& invariants

Prolongation

Most interesting group actions (Euclidean, a!ne,projective, etc.) are not free!

Freeness typically fails because the dimensionof the underlying manifold is not large enough, i.e.,m < r = dim G.

Thus, to make the action free, we must increasethe dimension of the space via some natural prolonga-tion procedure.

• An e"ective action can usually be made free by:

• Prolonging to derivatives (jet space)

G(n) : Jn(M, p) %& Jn(M, p)

=' di"erential invariants

• Prolonging to Cartesian product actions

G#n : M - · · ·-M %& M - · · ·-M

=' joint invariants

• Prolonging to “multi-space”

G(n) : M (n) %& M (n)

=' joint or semi-di"erential invariants=' invariant numerical approximations

Euclidean Plane CurvesSpecial Euclidean group: G = SE(2) = SO(2) ! R2

acts on M = R2 via rigid motions: w = R z + b

To obtain the classical (left) moving frame we invertthe group transformations:

y = cos' (x% a) + sin' (u% b)

v = % sin' (x% a) + cos' (u% b)

()

* w = R!1(z % b)

Assume for simplicity the curve is (locally) a graph:

C = {u = f(x)}

=' extensions to parametrized curves are straightforward

Prolong the action to Jn via implicit di"erentiation:

y = cos' (x% a) + sin' (u% b)

v = % sin' (x% a) + cos' (u% b)

vy =% sin' + ux cos'

cos' + ux sin'

vyy =uxx

(cos' + ux sin' )3

vyyy =(cos' + ux sin' )uxxx % 3u2

xx sin'

(cos' + ux sin' )5

...

Normalization: r = dim G = 3

y = cos' (x% a) + sin' (u% b) = 0

v = % sin' (x% a) + cos' (u% b) = 0


cos' + ux sin'= 0

vyy =uxx

(cos' + ux sin' )3

vyyy =(cos' + ux sin' )uxxx % 3u2

xx sin'

(cos' + ux sin' )5

...

Solve for the group parameters:

y = cos' (x% a) + sin' (u% b) = 0

v = % sin' (x% a) + cos' (u% b) = 0


cos' + ux sin'= 0

=' Left moving frame % : J1 %& SE(2)

a = x b = u ' = tan!1 ux

a = x b = u ' = tan!1 ux

Di"erential invariants

vyy =uxx

(cos' + ux sin' )3$%& ( =

uxx

(1 + u2x)3/2

vyyy = · · · $%&d(

ds=

(1 + u2x)uxxx % 3uxu2

xx

(1 + u2x)3

vyyyy = · · · $%&d2(

ds2% 3(3 = · · ·

=' recurrence formulae

Contact invariant one-form — arc length

dy = (cos'+ ux sin') dx $%& ds =+

1 + u2x dx

Dual invariant di"erential operator— arc length derivative

d

dy=

1

cos' + ux sin'

d

dx$%&

d

ds=

1+

1 + u2x

d

dx

Theorem. All di"erential invariants are functions ofthe derivatives of curvature with respect toarc length:

(,d(

ds,

d2(

ds2, · · ·

The Classical Picture:

z

t

n

Moving frame % : (x, u, ux) $%& (R, a) # SE(2)

R =1

+1 + u2

x

%1 %ux

ux 1

&

= ( t, n ) a =

%xu

&

Frenet frame

t =dx

ds=

%xs

ys

&

, n = t$ =

%% ys

xs

&

.

Frenet equations = Pulled-back Maurer–Cartan forms:

dx

ds= t,

dt

ds= (n,

dn

ds= %( t.

Equi-a!ne Curves G = SA(2)

z $%& A z + b A # SL(2), b # R2

Invert for left moving frame:

y = " (x% a)% $ (u% b)

v = % ! (x% a) + # (u% b)

()

* w = A!1(z % b)

# " % $ ! = 1

Prolong to J3 via implicit di"erentiation

dy = (" % $ ux) dx Dy =1

" % $ ux

Dx

Prolongation:

y = " (x% a)% $ (u% b)

v = % ! (x% a) + # (u% b)

vy = %! % # ux

" % $ ux

vyy = %uxx

(" % $ ux)3

vyyy = %(" % $ ux) uxxx + 3$ u2

xx

(" % $ ux)5

vyyyy = %uxxxx(" % $ ux)2 + 10$ (" % $ ux)uxx uxxx + 15$2 u3

xx

(" % $ ux)7

vyyyyy = . . .

Normalization: r = dim G = 5

y = " (x% a)% $ (u% b) = 0

v = % ! (x% a) + # (u% b) = 0

vy = %! % # ux

" % $ ux

= 0

vyy = %uxx

(" % $ ux)3= 1

vyyy = %(" % $ ux) uxxx + 3$ u2

xx

(" % $ ux)5= 0

vyyyy = %uxxxx(" % $ ux)2 + 10$ (" % $ ux)uxx uxxx + 15$2 u3

xx

(" % $ ux)7

vyyyyy = . . .

Equi-a!ne Moving Frame

% : (x, u, ux, uxx, uxxx) $%& (A,b) # SA(2)

A =

%# $! "

&

=

%3

+uxx % 1

3 u!5/3xx uxxx

ux3

+uxx u!1/3

xx % 13 u!5/3

xx uxxx

&

b =

%ab

&

=

%xu

&

Nondegeneracy condition: uxx )= 0.

Totally Singular Submanifolds

Definition. A p-dimensional submanifold N " M istotally singular if G(n) does not act freely on jnN for any n . 0.

Theorem. N is totally singular if and only if its symme-try group GN = { g | g · N " N } has dimension > p, and so GN

does not act freely on N itself.

Thus, the totally singular submanifolds are the only onesthat do not admit a moving frame of any order.

In equi-a!ne geometry, only the straight lines ( uxx / 0 )are totally singular since they admit a three-dimensional equi-a!ne symmetry group.

Equi-a!ne arc length

dy = (" % $ ux) dx $%& ds = 3

+uxx dx

Equi-a!ne curvature

vyyyy $%& ( =5uxxuxxxx % 3u2

xxx

9u8/3xx

vyyyyy $%&d(

ds

vyyyyyy $%&d2(

ds2% 5(2

The Classical Picture:

z

t

n

A =

%3

+uxx % 1

3 u!5/3xx uxxx

ux3

+uxx u!1/3

xx % 13 u!5/3

xx uxxx

&

= ( t, n ) b =

%xu

&

Frenet frame

t =dz

ds, n =

d2z

ds2.

Frenet equations = Pulled-back Maurer–Cartan forms:

dz

ds= t,

dt

ds= n,

dn

ds= ( t.

Equivalence & Invariants

• Equivalent submanifolds N * Nmust have the same invariants: I = I.

Constant invariants provide immediate information:

e.g. ( = 2 (' ( = 2

Non-constant invariants are not useful in isolation,because an equivalence map can drastically alter thedependence on the submanifold parameters:

e.g. ( = x3 versus ( = sinhx

However, a functional dependency or syzygy amongthe invariants is intrinsic:

e.g. (s = (3 % 1 (' (s = (3 % 1

• Universal syzygies — Gauss–Codazzi

• Distinguishing syzygies.

Theorem. (Cartan) Two submanifolds are (locally)equivalent if and only if they have identicalsyzygies among all their di"erential invariants.

Finiteness of Generators and Syzygies

0 There are, in general, an infinite number of di"er-ential invariants and hence an infinite numberof syzygies must be compared to establishequivalence.

1 But the higher order syzygies are all consequencesof a finite number of low order syzygies!

Example — Plane Curves

If non-constant, both ( and (s depend on a singleparameter, and so, locally, are subject to a syzygy:

(s = H(() (+)

But then

(ss =d

dsH(() = H %(()(s = H %(()H(()

and similarly for (sss, etc.

Consequently, all the higher order syzygies are generatedby the fundamental first order syzygy (+).

Thus, for Euclidean (or equi-a!ne or projective or . . . )plane curves we need only know a single syzygy between ( and(s in order to establish equivalence!

The Basis Theorem

Theorem. The di"erential invariant algebra I isgenerated by a finite number of di"erential invariants

I1, . . . , I!

and p = dimN invariant di"erential operators

D1, . . . ,Dp

meaning that every di"erential invariant can be locallyexpressed as a function of the generating invariantsand their invariant derivatives:

DJI" = Dj1Dj2

· · · Djn

I".

=' Lie, Tresse, Ovsiannikov, Kumpera

) Moving frames provides a constructive proof.

Signature Curves

Definition. The signature curve S " R2 of a curveC " R2 is parametrized by the two lowest orderdi"erential invariants

S =

' %

( ,d(

ds

& ,

" R2

Theorem. Two regular curves C and C are equiva-lent:

C = g · Cif and only if their signature curves are identical:

S = S

Symmetry and Signature

Theorem. The dimension of the symmetry group

GN = { g | g · N " N }

of a nonsingular submanifold N " M equals thecodimension of its signature:

dimGN = dim N % dimS

Corollary. For a nonsingular submanifold N " M ,

0 , dim GN , dim N

=' Only totally singular submanifolds can have largersymmetry groups!

Maximally Symmetric Submanifolds

Theorem. The following are equivalent:

• The submanifold N has a p-dimensional symmetry group

• The signature S degenerates to a point: dimS = 0

• The submanifold has all constant di"erential invariants

• N = H · {z0} is the orbit of a p-dimensional subgroup H " G

=' Euclidean geometry: circles, lines, helices, spheres, cylinders, planes, . . .

=' Equi-a!ne plane geometry: conic sections.

=' Projective plane geometry: W curves (Lie & Klein)

Discrete Symmetries

Definition. The index of a submanifold N equalsthe number of points in N which map to a genericpoint of its signature:

&N = min-

# $!1{w}... w # S

/

=' Self–intersections

Theorem. The cardinality of the symmetry group ofa submanifold N equals its index &N .

=' Approximate symmetries

The Index

$

%&

N S

The Curve x = cos t + 15 cos2 t, y = sin t + 1

10 sin2 t

-0.5 0.5 1

-0.5

0.5

1

The Original Curve

0.25 0.5 0.75 1 1.25 1.5 1.75 2

-2

-1

0

1

2

Euclidean Signature

0.5 1 1.5 2 2.5

-6

-4

-2

2

4

A!ne Signature

The Curve x = cos t + 15 cos2 t, y = 1

2 x + sin t + 110 sin2 t

-0.5 0.5 1

-1

-0.5

0.5

1

The Original Curve

0.5 1 1.5 2 2.5 3 3.5 4

-7.5

-5

-2.5

0

2.5

5

7.5

Euclidean Signature

0.5 1 1.5 2 2.5

-6

-4

-2

2

4

A!ne Signature

Canine Left Ventricle Signature

Original Canine HeartMRI Image

Boundary of Left Ventricle

Smoothed Ventricle Signature

10 20 30 40 50 60

20

30

40

50

60

10 20 30 40 50 60

20

30

40

50

60

10 20 30 40 50 60

20

30

40

50

60

-0.15 -0.1 -0.05 0.05 0.1 0.15 0.2

-0.06

-0.04

-0.02

0.02

0.04

0.06

-0.15 -0.1 -0.05 0.05 0.1 0.15 0.2

-0.06

-0.04

-0.02

0.02

0.04

0.06

-0.15 -0.1 -0.05 0.05 0.1 0.15 0.2

-0.06

-0.04

-0.02

0.02

0.04

0.06

Evolution of Invariants and Signatures

Basic question: If the submanifold evolves according toan invariant evolution equation, how do its di"erentialinvariants & signatures evolve?

Theorem. Under the curve shortening flow Ct = %(n,the signature curve (s = H(t,() evolves according to theparabolic equation

*H

*t= H2 H"" % (3H" + 4(2H

Signature Metrics

• Hausdor"

• Monge–Kantorovich transport

• Electrostatic repulsion

• Latent semantic analysis

• Histograms

• Gromov–Hausdor" & Gromov–Wasserstein

Signatures

s

(

Classical Signature%&

Original curve(

(s

Di"erential invariant signature

Occlusions

s

(

Classical Signature%&

Original curve(

(s

Di"erential invariant signature

The Ba"er Jigsaw Puzzle

The Ba"er Solved

=' Dan Ho"

Classical Invariant Theory

M = R2 \ {u = 0}

G = GL(2) =

' %# $! "

& ..... % = # " % $ ! )= 0

,

(x, u) $%&%#x + $

!x + ",

u

(!x + ")n

&

n )= 0, 1

Prolongation:

y =#x + $

! x + "+ = ! x + "

v = +!n u % = # " % $ !

vy =+ ux % n ! u

% +n!1

vyy =+2 uxx % 2(n% 1)! + ux + n(n% 1)!2 u

%2 +n!2

vyyy = · · ·

Normalization:

y =# x + $

! x + "= 0 + = ! x + "

v = +!n u = 1 % = # " % $ !

vy =+ ux % n ! u

% +n!1= 0

vyy =+2 uxx % 2(n% 1) ! + ux + n(n% 1)!2 u

%2 +n!2=

1

n(n% 1)

vyyy = · · ·

Moving frame:

# = u(1!n)/n2

H $ = %x u(1!n)/n2

H

! = 1n u(1!n)/n " = u1/n % 1

n xu(1!n)/n

Hessian:

H = n(n% 1)u uxx % (n% 1)2u2x )= 0

Note: H / 0 if and only if Q(x) = (a x + b)n

=' Totally singular forms

Di"erential invariants:

vyyy $%&J

n2(n% 1)* ( vyyyy $%&

K + 3(n% 2)

n3(n% 1)*

d(

ds

Absolute rational covariants:

J2 =T 2

H3K =

U

H2

H = 12(Q, Q)(2) = n(n% 1)QQ%% % (n% 1)2Q%2 ! QxxQyy %Q2

xy

T = (Q, H)(1) = (2n% 4)Q%H % nQH % ! QxHy %QyHx

U = (Q, T )(1) = (3n% 6)Q%T % nQT % ! QxTy %QyTx

deg Q = n deg H = 2n% 4 deg T = 3n% 6 deg U = 4n% 8

Signatures of Binary Forms

Signature curve of a nonsingular binary form Q(x):

SQ =

'

(J(x)2, K(x)) =

%T (x)2

H(x)3,

U(x)

H(x)2

&,

Nonsingular : H(x) )= 0 and (J %(x), K %(x)) )= 0.

Theorem. Two nonsingular binary forms are equiva-lent if and only if their signature curves are identical.

Maximally Symmetric Binary Forms

Theorem. If u = Q(x) is a polynomial, then thefollowing are equivalent:

• Q(x) admits a one-parameter symmetry group

• T 2 is a constant multiple of H3

• Q(x) 3 xk is complex-equivalent to a monomial

• the signature curve degenerates to a single point

• all the (absolute) di"erential invariants of Q areconstant

• the graph of Q coincides with the orbit of aone-parameter subgroup

Symmetries of Binary Forms

Theorem. The symmetry group of a nonzero binary formQ(x) )/ 0 of degree n is:

• A two-parameter group if and only if H / 0 if and only ifQ is equivalent to a constant. =' totally singular

• A one-parameter group if and only if H )/ 0 and T 2 = cH3

if and only if Q is complex-equivalent to a monomial xk,with k )= 0, n. =' maximally symmetric

• In all other cases, a finite group whose cardinality equalsthe index of the signature curve, and is bounded by

&Q ,'

6n% 12 U = cH2

4n% 8 otherwise

Symmetry–Preserving Numerical Methods

• Invariant numerical approximations to di"erentialinvariants.

• Invariantization of numerical integration methods.

=' Structure-preserving algorithms

Numerical approximation to curvature

ab

cA

B

C

Heron’s formula

0((A, B,C) = 4%

abc= 4

+s(s% a)(s% b)(s% c)

abc

s =a + b + c

2— semi-perimeter

Invariantization of Numerical Schemes

=' Pilwon Kim

Suppose we are given a numerical scheme for integratinga di"erential equation, e.g., a Runge–Kutta Method for ordi-nary di"erential equations, or the Crank–Nicolson method forparabolic partial di"erential equations.

If G is a symmetry group of the di"erential equation, thenone can use an appropriately chosen moving frame to invari-antize the numerical scheme, leading to an invariant numeri-cal scheme that preserves the symmetry group. In challengingregimes, the resulting invariantized numerical scheme can, withan inspired choice of moving frame, perform significantly betterthan its progenitor.

Invariant Runge–Kutta schemes

uxx + xux % (x + 1)u = sinx, u(0) = ux(0) = 1.

Comparison of symmetry reduction and invariantization for

uxx + xux % (x + 1)u = sinx, u(0) = ux(0) = 1.

Invariantization of Crank–Nicolsonfor Burgers’ Equation

ut = ,uxx + u ux

The Calculus of Variations

I[u ] =1

L(x, u(n)) dx — variational problem

L(x, u(n)) — Lagrangian

To construct the Euler-Lagrange equations: E(L) = 0

• Take the first variation:

"(L dx) =$

#,J

*L

*u#J

"u#J dx

• Integrate by parts:

"(Ldx) =$

#,J

*L

*u#J

DJ("u#) dx

/$

#,J

(%D)J *L

*u#J

"u# dx =q$

#=1

E#(L) "u# dx

Invariant Variational Problems

According to Lie, any G–invariant variational problem canbe written in terms of the di"erential invariants:

I[u ] =1

L(x, u(n)) dx =1

P ( . . . DKI# . . . ) !

I1, . . . , I! — fundamental di"erential invariants

D1, . . . ,Dp — invariant di"erential operators

DKI# — di"erentiated invariants

! = -1 4 · · · 4 -p — invariant volume form

If the variational problem is G-invariant, so

I[u ] =1

L(x, u(n)) dx =1

P ( . . . DKI# . . . ) !

then its Euler–Lagrange equations admit G as a symmetrygroup, and hence can also be expressed in terms of the di"er-ential invariants:

E(L) 3 F ( . . . DKI# . . . ) = 0

Main Problem:

Construct F directly from P .

(P. Gri!ths, I. Anderson )

Planar Euclidean group G = SE(2)

( =uxx

(1 + u2x)3/2

— curvature (di"erential invariant)

ds =+

1 + u2x dx — arc length

D =d

ds=

1+

1 + u2x

d

dx— arc length derivative

Euclidean–invariant variational problem

I[u ] =1

L(x, u(n)) dx =1

P ((,(s,(ss, . . . ) ds

Euler-Lagrange equations

E(L) 3 F ((,(s,(ss, . . . ) = 0

Euclidean Curve Examples

Minimal curves (geodesics):

I[u ] =1

ds =1 +

1 + u2x dx

E(L) = %( = 0=' straight lines

The Elastica (Euler):

I[u ] =1

12 (

2 ds =1 u2

xx dx

(1 + u2x)5/2

E(L) = (ss + 12 (

3 = 0=' elliptic functions

General Euclidean–invariant variational problem

I[u ] =1

L(x, u(n)) dx =1

P ((,(s,(ss, . . . ) ds

To construct the invariant Euler-Lagrange equations:

Take the first variation:

"(P ds) =$

j

*P

*(j

"(j ds + P "(ds)

Invariant variation of curvature:

"( = A"("u) A" = D2 + (2

Invariant variation of arc length:

"(ds) = B("u) ds B = %(

=' moving frame recurrence formulae

The Recurrence Formula

For any function or di"erential form &:

d &(&) = &(d&) +r$

k=1

.k 4 & [vk(&)]

v1, . . . ,vr — basis for g — infinitesimal generators

.1, . . . , .r — dual invariantized Maurer–Cartan forms

) ) The .k are uniquely determined by the recurrenceformulae for the phantom di"erential invariants

d &(&) = &(d&) +r$

k=1

.k 4 & [vk(&)]

) ) ) All identities, commutation formulae, syzygies, etc.,among di"erential invariants and, more generally,the invariant variational bicomplex follow from thisuniversal recurrence formula by letting & range overthe basic functions and di"erential forms!

) ) ) Therefore, the entire structure of the di"erential invari-ant algebra and invariant variational bicomplex can becompletely determined using only linear di"erential al-gebra; this does not require explicit formulas for themoving frame, the di"erential invariants, the invariantdi"erential forms, or the group transformations!

Integrate by parts:

"(P ds) / [ E(P )A("u)%H(P )B("u) ] ds

/ [A+E(P )% B+H(P ) ] "u ds = E(L) "u ds

Invariantized Euler–Lagrange expression

E(P ) =&$

n=0

(%D)n *P

*(n

D =d

ds

Invariantized Hamiltonian

H(P ) =$

i>j

(i!j (%D)j *P

*(i

% P

Euclidean–invariant Euler-Lagrange formula

E(L) = A+E(P )% B+H(P ) = (D2 + (2) E(P ) + (H(P ) = 0.

The Elastica:

I[u ] =1

12 (

2 ds P = 12 (

2

E(P ) = ( H(P ) = %P = % 12 (

2

E(L) = (D2 + (2) ( + ( (% 12 (

2 ) = (ss + 12 (

3 = 0

Evolution of Invariants and Signatures

G — Lie group acting on R2

C(t) — parametrized family of plane curves

G–invariant curve flow:

dC

dt= V = I t + J n

• I, J — di"erential invariants

• t — “unit tangent”

• n — “unit normal”

• The tangential component I t only a"ects the underlyingparametrization of the curve. Thus, we can set I to beanything we like without a"ecting the curve evolution.

Normal Curve Flows

Ct = J n

Examples — Euclidean–invariant curve flows

• Ct = n — geometric optics or grassfire flow;

• Ct = (n — curve shortening flow;

• Ct = (1/3 n — equi-a!ne invariant curve shortening flow:Ct = nequi!a!ne ;

• Ct = (s n — modified Korteweg–deVries flow;

• Ct = (ss n — thermal grooving of metals.

Intrinsic Curve Flows

Theorem. The curve flow generated by

v = I t + J n

preserves arc length if and only if

B(J) + D I = 0.

D — invariant arc length derivative

B — invariant arc length variation

"(ds) = B("u) ds

Normal Evolution of Di#erential Invariants

Theorem. Under a normal flow Ct = J n,

*(

*t= A"(J),

*(s

*t= A"s

(J).

Invariant variations:

"( = A"("u), "(s = A"s

("u).

A" = A — invariant variation of curvature;

A"s

= DA + ((s — invariant variation of (s.

Euclidean–invariant Curve Evolution

Normal flow: Ct = J n

*(

*t= A"(J) = (D2 + (2) J,

*(s

*t= A"s

(J) = (D3 + (2D + 3((s)J.

Warning : For non-intrinsic flows, *t and *s do not commute!

Theorem. Under the curve shortening flow Ct = %(n,the signature curve (s = H(t,() evolves according to theparabolic equation

*H

*t= H2 H"" % (3H" + 4(2H

Smoothed Ventricle Signature

10 20 30 40 50 60

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50

60

10 20 30 40 50 60

20

30

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10 20 30 40 50 60

20

30

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-0.15 -0.1 -0.05 0.05 0.1 0.15 0.2

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-0.15 -0.1 -0.05 0.05 0.1 0.15 0.2

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-0.15 -0.1 -0.05 0.05 0.1 0.15 0.2

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0.06

Intrinsic Evolution of Di#erential Invariants

Theorem.

Under an arc-length preserving flow,

(t = R(J) where R = A% (sD!1B (+)

In surprisingly many situations, (*) is a well-known integrableevolution equation, and R is its recursion operator!

=' Hasimoto

=' Langer, Singer, Perline

=' Marı–Be"a, Sanders, Wang

=' Qu, Chou, Anco, and many more ...

Euclidean plane curves

G = SE(2) = SO(2) ! R2

A = D2 + (2 B = %(

R = A% (sD!1B = D2 + (2 + (sD

!1 · (

(t = R((s) = (sss + 32 (

2(s

=' modified Korteweg-deVries equation

Equi-a!ne plane curves

G = SA(2) = SL(2) ! R2

A = D4 + 53 (D

2 + 53 (sD + 1

3 (ss + 49 (

2

B = 13 D

2 % 29 (

R = A% (sD!1B

= D4 + 53 (D

2 + 43 (sD + 1

3 (ss + 49 (

2 + 29 (sD

!1 · (

(t = R((s) = (5s + 53 ((sss + 5

3 (s(ss + 59 (

2(s

=' Sawada–Kotera equation

Recursion operator: 2R = R · (D2 + 13 ( + 1

3 (sD!1)

Euclidean space curves

G = SE(3) = SO(3) ! R3

A =

3

44445

D2s + ((2 % /2)

2/

(D2

s +3(/s % 2(s/

(2Ds +

(/ss % (s/s + 2(3/

(2

%2/Ds % /s

1

(D3

s %(s

(2D2

s +(2 % /2

(Ds +

(s/2 % 2(//s

(2

6

77778

B = (( 0 )

R = A%%(s

/s

&

D!1B%(t

/t

&

= R%(s

/s

&

=' vortex filament flow (Hasimoto)

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