Moving Framesand their Applications

Peter J. Olver

University of Minnesota

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

Technion, May, 2015

History of 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, . . .

Moving Frame — Space Curves

t

n

b

tangent normal binormal

t =dz

dsn =

zss" zss "

b = t# n

s — arc length

z — point on the curve

Frenet–Serret equations

dt

ds= !n

dn

ds= $! t+ " b

db

ds= $ " n

! — curvature " — torsion

Moving Frame — Space Curves

t

n

b

z

tangent normal binormal

t =dz

dsn =

zss" zss "

b = t# n

s — arc length

z — point on the curve

Frenet–Serret equations

dt

ds= !n

dn

ds= $! t+ " b

db

ds= $ " n

! — curvature " — torsion

“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 groups and groupoids

• 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 & symmetry detection

• 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?

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 in Mis 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

Geometric Construction

z

Oz

K

k

Normalization = choice of cross-section to the group orbits

Geometric Construction

z

Oz

K

k

g = #left(z)

Normalization = choice of cross-section to the group orbits

Geometric Construction

z

Oz

K

k

g = #right(z)

Normalization = choice of cross-section to the group orbits

K — cross-section to the group orbits

Oz — orbit through z & M

k & K -Oz — unique point in the intersection

• k is the canonical form of z

• the (nonconstant) coordinates of k are the fundamentalinvariants

g & G — unique group element mapping k to z=) freeness

Then #left(z) = g is a left moving frame: #left(h·z) = h·#left(z)

k = #left(z)!1 · z = #right(z) · z

Algebraic Construction

r = dimG . m = dimM

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 = dimG 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) producesthe fundamental invariants

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

=) These are the coordinates of the canonical formk & K.

Completeness of Invariants

Theorem. Every invariant I(z) can be (locally)uniquely written as a function of the fundamentalinvariants:

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”

Invariantization respects all algebraic operations:

$ [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

The Replacement Theorem

If I(z1, . . . , zm) is any invariant, then

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

This “Rewrite Rule” trivially proves that anyinvariant can be easily expressed (rewritten) in termsof the fundamental invariants!

The Rotation Group

G = SO(2) acting on R2

z = (x, u) '$( g · z = ( x cos%$ u sin% , x sin%+ u cos% )

=) Free on M = R2 \ {0}

Left moving frame:

w(g, z) = g!1 · z = (y, v)

y = x cos%+ u sin% v = $x sin%+ u cos%

Cross-section:

K = {u = 0, x > 0 }

Normalization equation:

v = $x sin%+ u cos% = 0

Left moving frame:

% = tan!1 u

x=) % = #(x, u) & SO(2)

Fundamental invariant

r = $(x) =/x2 + u2

Invariantization

$[F (x, u) ] = F (r, 0)

Prolongation

• The moving frame construction requires freeness of the groupaction

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

• Freeness typically fails because the dimension of the underlyingmanifold is not large enough, i.e., m < r = dimG.

• Thus, to make the action free, we must increase the dimen-sion of the space via some natural prolongation 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

• 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 Curves

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

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

To obtain the classical (left) moving frame we invert the grouptransformations:

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 = dimG = 3

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

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

vy =$ sin% + ux cos%

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

vy =$ sin% + ux cos%

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 — invariantization

vyy =uxx

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

uxx

(1 + u2x)

3/2= !

vyyy = · · · '$( $(uxxx) =(1 + u2

x)uxxx $ 3uxu2xx

(1 + u2x)

3=

d!

ds

vyyyy = · · · '$( $(uxxxx) = · · · =d2!

ds2$ 3!3

=) recurrence formulae

Invariant one-form — arc length

dy = (cos%+ ux sin%) dx '$( $(dx) =(1 + u2

x dx = ds

Dual invariant di"erential operator— arc length derivative

d

dy=

1

cos%+ ux sin%

d

dx'$(

d

ds=

1(1 + u2

x

d

dx

Theorem. All di"erential invariants are functions of thederivatives 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$ =

)$ ysxs

*

.

Frenet equations = Pulled-back Maurer–Cartan forms:

dx

ds= t,

dt

ds= !n,

dn

ds= $! t.

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

z '$( Az + 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 u3xx

(& $ ' ux)7

vyyyyy = . . .

Normalization: r = dimG = 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 u3xx

(& $ ' 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

!5/3xx uxxx

*

b =

)ab

*

=

)xu

*

Nondegeneracy condition: uxx *= 0.

Equi-a!ne arc length

dy = (& $ ' ux) dx '$( ds = $(dx) = 3

(uxx dx

Equi-a!ne curvature

vyyyy '$( $(uxxxx) =5 uxxuxxxx $ 3u2

xxx

9u8/3xx

= !

vyyyyy '$( $(uxxxxx) = · · · =d!

ds

vyyyyyy '$( $(uxxxxxx) = · · · =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

!5/3xx 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.

The Recurrence Formula

* While invariantization respects all algebraic operationsit does not commute with di"erentiation!

For any function or di"erential form #:

d $(#) = $(d#) +r+

k=1

+k 0 $ [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 0 $ [vk(#)]

* * * All identities, commutation formulae, syzygies, etc.,among di"erential invariants and, more generally, theinvariant di"erential forms follow from this universalrecurrence formula by letting # range over the basicfunctions 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!

The Basis Theorem

Theorem. The di"erential invariant algebraI(G) is generated by a finite number of di"erentialinvariants

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

· · · DjnI".

=) Lie, Tresse, Ovsiannikov, Kumpera

=) Moving frames provides a constructive proof.

TheDi"erential Invariant Algebra

Thus, remarkably, the structure of I(G) can be determinedwithout knowing the explicit formulae for either the movingframe, or the di"erential invariants, or the invariant di"er-ential operators!

The only required ingredients are the specification of the cross-section, and the standard formulae for the prolongedinfinitesimal generators.

Theorem. If G acts transitively on M , or if the infinitesimalgenerator coe!cients depend rationally in the coordinates,then all recurrence formulae are rational in the basicdi"erential invariants and so I(G) is a rational, non-commutative di"erential algebra.

Curves

Theorem. Let G be an ordinary# Lie group acting on the m-

dimensional manifold M . Then, locally, there exist m $ 1

generating di"erential invariants !1, . . . ,!m!1. Every other

di"erential invariant can be written as a function of the

generating di"erential invariants and their derivatives with

respect to the G-invariant arc length element ds.

# ordinary = transitive + no pseudo-stabilization.

=) m = 3 — curvature ! & torsion "

Minimal Generating Invariants

A set of di"erential invariants is a generating system if allother di"erential invariants can be written in terms of them andtheir invariant derivatives.

Euclidean space curves C % R3:• curvature ! and torsion "

Equi–a!ne space curves C % R3:• a!ne curvature ! and torsion "

Euclidean surfaces S % R3:• mean curvature H

* Gauss curvature K = $(D(4)H).

Equi–a!ne surfaces S % R3:• Pick invariant P .

Curves

Theorem. Let G be an ordinary# Lie group acting on the m-

dimensional manifold M . Then, locally, there exist m $ 1

generating di"erential invariants !1, . . . ,!m!1. Every other

di"erential invariant can be written as a function of the

generating di"erential invariants and their derivatives with

respect to the G-invariant arc length element ds.

# ordinary = transitive + no pseudo-stabilization.

=) m = 3 — curvature ! & torsion "

Euclidean Surfaces

Theorem.

The algebra of Euclidean di"erential invariants fora non-degenerate surface is generated by themean curvature through invariant di"erentiation.

K = $(H,D1H,D2H, . . . )

Euclidean Surfaces

Theorem.

The algebra of Euclidean di"erential invariants fora non-degenerate surface is generated by themean curvature through invariant di"erentiation.

K = $(H,D1H,D2H, . . . )

Euclidean ProofCommutation relation:

[D1,D2 ] = D1D2 $D2D1 = Z2D1 $ Z1D2,

Commutator invariants:

Z1 =D1!2

!1 $ !2

Z2 =D2!1

!2 $ !1

Codazzi relation:

K = !1!2 = $ (D1 + Z1)Z1 $ (D2 + Z2)Z2

=) Gauss’ Theorema Egregium

=) (Guggenheimer)

Euclidean ProofCommutation relation:

[D1,D2 ] = D1D2 $D2D1 = Z2D1 $ Z1D2,

Commutator invariants:

Z1 =D1!2

!1 $ !2

Z2 =D2!1

!2 $ !1

Codazzi relation:

K = !1!2 = $ (D1 + Z1)Z1 $ (D2 + Z2)Z2

=) Gauss’ Theorema Egregium

(Guggenheimer)

Euclidean ProofCommutation relation:

[D1,D2 ] = D1D2 $D2D1 = Z2D1 $ Z1D2,

Commutator invariants:

Z1 =D1!2

!1 $ !2

Z2 =D2!1

!2 $ !1

Codazzi relation:

K = !1!2 = $ (D1 + Z1)Z1 $ (D2 + Z2)Z2

=) Gauss’ Theorema Egregium

(Guggenheimer)

To determine the commutator invariants:

D1D2H $D2D1H = Z2D1H $ Z1D2H

D1D2DJH $D2D1DJH = Z2D1DJH $ Z1D2DJH(,)

Nondegenerate surface:

det

)D1H D2H

D1DJH D2DJH

*

*= 0,

Solve (,) for Z1, Z2 in terms of derivatives of H.

Q.E.D.

* A surface is mean curvature degenerate ifDjH = Fj(H) for j = 1, 2.

Totally umbilic and constant mean curvature surfaces,including minimal surfaces are degenerate. Geometry?

• 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

Joint Invariants

A joint invariant is an invariant of the k-foldCartesian product action of G on M # · · ·#M :

I(g · z1, . . . , g · zk) = I(z1, . . . , zk)

A joint di"erential invariant or semi-di"erentialinvariant is an invariant depending on the derivativesat several points z1, . . . , zk & N on the submanifold:

I(g · z(n)1 , . . . , g · z(n)k ) = I(z(n)1 , . . . , z(n)k )

Joint Euclidean Invariants

Theorem. Every joint Euclidean invariant is afunction of the interpoint distances

d(zi, zj) = " zi $ zj "

zi

zj

Joint Equi–A!ne Invariants

Theorem. Every planar joint equi–a!ne invariant isa function of the triangular areas

[ i j k ] = 12 (zi $ zj) 0 (zi $ zk)

zi

zj

zk

Joint Projective Invariants

Theorem. Every planar joint projective invariant isa function of the planar cross-ratios

[ zi, zj, zk, zl, zm ] =AB

C D

A B

C

D

Projective joint di"erential invariant:— tangent triangle ratio

z0 z1

z2

A

B C

z0 z1

z2

D

E

F

ABC

DE F

• 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

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

,!(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,then one can use an appropriately chosen moving frame toinvariantize the numerical scheme, leading to an invariant nu-merical scheme that preserves the symmetry group. In challeng-ing regimes, the resulting invariantized numerical scheme can,with an inspired choice of moving frame, perform significantlybetter than its progenitor.

! " # $ % &

!'

!(

!)

!&

!%

!$

*

+,-"!./011/2

3,4567

78894!

:894!

7894!

Invariant Runge–Kutta schemes

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

! " # $ % &!'

!(

!)

!&

!%

!$

!#

!"

!

*

+,-"!./011/2

34534346,76389:;8960<7

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 + uux

! !"# $!$"#

!$

!!"#

!

!"#

$

! !"# $!$"#

!$

!!"#

!

!"#

$

! !"# $!$"#

!$

!!"#

!

!"#

$

! !"# $!$"#

!$

!!"#

!

!"#

$

! !"# $!$"#

!$

!!"#

!

!"#

$

! !"# $!$"#

!$

!!"#

!

!"#

$

The Calculus of Variations

I[u ] =-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:

&(Ldx) =+

$,J

-L

-u$J

&u$J dx

• Integrate by parts:

&(Ldx) =+

$,J

-L

-u$J

DJ(&u$) dx

1+

$,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 ] =-L(x, u(n)) dx =

-P ( . . . DKI$ . . . ) !

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

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

DKI$ — di"erentiated invariants

! = .1 0 · · · 0 .p — invariant volume form

If the variational problem is G-invariant, so

I[u ] =-L(x, u(n)) dx =

-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) 2 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 + u2

x dx — arc length

D =d

ds=

1(1 + u2

x

d

dx— arc length derivative

Euclidean–invariant variational problem

I[u ] =-L(x, u(n)) dx =

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

Euler-Lagrange equations

E(L) 2 F (!,!s,!ss, . . . ) = 0

Euclidean Curve Examples

Minimal curves (geodesics):

I[u ] =-

ds =- (

1 + u2x dx

E(L) = $! = 0=) straight lines

The Elastica (Euler):

I[u ] =-

12 !

2 ds =- u2

xx dx

(1 + u2x)

5/2

E(L) = !ss +12 !

3 = 0=) elliptic functions

General Euclidean–invariant variational problem

I[u ] =-L(x, u(n)) dx =

-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

Integrate by parts:

&(P ds) 1 [ E(P )A(&u)$H(P )B(&u) ] ds1 [A,E(P )$ 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 ] =-

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= H2H"" $ !3H" + 4!2H

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

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 +

53 !s!ss +

59 !

2!s

=) Sawada–Kotera equation

Recursion operator: .R = R · (D2 + 13 !+ 1

3 !sD!1)

Euclidean space curves

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

A =

/

00001

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

2

33334

B = (! 0 )

R = A$)!s

"s

*

D!1B)!t

"t

*

= R)!s

"s

*

=) vortex filament flow (Hasimoto)