Dual invariant differential operator — arc length derivative d 1 d d 1 d dy = �− → ds = � cos φ + u x sin φ dx dx 1 + u 2 x All differential invariants are functions of Theorem. the derivatives of curvature with respect to arc length: d 2 κ dκ · · · κ, ds , ds 2 ,
n The Classical Picture: t z Moving frame ρ : ( x, u, u x ) �− → ( R, a ) ∈ SE(2) � � � � 1 1 − u x x R = = ( t , n ) a = � u x 1 u 1 + u 2 x
Frenet frame � � � � t = d x − y s x s n = t ⊥ = ds = , . y s x s Frenet equations = Pulled-back Maurer–Cartan forms: d x d t d n ds = t , ds = κ n , ds = − κ t .
G = SA(2) Equi-affine Curves b ∈ R 2 z �− → A z + b A ∈ SL(2) , Invert for left moving frame: y = δ ( x − a ) − β ( u − b ) w = A − 1 ( z − b ) v = − γ ( x − a ) + α ( u − b ) α δ − β γ = 1 Prolong to J 3 via implicit differentiation 1 dy = ( δ − β u x ) dx D y = D x δ − β u x
Prolongation: y = δ ( x − a ) − β ( u − b ) v = − γ ( x − a ) + α ( u − b ) v y = − γ − α u x δ − β u x u xx v yy = − ( δ − β u x ) 3 v yyy = − ( δ − β u x ) u xxx + 3 β u 2 xx ( δ − β u x ) 5 v yyyy = − u xxxx ( δ − β u x ) 2 + 10 β ( δ − β u x ) u xx u xxx + 15 β 2 u 3 xx ( δ − β u x ) 7 v yyyyy = . . .
Normalization: r = dim G = 5 y = δ ( x − a ) − β ( u − b ) = 0 v = − γ ( x − a ) + α ( u − b ) = 0 v y = − γ − α u x = 0 δ − β u x u xx v yy = − ( δ − β u x ) 3 = 1 v yyy = − ( δ − β u x ) u xxx + 3 β u 2 xx = 0 ( δ − β u x ) 5 v yyyy = − u xxxx ( δ − β u x ) 2 + 10 β ( δ − β u x ) u xx u xxx + 15 β 2 u 3 xx ( δ − β u x ) 7 v yyyyy = . . .
Equi-affine Moving Frame ρ : ( x, u, u x , u xx , u xxx ) �− → ( A, b ) ∈ SA(2) � � � � � − 1 3 u − 5 / 3 3 u xx u xxx α β xx A = = � γ δ u − 1 / 3 − 1 3 u − 5 / 3 u x 3 u xx u xxx xx xx � � � � a x b = = b u Nondegeneracy condition: u xx � = 0.
Totally Singular Submanifolds Definition. A p -dimensional submanifold N ⊂ M is totally singular if G ( n ) does not act freely on j n N for any n ≥ 0. Theorem. N is totally singular if and only if its symme- try group G N = { g | g · N ⊂ N } has dimension > p , and so G N does not act freely on N itself. Thus, the totally singular submanifolds are the only ones that do not admit a moving frame of any order. In equi-affine geometry, only the straight lines ( u xx ≡ 0 ) are totally singular since they admit a three-dimensional equi- affine symmetry group.
Equi-affine arc length � dy = ( δ − β u x ) dx �− → ds = u xx dx 3 Equi-affine curvature κ = 5 u xx u xxxx − 3 u 2 xxx v yyyy �− → 9 u 8 / 3 xx dκ v yyyyy �− → ds d 2 κ ds 2 − 5 κ 2 v yyyyyy �− →
n The Classical Picture: t z � � � � � − 1 3 u − 5 / 3 3 u xx u xxx x xx A = = ( t , n ) b = � u − 1 / 3 − 1 3 u − 5 / 3 u u x 3 u xx u xxx xx xx
Frenet frame n = d 2 z t = dz ds , ds 2 . Frenet equations = Pulled-back Maurer–Cartan forms: dz d t d n ds = t , ds = n , ds = κ t .
Equivalence & Invariants • Equivalent submanifolds N ≈ N must have the same invariants: I = I . Constant invariants provide immediate information: e.g. κ = 2 ⇐ ⇒ κ = 2 Non-constant invariants are not useful in isolation, because an equivalence map can drastically alter the dependence on the submanifold parameters: κ = x 3 e.g. versus κ = sinh x
Equivalence & Invariants • Equivalent submanifolds N ≈ N must have the same invariants: I = I . Constant invariants provide immediate information: e.g. κ = 2 ⇐ ⇒ κ = 2 Non-constant invariants are not useful in isolation, because an equivalence map can drastically alter the dependence on the submanifold parameters: κ = x 3 e.g. versus κ = sinh x
Equivalence & Invariants • Equivalent submanifolds N ≈ N must have the same invariants: I = I . Constant invariants provide immediate information: e.g. κ = 2 ⇐ ⇒ κ = 2 Non-constant invariants are not useful in isolation, because an equivalence map can drastically alter the dependence on the submanifold parameters: κ = x 3 e.g. versus κ = sinh x
However, a functional dependency or syzygy among the invariants is intrinsic: κ s = κ 3 − 1 s = κ 3 − 1 e.g. ⇐ ⇒ κ ¯ • Universal syzygies — Gauss–Codazzi • Distinguishing syzygies. Theorem. (Cartan) Two submanifolds are (locally) equivalent if and only if they have identical syzygies among all their differential invariants.
However, a functional dependency or syzygy among the invariants is intrinsic: κ s = κ 3 − 1 s = κ 3 − 1 e.g. ⇐ ⇒ κ ¯ • Universal syzygies — Gauss–Codazzi • Distinguishing syzygies. Theorem. (Cartan) Two submanifolds are (locally) equivalent if and only if they have identical syzygies among all their differential invariants.
However, a functional dependency or syzygy among the invariants is intrinsic: κ s = κ 3 − 1 s = κ 3 − 1 e.g. ⇐ ⇒ κ ¯ • Universal syzygies — Gauss–Codazzi • Distinguishing syzygies. Theorem. (Cartan) Two submanifolds are (locally) equivalent if and only if they have identical syzygies among all their differential invariants.
Finiteness of Generators and Syzygies ♠ There are, in general, an infinite number of differ- ential invariants and hence an infinite number of syzygies must be compared to establish equivalence. ♥ But the higher order syzygies are all consequences of a finite number of low order syzygies !
Example — Plane Curves If non-constant, both κ and κ s depend on a single parameter, and so, locally, are subject to a syzygy: κ s = H ( κ ) ( ∗ ) But then κ ss = d ds H ( κ ) = H � ( κ ) κ s = H � ( κ ) H ( κ ) and similarly for κ sss , etc. Consequently, all the higher order syzygies are generated by the fundamental first order syzygy ( ∗ ). Thus, for Euclidean (or equi-affine or projective or . . . ) plane curves we need only know a single syzygy between κ and κ s in order to establish equivalence !
Signature Curves The signature curve S ⊂ R 2 of a curve Definition. C ⊂ R 2 is parametrized by the two lowest order differential invariants � � � � κ , dκ R 2 S = ⊂ ds Two regular curves C and C are equiva- Theorem. lent: C = g · C if and only if their signature curves are identical: S = S
Signature Curves The signature curve S ⊂ R 2 of a curve Definition. C ⊂ R 2 is parametrized by the two lowest order differential invariants � � � � κ , dκ R 2 S = ⊂ ds Two regular curves C and C are equiva- Theorem. lent: C = g · C if and only if their signature curves are identical: S = S
Symmetry and Signature The dimension of the symmetry group Theorem. G N = { g | g · N ⊂ N } of a nonsingular submanifold N ⊂ M equals the codimension of its signature: dim G N = dim N − dim S For a nonsingular submanifold N ⊂ M , Corollary. 0 ≤ dim G N ≤ dim N = ⇒ Only totally singular submanifolds can have larger symmetry groups !
Maximally Symmetric Submanifolds The following are equivalent: Theorem. • The submanifold N has a p -dimensional symmetry group • The signature S degenerates to a point: dim S = 0 • The submanifold has all constant differential invariants • N = H ·{ z 0 } is the orbit of a p -dimensional subgroup H ⊂ G = ⇒ Euclidean geometry: circles, lines, helices, spheres, cylinders, planes, . . . = ⇒ Equi-affine plane geometry: conic sections. = ⇒ Projective plane geometry: W curves ( Lie & Klein )
Discrete Symmetries The index of a submanifold N equals Definition. the number of points in N which map to a generic point of its signature: � � � � # Σ − 1 { w } ι N = min � w ∈ S = ⇒ Self–intersections The cardinality of the symmetry group of Theorem. a submanifold N equals its index ι N . = ⇒ Approximate symmetries
The Index Σ − → S N
10 sin 2 t 5 cos 2 t , y = sin t + 1 The Curve x = cos t + 1 1 2 4 2 1 0.5 0.5 1 1.5 2 2.5 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 1 -0.5 0.5 -2 -1 -0.5 -4 -2 -6 The Original Curve Euclidean Signature Affine Signature
10 sin 2 t 5 cos 2 t , y = 1 The Curve x = cos t + 1 2 x + sin t + 1 4 7.5 1 5 2 0.5 2.5 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 3 3.5 4 -0.5 0.5 1 -2.5 -2 -5 -0.5 -4 -7.5 -1 -6 The Original Curve Euclidean Signature Affine Signature
Canine Left Ventricle Signature Original Canine Heart Boundary of Left Ventricle MRI Image
Smoothed Ventricle Signature 60 60 60 50 50 50 40 40 40 30 30 30 20 20 20 10 20 30 40 50 60 10 20 30 40 50 60 10 20 30 40 50 60 0.06 0.06 0.06 0.04 0.04 0.04 0.02 0.02 0.02 -0.15 -0.1 -0.05 0.05 0.1 0.15 0.2 -0.15 -0.1 -0.05 0.05 0.1 0.15 0.2 -0.15 -0.1 -0.05 0.05 0.1 0.15 0.2 -0.02 -0.02 -0.02 -0.04 -0.04 -0.04 -0.06 -0.06 -0.06
Evolution of Invariants and Signatures Basic question: If the submanifold evolves according to an invariant evolution equation, how do its differential invariants & signatures evolve? Under the curve shortening flow C t = − κ n , Theorem. the signature curve κ s = H ( t, κ ) evolves according to the parabolic equation ∂H ∂t = H 2 H κκ − κ 3 H κ + 4 κ 2 H
Signature Metrics • Hausdorff • Monge–Kantorovich transport • Electrostatic repulsion • Latent semantic analysis • Histograms • Gromov–Hausdorff & Gromov–Wasserstein
κ Signatures − → s Classical Signature κ s Original curve κ Differential invariant signature
κ Signatures − → s Classical Signature κ s Original curve κ Differential invariant signature
κ Occlusions − → s Classical Signature κ s Original curve κ Differential invariant signature
The Baffler Jigsaw Puzzle
The Baffler Solved = ⇒ Dan Hoff
Symmetry–Preserving Numerical Methods • Invariant numerical approximations to differential invariants. • Invariantization of numerical integration methods. = ⇒ Structure-preserving algorithms
Numerical approximation to curvature B b C a c A Heron’s formula � s ( s − a )( s − b )( s − c ) κ ( A, B, C ) = 4 ∆ abc = 4 � abc s = a + b + c — semi-perimeter 2
Invariantization of Numerical Schemes = ⇒ Pilwon Kim Suppose we are given a numerical scheme for integrating a differential equation, e.g., a Runge–Kutta Method for ordi- nary differential equations, or the Crank–Nicolson method for parabolic partial differential equations. If G is a symmetry group of the differential equation, then one 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 challenging regimes, the resulting invariantized numerical scheme can, with an inspired choice of moving frame, perform significantly better than its progenitor.
Invariant Runge–Kutta schemes u xx + x u x − ( x + 1) u = sin x, u (0) = u x (0) = 1 .
Comparison of symmetry reduction and invariantization for u xx + x u x − ( x + 1) u = sin x, u (0) = u x (0) = 1 .
Invariantization of Crank–Nicolson for Burgers’ Equation u t = ε u xx + u u x
The Calculus of Variations � L ( x, u ( n ) ) d x — variational problem I [ u ] = L ( x, u ( n ) ) — Lagrangian To construct the Euler-Lagrange equations: E ( L ) = 0 • Take the first variation: � ∂L δu α δ ( L d x ) = J d x ∂u α J α,J • Integrate by parts: ∂L � D J ( δu α ) d x δ ( L d x ) = ∂u α J α,J q � ( − D ) J ∂L � δu α d x = E α ( L ) δu α d x ≡ ∂u α J α,J α =1
Invariant Variational Problems According to Lie, any G –invariant variational problem can be written in terms of the differential invariants: � � P ( . . . D K I α . . . ) ω L ( x, u ( n ) ) d x = I [ u ] = I 1 , . . . , I � — fundamental differential invariants D 1 , . . . , D p — invariant differential operators D K I α — differentiated invariants ω = ω 1 ∧ · · · ∧ ω p — invariant volume form
If the variational problem is G -invariant, so � � P ( . . . D K I α . . . ) ω L ( x, u ( n ) ) d x = I [ u ] = then its Euler–Lagrange equations admit G as a symmetry group, and hence can also be expressed in terms of the differ- ential invariants: E ( L ) � F ( . . . D K I α . . . ) = 0 Main Problem: Construct F directly from P . ( P. Griffiths, I. Anderson )
Planar Euclidean group G = SE(2) u xx κ = — curvature (differential invariant) (1 + u 2 x ) 3 / 2 � 1 + u 2 ds = x dx — arc length 1 D = d d ds = — arc length derivative � dx 1 + u 2 x Euclidean–invariant variational problem � � L ( x, u ( n ) ) dx = I [ u ] = P ( κ, κ s , κ ss , . . . ) ds Euler-Lagrange equations E ( L ) � F ( κ, κ s , κ ss , . . . ) = 0
Euclidean Curve Examples Minimal curves (geodesics): � � � 1 + u 2 I [ u ] = ds = x dx E ( L ) = − κ = 0 = ⇒ straight lines The Elastica (Euler): � � u 2 xx dx 2 κ 2 ds = 1 I [ u ] = (1 + u 2 x ) 5 / 2 2 κ 3 = 0 E ( L ) = κ ss + 1 = ⇒ elliptic functions
General Euclidean–invariant variational problem � � L ( x, u ( n ) ) dx = I [ u ] = P ( κ, κ s , κ ss , . . . ) ds To construct the invariant Euler-Lagrange equations: Take the first variation: � ∂P δ ( P ds ) = δκ j ds + P δ ( ds ) ∂κ j j Invariant variation of curvature: A κ = D 2 + κ 2 δκ = A κ ( δu ) Invariant variation of arc length: δ ( ds ) = B ( δu ) ds B = − κ = ⇒ moving frame recurrence formulae
Integrate by parts: δ ( P ds ) ≡ [ E ( P ) A ( δu ) − H ( P ) B ( δu ) ] ds ≡ [ A∗E ( P ) − B∗H ( P ) ] δu ds = E ( L ) δu ds Invariantized Euler–Lagrange expression ∞ � ( −D ) n ∂P D = d E ( P ) = ∂κ n ds n =0 Invariantized Hamiltonian κ i − j ( −D ) j ∂P � H ( P ) = − P ∂κ i i>j Euclidean–invariant Euler-Lagrange formula E ( L ) = A∗E ( P ) − B∗H ( P ) = ( D 2 + κ 2 ) E ( P ) + κ H ( P ) = 0 .
The Elastica: � 2 κ 2 ds 1 P = 1 2 κ 2 I [ u ] = 2 κ 2 H ( P ) = − P = − 1 E ( P ) = κ E ( L ) = ( D 2 + κ 2 ) κ + κ ( − 1 2 κ 2 ) = κ ss + 1 2 κ 3 = 0
Evolution of Invariants and Signatures G — Lie group acting on R 2 C ( t ) — parametrized family of plane curves G –invariant curve flow: dC dt = V = I t + J n • I, J — differential invariants • t — “unit tangent” • n — “unit normal” • The tangential component I t only affects the underlying parametrization of the curve. Thus, we can set I to be anything we like without affecting the curve evolution.
Normal Curve Flows C t = J n Examples — Euclidean–invariant curve flows • C t = n — geometric optics or grassfire flow; • C t = κ n — curve shortening flow; • C t = κ 1 / 3 n — equi-affine invariant curve shortening flow: C t = n equi − affine ; • C t = κ s n — modified Korteweg–deVries flow; • C t = κ 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 δ ( ds ) = B ( δu ) ds
Normal Evolution of Differential Invariants Under a normal flow C t = J n , Theorem. ∂κ ∂κ s ∂t = A κ ( J ) , ∂t = A κ s ( J ) . Invariant variations: δκ = A κ ( δu ) , δκ s = A κ s ( δu ) . A κ = A — invariant variation of curvature; A κ s = D A + κ κ s — invariant variation of κ s .
Euclidean–invariant Curve Evolution Normal flow: C t = J n ∂κ ∂t = A κ ( J ) = ( D 2 + κ 2 ) J, ∂κ s ∂t = A κ s ( J ) = ( D 3 + κ 2 D + 3 κ κ s ) J. Warning : For non-intrinsic flows, ∂ t and ∂ s do not commute ! Under the curve shortening flow C t = − κ n , Theorem. the signature curve κ s = H ( t, κ ) evolves according to the parabolic equation ∂H ∂t = H 2 H κκ − κ 3 H κ + 4 κ 2 H
Smoothed Ventricle Signature 60 60 60 50 50 50 40 40 40 30 30 30 20 20 20 10 20 30 40 50 60 10 20 30 40 50 60 10 20 30 40 50 60 0.06 0.06 0.06 0.04 0.04 0.04 0.02 0.02 0.02 -0.15 -0.1 -0.05 0.05 0.1 0.15 0.2 -0.15 -0.1 -0.05 0.05 0.1 0.15 0.2 -0.15 -0.1 -0.05 0.05 0.1 0.15 0.2 -0.02 -0.02 -0.02 -0.04 -0.04 -0.04 -0.06 -0.06 -0.06
Intrinsic Evolution of Differential Invariants Theorem. Under an arc-length preserving flow, R = A − κ s D − 1 B κ t = R ( J ) where ( ∗ ) In surprisingly many situations, (*) is a well-known integrable evolution equation, and R is its recursion operator ! = ⇒ Hasimoto = ⇒ Langer, Singer, Perline = ⇒ Mar´ ı–Be ff a, Sanders, Wang = ⇒ Qu, Chou, Anco, and many more ...
Euclidean plane curves G = SE(2) = SO(2) � R 2 A = D 2 + κ 2 B = − κ R = A − κ s D − 1 B = D 2 + κ 2 + κ s D − 1 · κ 2 κ 2 κ s κ t = R ( κ s ) = κ sss + 3 = ⇒ modified Korteweg-deVries equation
Equi-affine plane curves G = SA(2) = SL(2) � R 2 A = D 4 + 5 3 κ D 2 + 5 9 κ 2 3 κ s D + 1 3 κ ss + 4 3 D 2 − 2 B = 1 9 κ R = A − κ s D − 1 B = D 4 + 5 3 κ D 2 + 4 9 κ 2 + 2 9 κ s D − 1 · κ 3 κ s D + 1 3 κ ss + 4 κ t = R ( κ s ) = κ 5 s + 5 3 κ κ sss + 5 3 κ s κ ss + 5 9 κ 2 κ s = ⇒ Sawada–Kotera equation R = R · ( D 2 + 1 3 κ + 1 � 3 κ s D − 1 ) Recursion operator:
Euclidean space curves G = SE(3) = SO(3) � R 3 s + ( κ 2 − τ 2 ) D 2 A = D s + κτ ss − κ s τ s + 2 κ 3 τ 2 τ s + 3 κτ s − 2 κ s τ κ D 2 κ 2 κ 2 − 2 τ D s − τ s s + κ 2 − τ 2 D s + κ s τ 2 − 2 κττ s 1 s − κ s κ D 3 κ 2 D 2 κ 2 κ B = ( κ 0 ) � � � � � � κ s κ t κ s D − 1 B R = A − = R τ s τ t τ s = ⇒ vortex filament flow (Hasimoto)
The Recurrence Formula For any function or differential form Ω : r � ν k ∧ ι [ v k ( Ω )] d ι ( Ω ) = ι ( d Ω ) + k =1 v 1 , . . . , v r — basis for g — infinitesimal generators ν 1 , . . . , ν r — dual invariantized Maurer–Cartan forms ⋆ ⋆ The ν k are uniquely determined by the recurrence formulae for the phantom di ff erential invariants
r � ν k ∧ ι [ v k ( Ω )] d ι ( Ω ) = ι ( d Ω ) + k =1 ⋆ ⋆ ⋆ All identities, commutation formulae, syzygies, etc., among di ff erential invariants and, more generally, the invariant variational bicomplex follow from this universal recurrence formula by letting Ω range over the basic functions and di ff erential forms ! ⋆ ⋆ ⋆ Therefore, the entire structure of the differential invari- ant algebra and invariant variational bicomplex can be completely determined using only linear differential al- gebra; this does not require explicit formulas for the moving frame, the differential invariants, the invariant differential forms, or the group transformations!
r � ν k ∧ ι [ v k ( Ω )] d ι ( Ω ) = ι ( d Ω ) + k =1 ⋆ ⋆ ⋆ All identities, commutation formulae, syzygies, etc., among di ff erential invariants and, more generally, the invariant variational bicomplex follow from this universal recurrence formula by letting Ω range over the basic functions and di ff erential forms ! ⋆ ⋆ ⋆ Therefore, the entire structure of the differential invari- ant algebra and invariant variational bicomplex can be completely determined using only linear differential al- gebra; this does not require explicit formulas for the moving frame, the differential invariants, the invariant differential forms, or the group transformations!
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