DDG 07 GENERALIZED CURVATURES J.M. Morvan Institut Camille Jordan - - PDF document

DDG 07 GENERALIZED CURVATURES J.M. Morvan Institut Camille Jordan Universit Claude Bernard Lyon 1, with ...

• V. Borrelli,
• D. Cohen-Steiner,
• B. Thibert.

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... An Introduction ... "In what sense do two sets have to be close to each other, in order to guarantee that their curvature measures are close to each

• ther ?"
• J. Milnor, (1994)

... Enigmatic question ...

• What are the curvature measures of a

set ?

• What does mean : close to each other ?

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... TWO QUESTIONS ...

• Does a geometric property dened on the

class of smooth objects have an analo- gous on the class of discrete objects?

• Dene a frame in which one can compare

geometric properties of a smooth object and the corresponding geometric proper- ties of a discrete object close to it .

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... EXAMPLE ...

• On a smooth curve, one can dene its

length l, its curvature k, its torsion t ...

• On a smooth surface, one can dene its

area A, its Gauss curvature G, its mean curvature H, its second fundamental form h, the lines of curvatures ... Can one dene these invariants on discrete

• bjects like polyhedron for instance ?

The main problem is to nd GOOD DEFINITIONS

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HISTORY

• 1. The convex case, Steiner, 1840
• 2. The smooth case, Weyl, 1939
• 3. The introduction of geometric measure,

Federer, 1958

• 4. The introduction of cohomology, Wint-

gen, 1978

• 5. The recent works, (Zahle, Fu, etc... 1980−

2007)

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THE CURVATURES OF A SMOOTH HYPERSURFACE M : (oriented) smooth hypersurface of the (oriented) Euclidean space EN, < . >, ˜ ∇.

• Let ξ be the unit normal vector eld to

M.

• The Weingarten tensor (shape operator)

A dened on M : A(X) = −˜ ∇Xξ. (1)

• The second fundamental form h of M :

< A(X), Y >= h(X, Y ). (2)

• The principal curvatures

(λ1, ..., λi, ..., λ(N−1))

• f M at m, and the principal vector elds,

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Denition Let M be a smooth hypersurface

• f EN. For every , k = 0, ..., (N −1), the k−th-

elementary symmetric function of the princi- pal curvatures of M Ξk = {λi1, ..., λik} is called the k−th-mean curvature of M. Remark : One has det (I + tA) =

k=N−1

• k=0

Ξktk, (3)

• Ξ0 = 1;
• Ξ1 is the trace of h, (H = 1

nX1 is called

the mean curvature of M).

• ΞN−1 is the determinant of h, (usually

denoted by G), called the Gauss curva- ture of M.

CURVATURE MEASURES OF A SMOOTH HYPERSURFACE Let U be a domain of M. ϕk(U) =

• U ΞkdvM

is called the k-th Lipschitz-Killing curva- ture of M.

• ϕ0(U) is the classical (N − 1)-volume of

U;

• ϕN(U) is the total Gauss curvature of U.

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Go back to the problem : How to dene these curvature measures for discrete subsets of EN ? A IMPORTANT TOOL : THE VOLUME By computing the volume of some tubular neighborhoods of a subset X, we can exhibit some geometric invariants of X...

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THE CONVEX CASE

• K convex body,
• Kǫ = {m ∈ EN, d(m, K) ≤ ǫ},

ε

A

• ∂K: convex hypersurface,
• ∂Kǫ: parallel hypersurface at distance ǫ.

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THE CONVEX CASE... STEINER FORMULA. Theorem - Let K be a convex body. Then, V (Kǫ) =

N

• k=0

Φk(K)ǫk, ∀ǫ ≥ 0. AN ADDITIVITY PROPERTY Theorem: If K1, K2 an K1 ∪ K2 are convex, then ∀k,

Φk(K1 ∪ K2) = Φk(K1) + Φk(K2) − Φk(K1 ∩ K2).

CONTINUITY OF THE Φk IN THE CONVEX WORLD Theorem: If a sequence of Kn tends to K in the Hausdor sense, then ∀k, lim

n→∞ Φk(Kn) = Φk(K).

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Question: Compute explicitly the Φ′

ks ?

We will test on

• smooth objects,
• polyhedra.

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THE CONVEX CASE... STEINER FORMULA FOR SMOOTH BODIES Steiner formula can be stated as follows : VolN(Kǫ) =

k=N

• k=0

Φk(K)ǫk, with Φk(K) = CNϕk(K) = CN

• ∂K ΞkdvM.

In particular, if K is a compact convex body in E3 with smooth boundary, then Vol3(Kǫ) = Vol3(K)+A(∂K)ǫ+(

• ∂K Hda)ǫ2+1

3(

• ∂K Gda)ǫ3,

where H, (resp. G), denotes the mean cur- vature, (resp. Gauss curvature) of ∂K.

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THE CONVEX CASE... STEINER FORMULA FOR POLYHEDRA We give the following Denition 1 Let σl be a l-dimensional face

• f a k-simplex σk, (l < k).

Let q ∈ int(σl). The following notions are independent of q.

• 1. The normal cone C⊥(σl, σk) to σl :

C⊥(σl, σk) = {x ∈ int(σk) : qx ∈ σl⊥}.

• 2. The basis of the normal cone C⊥(σl, σk)

to σl is L(σl, σk) = C⊥(σl, σk) ∩ Sk−l−1, where

Sk−l−1 is the unit sphere centered

at q.

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3

q

σ e

• 3. The internal dihedral angle
• (σl, σk) is the

measure of L(σl, σk). The normalized internal dihedral angle (σl, σk) : (σl, σk) =

• (σl, σk)

sk−l−1 .

• 4. The external dihedral angle
• (σl, σk)∗ is

the measure of the subset of

Sk−l−1 ob-

tained by intersecting Sk−l−1 with the half- lines whose origin is q, and making an an- gle greater than π

2 with the interior of σk.

t v

The normalized external dihedral angle (σl, σk)∗ : (σl, σk)∗ =

• (σl, σk)∗

sk−l−1 .

THE CONVEX CASE... STEINER FORMULA FOR POLYHEDRA Theorem - Let K be a convex body of EN whose boundary ∂K is a polyhedron. Then, VolN(Kǫ) =

k=N

• k=0

Φk(K)ǫk, with Φk(K) =

• σN−k⊂σN⊂K

VolN−k(σN−k)(σN−k, σN)∗. where

• σN−k denotes a generic (N − k)-face of

∂K,

• (σN−k, σN)∗ denotes the normalized diedral

external angle,

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THE CONVEX CASE... STEINER FORMULA FOR POLYHEDRA IN E3

Vol3(Kǫ) = Vol3(K)+A(∂K)ǫ+(

• a

∠(a)l(a))ǫ2+ 4

3πǫ3. Remember the smooth case : Vol3(Kǫ) = Vol3(K)+A(∂K)ǫ+(

• ∂K

Hda)ǫ2+1 3(

• ∂K

Gda)ǫ3,

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Example : One dene the global mean curvature

• of a smooth compact convexe surface S

by Φ2(S) =

• S Hda,
• of a compact convex polyhedron P by:

Φ2(P) =

• a

∠(a).l(a).

Then

• 1. In both cases,

Φ2(A∪B) = Φ2(A)+Φ2(B)−Φ2(A∩B).

• 2. Moreover, if Pn tends to S in the Haus-

dor sense then lim

• n

∠(an).l(an) =

• S Hda.

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HOWEVER, ALL THIS THEORY FAILS FOR NON CONVEX SUBSETS BECAUSE :

• THE ADDITIVITY FORMULA IS NOT

SATISFIED ;

• THE CONTINUITY PROPERTY IS

NOT SATISFIED.

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... A CLASSICAL PROBLEM ... NON CONTINUITY OF THE AREA The situation is completely dierent for sur- faces and their areas.... Let M be a C∞ sur- face of E3. One can dene its area A(M). It satises A(M1∪M2) = A(M1)+A(M2)−A(M1∩M2). Let P be a polyhedron. One can dene its area which satises the same property. What about the continuity property?

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THE LANTERN OF SCHWARZ

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THE LANTERN OF SCHWARZ Let C be a cylinder of height l and radius R. Let Pn,N be the lantern with N slices and 2n triangles on each slice. The sequence Pn,n2. The area of Pn,n2 sat- ises: A(Pn,n2) → 2πR

• R2π4

4 + l2, Then: A(Pn,n2) → A′ = A(C). However δ(Pn,n2, C) → 0 (where δ is Hausdor distance).

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NON ADDITIVITY OF Vol Kǫ Vol (I1 ∪ I2)ǫ = Vol (I1)ǫ + Vol (I2)ǫ − Vol (I1 ∩ I2)ǫ.

r p q

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... ANOTHER POINT OF VIEW ... THE NORMAL CYCLE Let M smooth hypersurface in EN. The good object to study is the Gauss map. G : M → EN × SN−1 ⊂ EN × EN ≃ TEN m → G(m) = (m, ξ) CRUCIAL FACT : integrating particular dierential forms of EN × SN−1 on G(U) gives all the Lipschitz curvatures of U !!

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INVARIANT (N − 1)-FORMS ON EN × SN−1 Let G the group of rigid motions of EN. G acts on EN × SN−1 in a natural way. ω ∈ ΛN−1(EN × SN−1) is invariant (by G) if φ∗(ω) = ω, for all φ ∈ G. One can build a basis ω0, ..., ωN of the space

• f invariant dierential forms on EN × SN−1.

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INVARIANT 2-FORMS ON E3 × S2 The vector space of invariant 2-forms on E3× S2 has dimension 4. It is spent by

ωA = y1dx2 ∧ dx3 + y2dx3 ∧ dx1 + y3dx1 ∧ dx2. ωG = y1dy2 ∧ dy3 + y2dy3 ∧ dy1 + y3dy1 ∧ dy2. ωH = y1(dx2 ∧ dy3 + dy2 ∧ dx3)+ y2(dx3 ∧ dy1 + dy3 ∧ dx1)+ y3(dx1 ∧ dy2 + dy1 ∧ dx2). Ω = (y2

2 + y2 3)dx1 ∧ dy1 − y1y2dx1 ∧ dy2 − y1y3dx1 ∧ dy3

−y1y2dx2 ∧ dy1 + (y2

1 + y2 3)dx2 ∧ dy2n − y2y3dx2 ∧ dy3

−y1y3dx1 ∧ dy3 − y2y3dx3 ∧ dy2 + (y2

2 + y2 3)dx3 ∧ dy3.

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M smooth surface of E3

• G(U) ωA = A(U);
• G(U) ωG =
• U G;
• G(U) ωH =
• U H;
• G(U) Ω = 0.

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THE NORMAL CYCLE... (the case N=3) X → N(X) "Theorem" - To "each" compact sub- set X of EN, one can associate canonically an unique interesting rectiable Legendrian (N − 1)-current N(X) of EN × SN−1. More-

• ver, the map N satises the following addi-

tivity formula : If A and B are two compact subsets of E3 then, N(A ∪ B) = N(A) + N(B) − N(A ∩ B).

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THE NORMAL CYCLE... Example ... The normal cycle of a surface If M is a smooth surface, N(M) = {(m, v) ∈ E3 × S2 : v ∈ T ⊥

m(M)}.

(N(M) is the normal bundle in the usual sense).

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THE NORMAL CYCLE ... The case of a triangulated polyhedron of E3 ... Let P be any triangulated polyhedron of E3. One denes N(P) by decomposing P in a union of simplices and using the additivity formula N(A ∪ B) = N(A) + N(B) − N(A ∩ B). If σ is a k-simplex, N(σ) is simply the unit normal cone of σ.

r p q

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THE NORMAL CYCLE ... CURVATURE MEASURES OF A SUBSET X ... Let U ⊂ X ⊂ EN. Φk(U) =< N(X), χ(U × E3)ωk > .

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SECOND FUNDAMENTAL MEASURE...in E3 Let X, Y be two constant vectors of E3, and (p, ξ) un point de E3 × E3. We build the 2- form on E3 × E3 :

hX,Y

(p,ξ) = (ξ × X) ∧ Y.

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SECOND FUNDAMENTAL MEASURE...in E3 Let K be a subset of E3, and U ⊂ X. Denition

hX,Y

K

(U) =< N(K), χU×E3hX,Y

K

> Theorem The map E3 × E3 → R (X, Y ) → hX,Y

K

(U) is bilinear and symmetric.

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SECOND FUNDAMENTAL MESURE... For a smooth surface S,

hX,Y (U) =

• U∩S

hS(prTSX, prTSY )dvS,

where hS is the usual second fundamental form of S. Let T be a triangulated surface,

hX,Y

T

(U) =

• e

∠(e)l(e ∩ U) <

e, X >< e, Y > .

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AN APPROXIMATION RESULT "In what sense do two sets have to be close to each other, in order to guarantee that their curvature measures are close to each

• ther ?"
• J. Milnor, (1994)

A possible answer : Let K ⊂ EN and K′ ⊂

EN such that N(K) and N(K′) are close as

• currents. Then the corresponding curvature

measures are close. For instance : if K is smooth, U ⊂ K, U′ = pr U, Theorem -

|Φk

K(U) − Φk K′(U′)| ≤

C(δU + αU)[2(1 + ||hU)||) 1 − δU||hU|| ]N−1,

where C is a constant depending on the mass

• f the normal cycles of K and K′.

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THANK YOU

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