Some existence results for Chebyshev nets with singularities

Yannick Masson (Cermics, University Paris-Est)

joint work with O. Baverel, A. Ern, L. Hauswirth, A. Lebée, L. Monasse

9th International Conference on Mathematical Methodsfor Curves and Surfaces

June, 23rd 2016

Chebyshev nets with singularities

A motivation: Elastic gridshells

Chebyshev nets with singularities

A motivation: Elastic gridshells

Initial grid

(x , y , z) = Φ(u, v)

Shaped grid

Construction obtained by deformation of inextensible beams

Mathematical model: Chebyshev net parametrization

Φ : Ω ⊂ R2 → S ⊂ R3, | ∂u Φ| = | ∂v Φ| = 1

Metric induced by Φ: ds2 = du2 + 2 cos[ω(u, v)]dudv + dv2

Restriction on surfaces homeomorphic to the plane

Chebyshev nets with singularities

A motivation: Elastic gridshells

Initial grid

(x , y , z) = Φ(u, v)

Shaped grid

Construction obtained by deformation of inextensible beams

Mathematical model: Chebyshev net parametrization

Φ : Ω ⊂ R2 → S ⊂ R3, | ∂u Φ| = | ∂v Φ| = 1

Metric induced by Φ: ds2 = du2 + 2 cos[ω(u, v)]dudv + dv2

Restriction on surfaces homeomorphic to the plane

Chebyshev nets with singularities

Hazzidakis formula, folds and cusps

Cauchy boundary conditions giving

existence and uniqueness: two curves

Hazzidakis formula :

ωA + ωC = ωB + ωD −∫ABCD

K∣∣∣∣∫ABCD

K

∣∣∣∣ ≥ 2π ⇒ folds (angle π) and

cusps (angle 0) appear

AB

CD

Objective : Meshing surfaces without cusps and folds

Chebyshev nets with singularities

Hazzidakis formula, folds and cusps

Cauchy boundary conditions giving

existence and uniqueness: two curves

Hazzidakis formula :

ωA + ωC = ωB + ωD −∫ABCD

K

∣∣∣∣∫ABCD

K

∣∣∣∣ ≥ 2π ⇒ folds (angle π) and

cusps (angle 0) appear

ωA π − ωB

ωCπ − ωD

AB

CD

Objective : Meshing surfaces without cusps and folds

Chebyshev nets with singularities

Hazzidakis formula, folds and cusps

Cauchy boundary conditions giving

existence and uniqueness: two curves

Hazzidakis formula :

ωA + ωC = ωB + ωD −∫ABCD

K∣∣∣∣∫ABCD

K

∣∣∣∣ ≥ 2π ⇒ folds (angle π) and

cusps (angle 0) appear

ωA π − ωB

ωCπ − ωD

AB

CD

Objective : Meshing surfaces without cusps and folds

Chebyshev nets with singularities

Hazzidakis formula, folds and cusps

Cauchy boundary conditions giving

existence and uniqueness: two curves

Hazzidakis formula :

ωA + ωC = ωB + ωD −∫ABCD

K∣∣∣∣∫ABCD

K

∣∣∣∣ ≥ 2π ⇒ folds (angle π) and

cusps (angle 0) appear

ωA π − ωB

ωCπ − ωD

AB

CD

Objective : Meshing surfaces without cusps and folds

Chebyshev nets with singularities

Smooth Chebyshev nets

Let K± = max(0,±K ).

Theorem (Samelson, Dayawansa, 1995)

Let∫SK± < π

2. Then there exist global smooth Chebyshev coordinates on S.

Considering diagonal curves

⇒ dual coordinates

Cauchy boundary conditions:

dual curve + angle distribution

Hazzidakis formula in the dual

coordinates:

ωC =ωA + ωB

2+

∫AB

kAB −∫ABC

K

kAB : geodesic curvature of the dual curve dual curve

primal

curveprimal

curve

A B

C

Theorem (Y. M., L. Monasse, 2016)

Let∫S|K | < 2π. Then there exist global smooth Chebyshev coordinates on S.

Chebyshev nets with singularities

Smooth Chebyshev nets

Let K± = max(0,±K ).

Theorem (Samelson, Dayawansa, 1995)

Let∫SK± < π

2. Then there exist global smooth Chebyshev coordinates on S.

Considering diagonal curves

⇒ dual coordinates

Cauchy boundary conditions:

dual curve + angle distribution

Hazzidakis formula in the dual

coordinates:

ωC =ωA + ωB

2+

∫AB

kAB −∫ABC

K

kAB : geodesic curvature of the dual curve

dual curve

primal

curveprimal

curve

A B

C

angle

distribution

Theorem (Y. M., L. Monasse, 2016)

Let∫S|K | < 2π. Then there exist global smooth Chebyshev coordinates on S.

Chebyshev nets with singularities

Smooth Chebyshev nets

Let K± = max(0,±K ).

Theorem (Samelson, Dayawansa, 1995)

Let∫SK± < π

2. Then there exist global smooth Chebyshev coordinates on S.

Considering diagonal curves

⇒ dual coordinates

Cauchy boundary conditions:

dual curve + angle distribution

Hazzidakis formula in the dual

coordinates:

ωC =ωA + ωB

2+

∫AB

kAB −∫ABC

K

kAB : geodesic curvature of the dual curve dual curve

primal

curveprimal

curve

π−ωA2

π−ωB2

ωC

A B

C

Theorem (Y. M., L. Monasse, 2016)

Let∫S|K | < 2π. Then there exist global smooth Chebyshev coordinates on S.

Chebyshev nets with singularities

Smooth Chebyshev nets

Let K± = max(0,±K ).

Theorem (Samelson, Dayawansa, 1995)

Let∫SK± < π

2. Then there exist global smooth Chebyshev coordinates on S.

Considering diagonal curves

⇒ dual coordinates

Cauchy boundary conditions:

dual curve + angle distribution

Hazzidakis formula in the dual

coordinates:

ωC =ωA + ωB

2+

∫AB

kAB −∫ABC

K

kAB : geodesic curvature of the dual curve dual curve

primal

curveprimal

curve

π−ωA2

π−ωB2

ωC

A B

C

Theorem (Y. M., L. Monasse, 2016)

Let∫S|K | < 2π. Then there exist global smooth Chebyshev coordinates on S.

Chebyshev nets with singularities

Nonsmooth Chebyshev nets and singularities

Sector: Domain on the surface delimited by two smooth curves

(and possibly the surface's boundary).

Theorem (Burago, Ivanov, Malev, 2005)

Suppose∫SK± < 2π. Then there exist global (possibly non C 1) Chebyshev

coordinates on S.

Obtained by splitting the surface into 4 sectors parametrizedindependently

→ degrees of freedom: the curves delimiting the sectors

⇒ Generalization into bifurcation singularities:

junction of N sectors at one point

Motivation: splitting the constraint on curvature acting on

boundary conditions (Hazzidakis formula)

Chebyshev nets with singularities

Nonsmooth Chebyshev nets and singularities

Sector: Domain on the surface delimited by two smooth curves

(and possibly the surface's boundary).

Theorem (Burago, Ivanov, Malev, 2005)

Suppose∫SK± < 2π. Then there exist global (possibly non C 1) Chebyshev

coordinates on S.

Obtained by splitting the surface into 4 sectors parametrizedindependently

→ degrees of freedom: the curves delimiting the sectors

⇒ Generalization into bifurcation singularities:

junction of N sectors at one point

Motivation: splitting the constraint on curvature acting on

boundary conditions (Hazzidakis formula)

Chebyshev nets with singularities

Nonsmooth Chebyshev nets and singularities

Sector: Domain on the surface delimited by two smooth curves

(and possibly the surface's boundary).

Theorem (Burago, Ivanov, Malev, 2005)

Suppose∫SK± < 2π. Then there exist global (possibly non C 1) Chebyshev

coordinates on S.

Obtained by splitting the surface into 4 sectors parametrizedindependently

→ degrees of freedom: the curves delimiting the sectors

⇒ Generalization into bifurcation singularities:

junction of N sectors at one point

Motivation: splitting the constraint on curvature acting on

boundary conditions (Hazzidakis formula)

Chebyshev nets with singularities

Nonsmooth Chebyshev nets and singularities

Sector: Domain on the surface delimited by two smooth curves

(and possibly the surface's boundary).

Theorem (Burago, Ivanov, Malev, 2005)

Suppose∫SK± < 2π. Then there exist global (possibly non C 1) Chebyshev

coordinates on S.

Obtained by splitting the surface into 4 sectors parametrizedindependently

→ degrees of freedom: the curves delimiting the sectors

⇒ Generalization into bifurcation singularities:

junction of N sectors at one point

Motivation: splitting the constraint on curvature acting on

boundary conditions (Hazzidakis formula)

Chebyshev nets with singularities

Chebyshev nets with singularities

Chebyshev parametrization with bifurcation singularities:

Φ : D → S with D a polyhedral surface

Φ

Φ

Chebyshev nets with singularities

Chebyshev nets with singularities

Chebyshev nets with singularities

Chebyshev nets with singularities

Nonsmooth parametrization but "smoothable" upon introducing a small

hole in S

Smoothing

Problem: nding the skeleton of the net (red curves)

Chebyshev nets with singularities

Chebyshev nets with singularities

Nonsmooth parametrization but "smoothable" upon introducing a small

hole in S

Smoothing

Problem: nding the skeleton of the net (red curves)

Chebyshev nets with singularities

Existence of Chebyshev nets with singularities

Theorem (Y. M., L. Monasse, 2016)

Suppose S satises∫SK− <∞,

∫SK+ < 2π. Then there exist global

Chebyshev coordinates with bifurcations Φ : D → S. Moreover, these

coordinates can be smoothed introducing small holes in the surface

Chebyshev coordinates without cusps and folds are bi-Lipschitz

coordinates

Theorem (Bonk and Lang, 2002)

There exists a bi-Lipschitz mapping under the sharp conditions∫SK− <∞

and∫SK+ < 2π

Chebyshev nets are obtained under the same conditions on S

⇒ S is bi-Lipschitz to some nonpositively curved polyhedral surface

Chebyshev nets with singularities

Existence of Chebyshev nets with singularities

Theorem (Y. M., L. Monasse, 2016)

Suppose S satises∫SK− <∞,

∫SK+ < 2π. Then there exist global

Chebyshev coordinates with bifurcations Φ : D → S. Moreover, these

coordinates can be smoothed introducing small holes in the surface

Chebyshev coordinates without cusps and folds are bi-Lipschitz

coordinates

Theorem (Bonk and Lang, 2002)

There exists a bi-Lipschitz mapping under the sharp conditions∫SK− <∞

and∫SK+ < 2π

Chebyshev nets are obtained under the same conditions on S

⇒ S is bi-Lipschitz to some nonpositively curved polyhedral surface

Chebyshev nets with singularities

Existence of Chebyshev nets with singularities

Theorem (Y. M., L. Monasse, 2016)

Suppose S satises∫SK− <∞,

∫SK+ < 2π. Then there exist global

Chebyshev coordinates with bifurcations Φ : D → S. Moreover, these

coordinates can be smoothed introducing small holes in the surface

Chebyshev coordinates without cusps and folds are bi-Lipschitz

coordinates

Theorem (Bonk and Lang, 2002)

There exists a bi-Lipschitz mapping under the sharp conditions∫SK− <∞

and∫SK+ < 2π

Chebyshev nets are obtained under the same conditions on S

⇒ S is bi-Lipschitz to some nonpositively curved polyhedral surface

Chebyshev nets with singularities

Existence of Chebyshev nets with singularities

Theorem (Y. M., L. Monasse, 2016)

Suppose S satises∫SK− <∞,

∫SK+ < 2π. Then there exist global

Chebyshev coordinates with bifurcations Φ : D → S. Moreover, these

coordinates can be smoothed introducing small holes in the surface

Chebyshev coordinates without cusps and folds are bi-Lipschitz

coordinates

Theorem (Bonk and Lang, 2002)

There exists a bi-Lipschitz mapping under the sharp conditions∫SK− <∞

and∫SK+ < 2π

Chebyshev nets are obtained under the same conditions on S

⇒ S is bi-Lipschitz to some nonpositively curved polyhedral surface

Chebyshev nets with singularities

Sketch of the proof

We follow the proof of Burago et al.

γ0

ψ1

γ1

ψ2

γ2 ψ3

γ3

ψ4

γ4

Proposition (Y. M., L. Monasse, 2016)

Let the geodesic polygonal curve with N

vertices satisfy∫S

K+ < π −N∑i=1

ψi ,

∫S

K− < maxi≤N

ψi . (1)

Then there exist Chebyshev coordinates on S

such that the (γi )0≤i≤N are coordinate curves.

We split the surface with Ham-Sandwich theorem into two half surfaces

containing the same total positive and negative curvature.

We split recursivly each half surface to obtain domains delimited by

geodesic polygonal curve with N vertices satisfying the condition on

curvature.

Convergence of the algorithm thanks to an estimation on minimal

(maximal) angle of each geodesic polygonal curve

Chebyshev nets with singularities

Sketch of the proof

We follow the proof of Burago et al.

γ0

ψ1

γ1

ψ2

γ2 ψ3

γ3

ψ4

γ4

Proposition (Y. M., L. Monasse, 2016)

Let the geodesic polygonal curve with N

vertices satisfy∫S

K+ < π −N∑i=1

ψi ,

∫S

K− < maxi≤N

ψi . (1)

Then there exist Chebyshev coordinates on S

such that the (γi )0≤i≤N are coordinate curves.

We split the surface with Ham-Sandwich theorem into two half surfaces

containing the same total positive and negative curvature.

We split recursivly each half surface to obtain domains delimited by

geodesic polygonal curve with N vertices satisfying the condition on

curvature.

Convergence of the algorithm thanks to an estimation on minimal

(maximal) angle of each geodesic polygonal curve

Chebyshev nets with singularities

Sketch of the proof

We follow the proof of Burago et al.

γ0

ψ1

γ1

ψ2

γ2

σS1 S2

ψ3

γ3

ψ4

γ4

Proposition (Y. M., L. Monasse, 2016)

Let the geodesic polygonal curve with N

vertices satisfy∫S

K+ < π −N∑i=1

ψi ,

∫S

K− < maxi≤N

ψi . (1)

Then there exist Chebyshev coordinates on S

such that the (γi )0≤i≤N are coordinate curves.

We split the surface with Ham-Sandwich theorem into two half surfaces

containing the same total positive and negative curvature.

We split recursivly each half surface to obtain domains delimited by

geodesic polygonal curve with N vertices satisfying the condition on

curvature.

Convergence of the algorithm thanks to an estimation on minimal

(maximal) angle of each geodesic polygonal curve

Chebyshev nets with singularities

Sketch of the proof

We follow the proof of Burago et al.

γ0

ψ1

γ1

ψ2

γ2

σS1 S2

ψ3

γ3

ψ4

γ4

Proposition (Y. M., L. Monasse, 2016)

Let the geodesic polygonal curve with N

vertices satisfy∫S

K+ < π −N∑i=1

ψi ,

∫S

K− < maxi≤N

ψi . (1)

Then there exist Chebyshev coordinates on S

such that the (γi )0≤i≤N are coordinate curves.

We split the surface with Ham-Sandwich theorem into two half surfaces

containing the same total positive and negative curvature.

We split recursivly each half surface to obtain domains delimited by

geodesic polygonal curve with N vertices satisfying the condition on

curvature.

Convergence of the algorithm thanks to an estimation on minimal

(maximal) angle of each geodesic polygonal curve

Chebyshev nets with singularities

Application to an academic surface

Chebyshev nets with singularities

Application to an academic surface

Chebyshev nets with singularities

Perspectives on Chebyshev nets

Theoretical open questions:

⇒ Existence of Chebyshev nets whenever∫SK+ > 2π

⇒ Existence of Chebyshev nets on surfaces with nontrivial topology

Applied perspective:

⇒ Ecient algorithm for constructions of Chebyshev nets with singularities

⇒ Mechanical study of gridshells with singularities (stability,

construction,...)

Chebyshev nets with singularities