Triangulating the 3D Periodic Space (work in progress)

Triangulating the 3D Periodic Space (work in progress)

Manuel Caroli, Nico Kruithof, Monique Teillaud

NYU - June 2008

Outline

1 Triangulations

2 Periodic Space

3 Coverings

4 Implementation

5 Benchmarks

6 Conclusion

Outline

1 Triangulations

2 Periodic Space

3 Coverings

4 Implementation

5 Benchmarks

6 Conclusion

Triangulations Periodic Space Coverings Implementation Benchmarks Conclusion

Definition

Triangulation : simplicial complex

= finite collection K of simplices such that:

if σ ∈ K and τ is a face of σ, then τ ∈ K , if σ 1 , σ 2 ∈ K and σ 1 ∩ σ 2 6= ∅,

then σ ₁ ∩ σ ₂ is a face of both σ ₁ and σ ₂ .

K simplicial complex is a triangulation of S if each point in S is a vertex of K

S

σ∈K σ is homeomorphic to X .

Definition

Triangulation : simplicial complex

= finite collection K of simplices such that:

if σ ∈ K and τ is a face of σ, then τ ∈ K , if σ 1 , σ 2 ∈ K and σ 1 ∩ σ 2 6= ∅,

then σ ₁ ∩ σ ₂ is a face of both σ ₁ and σ ₂ .

X topological space, S set of points

K simplicial complex is a triangulation of S if each point in S is a vertex of K

S

σ∈K σ is homeomorphic to X .

Example

point set S

Example

point set S

Example

Delaunay triangulation T

Outline

1 Triangulations

2 Periodic Space

3 Coverings

4 Implementation

5 Benchmarks

6 Conclusion

The 2D Periodic Space T ²

The 2D Periodic Space T ²

The 2D Periodic Space T ²

Homeomorphic to the surface of a torus in 3D

The 3D Periodic Space T ³

Homeomorphic to the 3D hypersurface of a torus in 4D

Mapping T ³ into R ³

g : T ³ −→ R ³

p 7−→ {p + (i , j, k ) | i, j, k ∈ Z }

Mapping T ³ into R ³

g : T ³ −→ R ³

p 7−→ {p + (i , j, k ) | i, j, k ∈ Z }

Outline

1 Triangulations

2 Periodic Space

3 Coverings

4 Implementation

5 Benchmarks

6 Conclusion

Triangulation of one Point of T ²

Triangulation of one Point of T ²

Triangulation of one Point of T ²

Triangulation of one Point of T ²

Not a simplicial complex

Triangulation of one Point of T ³

Not a simplicial complex

Coverings

T ³ n = n ³ -sheeted covering

h n : T ³ −→ T ³ n

p 7−→ {p + (i, j, k ) | i, j, k ∈ {0, . . . , n − 1}}

Coverings

T ³ n = n ³ -sheeted covering

h n : T ³ −→ T ³ n

p 7−→ {p + (i, j, k ) | i, j, k ∈ {0, . . . , n − 1}}

1-sheeted covering

Coverings

T ³ n = n ³ -sheeted covering

h n : T ³ −→ T ³ n

p 7−→ {p + (i, j, k ) | i, j, k ∈ {0, . . . , n − 1}}

2-sheeted covering

Coverings

T ³ n = n ³ -sheeted covering

h n : T ³ −→ T ³ n

p 7−→ {p + (i, j, k ) | i, j, k ∈ {0, . . . , n − 1}}

4-sheeted covering

Coverings

T ³ n = n ³ -sheeted covering

h n : T ³ −→ T ³ n

p 7−→ {p + (i, j, k ) | i, j, k ∈ {0, . . . , n − 1}}

∞-sheeted covering = R ² = universal covering

2 ^d -sheeted covering

S ⊂ T ^d , d = 1, 2.

h ₂ (T ) has no self-edges.

Proof:

case |S| = 1

2 ^d -sheeted covering

S ⊂ T ^d , d = 1, 2.

h ₂ (T ) has no self-edges.

Proof:

Largest empty circle

2 ^d -sheeted covering

S ⊂ T ^d , d = 1, 2.

h ₂ (T ) has no self-edges.

Proof:

Delaunay property ⇒ longest possible edge

2 ^d -sheeted covering

S ⊂ T ^d , d = 1, 2.

h ₂ (T ) has no self-edges.

Proof:

Same for T ³ with h ₂ inducing an 8-sheeted covering

3 ^d -sheeted covering

S ⊂ T ^d , d = 1, 2.

h ₃ (T ) is a simplicial complex.

Proof: (work in progress)

Triangulations Periodic Space Coverings Implementation Benchmarks Conclusion

1-sheeted covering

S ⊂ T ² .

|Longest edge| <

√ 3

2 ⇒ T has no self-edge.

Proof:

1

√ 3 2

Largest emtpy circle has diameter < 1.

|Longest edge| < ² ₃ ⇒ T has no self-edge.

1-sheeted covering

S ⊂ T ² .

|Longest edge| <

√ 3

2 ⇒ T has no self-edge.

Proof:

1

√ 3 2

Largest emtpy circle has diameter < 1.

S ⊂ T ³ .

|Longest edge| <

q 2

3 ⇒ T has no self-edge.

1-sheeted covering

S ⊂ T ³ .

|Longest edge| < ^{√ 1}

6 ⇒ T is a simplicial complex.

1-sheeted covering

Consider S ⁰ , S ⊂ S ⁰ ⊂ T ³ ,

and T ⁰ triangulation of S ⁰ in some covering Edge length criterion fulfilled for T

⇒ T ⁰ simplicial complex in T ³ .

Proof:

Adding further points cannot increase the largest empty

spheres size.

Algorithm

Compute initially in T ³ ₃

Maintain a data structure D storing pointers to all edges that are longer than ^{√ 1}

6

Once D is empty: switch back to T ³ ₁ .

Outline

1 Triangulations

2 Periodic Space

3 Coverings

4 Implementation

5 Benchmarks

6 Conclusion

Offsets

Offsets

Offsets

00 00

00 00

01

01

Offsets

Storage

Storage

actually stored

Outline

1 Triangulations

2 Periodic Space

3 Coverings

4 Implementation

5 Benchmarks

6 Conclusion

R ³ vs. T ³

Delaunay triangulation ( R ³ vs. T ³ ):

No. of points R

T

factor

1000 0.0260 0.0420 1.62i 10000 0.212 0.325 1.53i 100000 2.20 3.13 1.42i 1000000 26.4 34.7 1.31i

time in seconds

Triangulation in T ³

actually computed in the 1-sheeted covering only

(initially dummy points - then removed)

Outline

1 Triangulations

2 Periodic Space

3 Coverings

4 Implementation

5 Benchmarks

6 Conclusion

Conclusion and Outlook

Soon in

New design of C GAL 3D Triangulation package

VIDEO

Applications in Simulation Astronomy Fluid dynamics Crystallography . . .

another Orbifold (of S ² ):

Projective plane [Aanjaneya-T.’07]

state of the art omitted