Triangulating the 3D Periodic Space
Manuel Caroli, Monique Teillaud Contributions by Triangulating the 3D Periodic Space Nico Kruithof Applications
Flat torus Manuel Caroli, Monique Teillaud Triangulations Contributions by Nico Kruithof Covering Spaces Algorithm
Extensions and Future work
TGDA – July 2009 – Paris Triangulating the Outline 3D Periodic Space Manuel Caroli, Monique Teillaud Contributions by Applications Nico Kruithof
Applications
Flat torus Flat torus Triangulations
Covering Spaces Triangulations Algorithm Extensions and Future work Covering Spaces
Algorithm
Extensions and Future work Triangulating the Outline 3D Periodic Space Manuel Caroli, Monique Teillaud Contributions by Applications Nico Kruithof
Applications
Flat torus Flat torus Triangulations
Covering Spaces Triangulations Algorithm Extensions and Future work Covering Spaces
Algorithm
Extensions and Future work Triangulating the Astronomy / Cosmic web 3D Periodic Space Manuel Caroli, Monique Teillaud Contributions by Nico Kruithof
Applications
Flat torus
Triangulations
Covering Spaces
Algorithm
Extensions and Future work
R. v. d. Weijgaert (Kapteyn Institute, Groningen) G. Vegter (Groningen) Associate Team “OrbiCG” (INRIA – Groningen) http://www-sop.inria.fr/geometrica/collaborations/OrbiCG C. Dullemond, MPI for Astronomy (Heidelberg) Triangulating the Material engineering / Bone scaffolding 3D Periodic Space Manuel Caroli, Monique Teillaud Contributions by Nico Kruithof
Applications
Flat torus
Triangulations
Covering Spaces
Algorithm
Extensions and Future work
M. Moesen (Leuven) Triangulating the Mechanics of granular materials 3D Periodic Space Manuel Caroli, Monique Teillaud Contributions by Nico Kruithof
Applications
Flat torus
Triangulations
Covering Spaces
Algorithm
Extensions and Future work
N. P. Kruyt (Twente)
CGAL Workshop on Periodic Spaces http://www.cgal.org/Events/PeriodicSpacesWorkshop Triangulating the And more. . . 3D Periodic Space Manuel Caroli, Monique Teillaud Contributions by I Physics of condensed matter (V. Robins (Canberra)) Nico Kruithof
I Structural biology Applications (D. Weiss (Stanford), J. Bernauer (INRIA Sophia)) Flat torus Triangulations I Crystallography Covering Spaces Fluid dynamics I Algorithm I Modeling of foams Extensions and Future work I Kelvin conjecture (R. Gabrielli (Bath))
I ... Triangulating the Outline 3D Periodic Space Manuel Caroli, Monique Teillaud Contributions by Applications Nico Kruithof
Applications
Flat torus Flat torus Triangulations
Covering Spaces Triangulations Algorithm Extensions and Future work Covering Spaces
Algorithm
Extensions and Future work 2 Triangulating the The 2D Periodic Space T 3D Periodic Space Manuel Caroli, Monique Teillaud Contributions by Nico Kruithof
Applications
Flat torus
Triangulations
Covering Spaces
Algorithm
Extensions and Future work 2 Triangulating the The 2D Periodic Space T 3D Periodic Space Manuel Caroli, Monique Teillaud Contributions by Nico Kruithof
Applications
Flat torus
Triangulations
Covering Spaces
Algorithm
Extensions and Future work 2 Triangulating the The 2D Periodic Space T 3D Periodic Space Manuel Caroli, Monique Teillaud Contributions by Nico Kruithof
Applications
Flat torus
Triangulations
Covering Spaces
Algorithm
Extensions and Future work
Homeomorphic to the surface of a torus in 3D 3 Triangulating the The 3D Periodic Space T 3D Periodic Space Manuel Caroli, Monique Teillaud Flat torus: Contributions by Nico Kruithof 3 3 3 T = R /Z Applications π : R3 → T3 quotient map Flat torus Triangulations T3 homeomorphic to the 3D hypersurface of a torus in 4D Covering Spaces Algorithm
Extensions and Future work original domain D = [0, 1) × [0, 1) × [0, 1)
D contains exactly one representative for each element of T3 3 Triangulating the Mapping into R 3D Periodic Space Manuel Caroli, Monique Teillaud 3 3 Contributions by ϕ : D × Z → R Nico Kruithof
(p, ζ) 7→ p + ζ is bijective Applications
Flat torus ζ = offset of a periodic copy Triangulations
Covering Spaces
Algorithm
Extensions and Future work 3 Triangulating the Mapping into R 3D Periodic Space Manuel Caroli, Monique Teillaud 3 3 Contributions by ϕ : D × Z → R Nico Kruithof
(p, ζ) 7→ p + ζ is bijective Applications
Flat torus ζ = offset of a periodic copy Triangulations
Covering Spaces
Algorithm
Extensions and Future work Triangulating the Outline 3D Periodic Space Manuel Caroli, Monique Teillaud Contributions by Applications Nico Kruithof
Applications
Flat torus Flat torus Triangulations
Covering Spaces Triangulations Algorithm Extensions and Future work Covering Spaces
Algorithm
Extensions and Future work Triangulating the Definition 3D Periodic Space Manuel Caroli, simplicial complex Monique Teillaud Contributions by = collection K of simplices such that: Nico Kruithof
I if σ ∈ K and τ is a face of σ, then τ ∈ K , Applications Flat torus I if σ1, σ2 ∈ K and σ1 ∩ σ2 6= ∅, Triangulations then σ1 ∩ σ2 is a face of both σ1 and σ2. Covering Spaces
I (local finiteness) Algorithm
Every point in a simplex of K has a neighborhood Extensions and that intersects at most finitely many simplices in K Future work Triangulating the Definition 3D Periodic Space Manuel Caroli, simplicial complex Monique Teillaud Contributions by = collection K of simplices such that: Nico Kruithof
I if σ ∈ K and τ is a face of σ, then τ ∈ K , Applications Flat torus I if σ1, σ2 ∈ K and σ1 ∩ σ2 6= ∅, Triangulations then σ1 ∩ σ2 is a face of both σ1 and σ2. Covering Spaces
I (local finiteness) Algorithm
Every point in a simplex of K has a neighborhood Extensions and that intersects at most finitely many simplices in K Future work Triangulating the Definition 3D Periodic Space Manuel Caroli, simplicial complex Monique Teillaud Contributions by = collection K of simplices such that: Nico Kruithof
I if σ ∈ K and τ is a face of σ, then τ ∈ K , Applications Flat torus I if σ1, σ2 ∈ K and σ1 ∩ σ2 6= ∅, Triangulations then σ1 ∩ σ2 is a face of both σ1 and σ2. Covering Spaces
I (local finiteness) Algorithm
Every point in a simplex of K has a neighborhood Extensions and that intersects at most finitely many simplices in K Future work Triangulating the Definition 3D Periodic Space Manuel Caroli, simplicial complex Monique Teillaud Contributions by = collection K of simplices such that: Nico Kruithof
I if σ ∈ K and τ is a face of σ, then τ ∈ K , Applications Flat torus I if σ1, σ2 ∈ K and σ1 ∩ σ2 6= ∅, Triangulations then σ1 ∩ σ2 is a face of both σ1 and σ2. Covering Spaces
I (local finiteness) Algorithm
Every point in a simplex of K has a neighborhood Extensions and that intersects at most finitely many simplices in K Future work Triangulating the Definition 3D Periodic Space Manuel Caroli, simplicial complex Monique Teillaud Contributions by = collection K of simplices such that: Nico Kruithof
I if σ ∈ K and τ is a face of σ, then τ ∈ K , Applications Flat torus I if σ1, σ2 ∈ K and σ1 ∩ σ2 6= ∅, Triangulations then σ1 ∩ σ2 is a face of both σ1 and σ2. Covering Spaces
I (local finiteness) Algorithm
Every point in a simplex of K has a neighborhood Extensions and that intersects at most finitely many simplices in K Future work Triangulating the Definition 3D Periodic Space Manuel Caroli, Star Monique Teillaud Contributions by Simplex σ, Simplicial complex K Nico Kruithof Star of σ in K : Applications Flat torus
St(σ) = {τ ∈ K | σ face of τ} Triangulations
Covering Spaces
Algorithm
Extensions and Future work
St(v) S |St(v)| := σ∈St(v) σ Triangulating the Definition 3D Periodic Space Manuel Caroli, Triangulation Monique Teillaud Contributions by X topological space, S set of points Nico Kruithof simplicial complex K is a triangulation of S if Applications Flat torus I each point in S is a vertex of K S Triangulations I σ∈K σ is homeomorphic to X. Covering Spaces
Algorithm Delaunay Triangulation Extensions and The circumsphere of each tetrahedron does not contain Future work any other point of S. Triangulating the Example 3D Periodic Space Manuel Caroli, Monique Teillaud point set S Contributions by Nico Kruithof
Applications
Flat torus
Triangulations
Covering Spaces
Algorithm
Extensions and Future work Triangulating the Example 3D Periodic Space Manuel Caroli, Monique Teillaud point set S Contributions by Nico Kruithof
Applications
Flat torus
Triangulations
Covering Spaces
Algorithm
Extensions and Future work Triangulating the Example 3D Periodic Space Manuel Caroli, Monique Teillaud Delaunay triangulation T Contributions by Nico Kruithof
Applications
Flat torus
Triangulations
Covering Spaces
Algorithm
Extensions and Future work 3 P = {(pi , ζi ) ∈ D × Z , i = 1,..., k + 1}, k ≤ 3 Ch(P) = convex hull of ϕ(P) = {pi + ζi }
If π|Ch(P) is a homeomorphism, then π(Ch(P)) is a k-simplex in T3. A
0 1 2 p0, 2 p0, 2 p0, 2 1 2 p1, 2 p1, 2 0 0 p2, 2 p1, 2 2 1 p2, p2, 2 2
0 1 2 p0, p0, 1 1 p0, 1 0 1 2 p1, 1 p1, 1 p1, 1
0 1 2 p2, 1 p2, 1 p2, 1
0 1 2 p0, 0 p0, 0 p0, 0 0 1 2 p1, 0 p1, 0 p1, 0 1 p2, 0 0 2 p2, 0 p2, 0 B
3 Triangulating the Simplex in T ? 3D Periodic Space Manuel Caroli, 3 Monique Teillaud k-simplex in R : Convex hull of k + 1 points Contributions by Nico Kruithof
Applications
Flat torus
Triangulations
Covering Spaces
Algorithm
Extensions and Future work 3 Triangulating the Simplex in T ? 3D Periodic Space Manuel Caroli, 3 Monique Teillaud P = {(pi , ζi ) ∈ D × Z , i = 1,..., k + 1}, k ≤ 3 Contributions by Nico Kruithof Ch(P) = convex hull of ϕ(P) = {pi + ζi } Applications
If π|Ch(P) is a homeomorphism, Flat torus then π(Ch(P)) is a k-simplex in T3. Triangulations A Covering Spaces
Algorithm 0 1 2 p0, 2 p0, 2 p0, 2 1 2 Extensions and p1, 2 p1, 2 0 Future work 0 p2, 2 p1, 2 2 1 p2, p2, 2 2
0 1 2 p0, p0, 1 1 p0, 1 0 1 2 p1, 1 p1, 1 p1, 1
0 1 2 p2, 1 p2, 1 p2, 1
0 1 2 p0, 0 p0, 0 p0, 0 0 1 2 p1, 0 p1, 0 p1, 0 1 p2, 0 0 2 p2, 0 p2, 0 B 3 Triangulating the Delaunay triangulation in T ? 3D Periodic Space Manuel Caroli, Monique Teillaud Idea: Contributions by Consider the image under π of an Nico Kruithof infinite periodic Delaunay triangulation in R3 Applications Flat torus
Triangulations
Covering Spaces
Algorithm
Extensions and Future work 3 Triangulating the Delaunay triangulation in T ? 3D Periodic Space Manuel Caroli, Monique Teillaud Idea: Contributions by Consider the image under π of an Nico Kruithof infinite periodic Delaunay triangulation in R3 Applications Flat torus
Triangulations
Covering Spaces
Algorithm
Extensions and Future work 3 Triangulating the Delaunay triangulation in T ? 3D Periodic Space Manuel Caroli, Monique Teillaud S finite point set in D Contributions by Nico Kruithof
3 Applications DTT(S) Delaunay triangulation of π(S) in T Flat torus = projection under π Triangulations
ϕ 3 3 Covering Spaces of the Delaunay triangulation DTR(S ) of ϕ(S × Z ) in R Algorithm only if π(DT (Sϕ)) is a simplicial complex Extensions and R Future work Triangulating the Delaunay triangulation of a periodic point set 3D Periodic Space Manuel Caroli, Monique Teillaud Contributions by Nico Kruithof
Applications
Flat torus
Triangulations
Covering Spaces
Algorithm
Extensions and Future work
S Triangulating the Delaunay triangulation of a periodic point set 3D Periodic Space Manuel Caroli, Monique Teillaud Contributions by Nico Kruithof
Applications
Flat torus
Triangulations
Covering Spaces
Algorithm
Extensions and Future work
ϕ(S × Z) Triangulating the Delaunay triangulation of a periodic point set 3D Periodic Space Manuel Caroli, Monique Teillaud Contributions by Nico Kruithof
Applications
Flat torus
Triangulations
Covering Spaces
Algorithm
Extensions and Future work
Why is DT (ϕ(S × Z)) a triangulation? Triangulating the Motivation for definition of DTT(S) 3D Periodic Space Manuel Caroli, S 3 Monique Teillaud σ∈DT (S) σ homeomorphic to T Contributions by T Nico Kruithof Is π(DT (Sϕ)) a simplicial complex ? R Applications
Flat torus Theorem Triangulations ϕ Covering Spaces If for all vertices v of DTR(S ) Algorithm π| is a homeomorphism, |St(v)| Extensions and ϕ Future work then π(DTR(S )) is a simplicial complex. Triangulating the Proof of the theorem 3D Periodic Space Manuel Caroli, Monique Teillaud Theorem Contributions by ϕ Nico Kruithof If for all vertices v of DTR(S ), π||St(v)| is a homeomorphism, ϕ then π(DTR(S )) is a simplicial complex. Applications Flat torus
Proof sketch: Triangulations ϕ Covering Spaces σ simplex in DTR(S ) Algorithm If π|σ homeomorphism ⇒ simplices “match” under π Extensions and Future work Degenerate cases solved by symbolic perturbation, invariant by translations [Devillers-Teillaud 03] Triangulating the Proof of the theorem 3D Periodic Space Manuel Caroli, Monique Teillaud Theorem Contributions by ϕ Nico Kruithof If for all vertices v of DTR(S ), π||St(v)| is a homeomorphism, ϕ then π(DTR(S )) is a simplicial complex. Applications Flat torus
Proof sketch: Triangulations 1. simplices “match” under π Covering Spaces Algorithm π(DT (Sϕ)) is finite Extensions and R Future work ϕ follows from local finiteness of DTR(S ) and 1. Triangulating the Proof of the theorem 3D Periodic Space Manuel Caroli, Monique Teillaud Theorem Contributions by ϕ Nico Kruithof If for all vertices v of DTR(S ), π||St(v)| is a homeomorphism, ϕ then π(DTR(S )) is a simplicial complex. Applications Flat torus
Proof sketch: Triangulations 1. simplices “match” under π Covering Spaces Algorithm 2. π(DT (Sϕ)) is finite R Extensions and Future work ϕ σR, τR simplices in DTR(S ) π maintains the incidence relation, i.e.
τR face of σR ⇒ π(τR) face of π(σR) Triangulating the Proof of the theorem 3D Periodic Space Manuel Caroli, Monique Teillaud Theorem Contributions by ϕ Nico Kruithof If for all vertices v of DTR(S ), π||St(v)| is a homeomorphism, ϕ then π(DTR(S )) is a simplicial complex. Applications Flat torus
Proof sketch: Triangulations 1. simplices “match” under π Covering Spaces Algorithm 2. π(DT (Sϕ)) is finite R Extensions and Future work 3. π(τR) face of π(σR)
ϕ σ, τ simplices in π(DTR(S )) Lemma ϕ σ ∩ τ consists of simplices in π(DTR(S )) Proof using 1. Triangulating the Proof of the theorem 3D Periodic Space Manuel Caroli, Monique Teillaud Theorem Contributions by ϕ Nico Kruithof If for all vertices v of DTR(S ), π||St(v)| is a homeomorphism, ϕ then π(DTR(S )) is a simplicial complex. Applications Flat torus
Proof sketch: Triangulations 1. simplices “match” under π Covering Spaces Algorithm 2. π(DT (Sϕ)) is finite R Extensions and Future work 3. π(τR) face of π(σR) ϕ 4. σ ∩ τ ⊆ π(DTR(S ))
ϕ σ, τ simplices in π(DTR(S )) Remains to show: #(σ ∩ τ) = 1
Use π|St(v)| is a homeomorphism Triangulating the Proof of the theorem 3D Periodic Space Manuel Caroli, Monique Teillaud Theorem Contributions by ϕ Nico Kruithof If for all vertices v of DTR(S ), π||St(v)| is a homeomorphism, ϕ then π(DTR(S )) is a simplicial complex. Applications Flat torus
Proof sketch: Triangulations 1. simplices “match” under π Covering Spaces Algorithm 2. π(DT (Sϕ)) is finite R Extensions and Future work 3. π(τR) face of π(σR) ϕ 4. σ ∩ τ ⊆ π(DTR(S )) ϕ 5. σ ∩ τ ∈ π(DTR(S )) Triangulating the Proof of the theorem 3D Periodic Space Manuel Caroli, Monique Teillaud Theorem Contributions by ϕ Nico Kruithof If for all vertices v of DTR(S ), π||St(v)| is a homeomorphism, ϕ then π(DTR(S )) is a simplicial complex. Applications Flat torus
Proof sketch: Triangulations 1. simplices “match” under π Covering Spaces Algorithm 2. π(DT (Sϕ)) is finite local finiteness R Extensions and Future work 3. π(τR) face of π(σR) criterion 1 ϕ 4. σ ∩ τ ⊆ π(DTR(S )) ϕ 5. σ ∩ τ ∈ π(DTR(S )) criterion 2 Triangulating the Another criterion 3D Periodic Space Manuel Caroli, Monique Teillaud Theorem Contributions by ϕ Nico Kruithof π(DTR(S )) is a simplicial complex (if and) only if Applications no cycles of length 2 in its 1-skeleton Flat torus Triangulations
Covering Spaces
Algorithm
Extensions and Future work
p ϕ Triangulating the π(DTR(S )) is not always a simplicial complex 3D Periodic Space Manuel Caroli, Monique Teillaud Contributions by Nico Kruithof
Applications
Flat torus
Triangulations
Covering Spaces
Algorithm
Extensions and Future work
One point in T2 ϕ Triangulating the π(DTR(S )) is not always a simplicial complex 3D Periodic Space Manuel Caroli, Monique Teillaud Contributions by Nico Kruithof
Applications
Flat torus
Triangulations
Covering Spaces
Algorithm
Extensions and Future work
One point in T2 ϕ Triangulating the π(DTR(S )) is not always a simplicial complex 3D Periodic Space Manuel Caroli, Monique Teillaud Contributions by Nico Kruithof
Applications
Flat torus
Triangulations
Covering Spaces
Algorithm
Extensions and Future work
One point in T2 ϕ Triangulating the π(DTR(S )) is not always a simplicial complex 3D Periodic Space Manuel Caroli, Monique Teillaud Contributions by Nico Kruithof
Applications
Flat torus
Triangulations
Covering Spaces
Algorithm
Extensions and Future work
One point in T2: self-edges ϕ Triangulating the π(DTR(S )) is not always a simplicial complex 3D Periodic Space Manuel Caroli, Monique Teillaud Contributions by Nico Kruithof
Applications
Flat torus
Triangulations
Covering Spaces
Algorithm
Extensions and Future work
One point in T3: self-edges ϕ Triangulating the π(DTR(S )) is not always a simplicial complex 3D Periodic Space Manuel Caroli, Monique Teillaud Contributions by Nico Kruithof
Applications
Flat torus
Triangulations
Covering Spaces
Algorithm
Extensions and Future work
Cycle of length 2 Triangulating the Outline 3D Periodic Space Manuel Caroli, Monique Teillaud Contributions by Applications Nico Kruithof
Applications
Flat torus Flat torus Triangulations
Covering Spaces Triangulations Algorithm Extensions and Future work Covering Spaces
Algorithm
Extensions and Future work Triangulating the Covering Space 3D Periodic Space Manuel Caroli, Monique Teillaud X a topological space. Contributions by Nico Kruithof ρ : X˜ → X is a covering map, and ˜ is a covering space of if: Applications X X Flat torus ∀x ∈ X Triangulations Covering Spaces I ∃Vx open neighborhood of x Algorithm −1 I ∃ a decomposition of ρ (Vx ) as a family {Uαx }, Extensions and ˜ Future work Uαx ⊂ X pairwise disjoint s.t. ρ| is a homeomorphism for each α . Uαx x ˜ If h = maxx∈X|Uαx | is finite, then X = h-sheeted covering space. Triangulating the Covering Space 3D Periodic Space Manuel Caroli, Monique Teillaud Contributions by Nico Kruithof
Applications
Flat torus
Triangulations
Covering Spaces
Algorithm
Extensions and Future work
1-sheeted covering Triangulating the Covering Space 3D Periodic Space Manuel Caroli, Monique Teillaud Contributions by Nico Kruithof
Applications
Flat torus
Triangulations
Covering Spaces
Algorithm
Extensions and Future work
2-sheeted covering Triangulating the Covering Space 3D Periodic Space Manuel Caroli, Monique Teillaud Contributions by Nico Kruithof
Applications
Flat torus
Triangulations
Covering Spaces
Algorithm
Extensions and Future work
4-sheeted covering Triangulating the Covering Space 3D Periodic Space Manuel Caroli, Monique Teillaud Contributions by Nico Kruithof
Applications
Flat torus
Triangulations
Covering Spaces
Algorithm
Extensions and Future work
∞-sheeted covering = R2 = universal covering ϕ π27(DTR(S )) is always a simplicial complex
Proof uses [Dolbilin-Huson 97] but the simplicial complex is homeomorphic to T3
3 Triangulating the T27 3D Periodic Space Manuel Caroli, Monique Teillaud Contributions by Nico Kruithof
Applications
Flat torus
3 3 Triangulations T27 = 3 -sheeted covering Covering Spaces
Algorithm 3 3 3 T27 = R /3Z Extensions and Future work 3 3 π27 : R → T27 3 Triangulating the T27 3D Periodic Space Manuel Caroli, Monique Teillaud Contributions by Nico Kruithof
Applications
Flat torus
3 3 Triangulations T27 = 3 -sheeted covering Covering Spaces
Algorithm 3 3 3 T27 = R /3Z Extensions and Future work 3 3 π27 : R → T27
ϕ π27(DTR(S )) is always a simplicial complex
Proof uses [Dolbilin-Huson 97] but the simplicial complex is homeomorphic to T3 Triangulating the Outline 3D Periodic Space Manuel Caroli, Monique Teillaud Contributions by Applications Nico Kruithof
Applications
Flat torus Flat torus Triangulations
Covering Spaces Triangulations Algorithm Extensions and Future work Covering Spaces
Algorithm
Extensions and Future work As soon as possible = ϕ I not just when π(DTR(S )) is a simplicial complex ϕ I but when π(DTR(T )) is guaranteed to be a simplicial complex for any T ⊇ S
Triangulating the Incremental Algorithm 3D Periodic Space Manuel Caroli, Monique Teillaud Contributions by 3 Nico Kruithof I Compute initially in T27 27 copies of each point Applications
Flat torus
I As soon as possible Triangulations Compute in T3 Covering Spaces Algorithm
Extensions and Future work Triangulating the Incremental Algorithm 3D Periodic Space Manuel Caroli, Monique Teillaud Contributions by 3 Nico Kruithof I Compute initially in T27 27 copies of each point Applications
Flat torus
I As soon as possible Triangulations Compute in T3 Covering Spaces Algorithm
Extensions and As soon as possible = Future work ϕ I not just when π(DTR(S )) is a simplicial complex ϕ I but when π(DTR(T )) is guaranteed to be a simplicial complex for any T ⊇ S Triangulating the Ball diameter criterion 3D Periodic Space
ϕ Manuel Caroli, ∆ tetrahedron in π(DT (S )) Monique Teillaud R Contributions by Nico Kruithof Circumsphere diameter of any ∆ < 1 2 Applications
ϕ Flat torus ⇒ π(DTR(S )) is a triangulation Triangulations
Covering Spaces
Algorithm
Extensions and Future work Triangulating the Edge length criterion 3D Periodic Space Manuel Caroli, ϕ Monique Teillaud e edge in π(DTR(S )) Contributions by Nico Kruithof Length of any e < √1 6 Applications Flat torus ⇒ π(DT (Sϕ)) is a triangulation R Triangulations
Covering Spaces
Algorithm
Extensions and √ Future work 3 2 1 2 Triangulating the Properties 3D Periodic Space Manuel Caroli, Monique Teillaud I Structure always a Delaunay triangulation Contributions by Nico Kruithof [Thompson 02] computes a “tessellation” not always a triangulation Applications Flat torus not always Delaunay Triangulations
Covering Spaces I No duplication of points if possible Algorithm Extensions and [Dolbilin-Huson 97] and others always 27 copies Future work
Vertex removal works too Triangulating the Theoretical Analysis 3D Periodic Space Manuel Caroli, Monique Teillaud Contributions by Nico Kruithof 2 Randomized worst-case optimal O(n ) Applications
with adapted Delaunay hierarchy [Devillers 02] Flat torus
Triangulations
Covering Spaces
Algorithm
Extensions and Future work Triangulating the Experimental Analysis 3D Periodic Space Manuel Caroli, Monique Teillaud Optimizations Contributions by Nico Kruithof I Spatial sorting from CGAL [Delage] Applications I Optional insertion of dummy points to force Flat torus 1-sheeted covering Triangulations
Covering Spaces
- Data from research in cosmology Algorithm - Random points Extensions and Future work 1 million random points in 23 seconds 2.33 GHz Intel Core 2 Duo processor
Factor ' 1.6 compared to Delaunay triangulation in R3 on large data sets in CGAL [Pion-Teillaud] Triangulating the Outline 3D Periodic Space Manuel Caroli, Monique Teillaud Contributions by Applications Nico Kruithof
Applications
Flat torus Flat torus Triangulations
Covering Spaces Triangulations Algorithm Extensions and Future work Covering Spaces
Algorithm
Extensions and Future work Triangulating the Extensions 3D Periodic Space Manuel Caroli, I General cube: same Monique Teillaud Contributions by Nico Kruithof I Weighted Delaunay triangulation: adapted number of sheets Applications Flat torus I Iso-cuboid: adapted number of sheets Triangulations Non-orthogonal translations: similar I Covering Spaces
Algorithm
Extensions and Future work Triangulating the Extensions 3D Periodic Space Manuel Caroli, I General cube: same Monique Teillaud Contributions by Nico Kruithof I Weighted Delaunay triangulation: adapted number of sheets Applications Flat torus I Iso-cuboid: adapted number of sheets Triangulations Non-orthogonal translations: similar I Covering Spaces
Algorithm
Extensions and Future work
parallelogram: find shortest translation vectors Triangulating the Extensions 3D Periodic Space Manuel Caroli, I General cube: same Monique Teillaud Contributions by Nico Kruithof I Weighted Delaunay triangulation: adapted number of sheets Applications Flat torus I Iso-cuboid: adapted number of sheets Triangulations Non-orthogonal translations: similar I Covering Spaces
Algorithm
Extensions and Future work
parallelogram: find shortest translation vectors Triangulating the Extensions 3D Periodic Space Manuel Caroli, I General cube: same Monique Teillaud Contributions by Nico Kruithof I Weighted Delaunay triangulation: adapted number of sheets Applications Flat torus I Iso-cuboid: adapted number of sheets Triangulations Non-orthogonal translations: similar I Covering Spaces
Algorithm
Extensions and Future work
parallelogram: find shortest translation vectors Triangulating the Extensions 3D Periodic Space Manuel Caroli, I General cube: same Monique Teillaud Contributions by Nico Kruithof I Weighted Delaunay triangulation: adapted number of sheets Applications Flat torus I Iso-cuboid: adapted number of sheets Triangulations Non-orthogonal translations: similar I Covering Spaces
Algorithm
Extensions and Future work
parallelogram: find shortest translation vectors Triangulating the Extensions 3D Periodic Space Manuel Caroli, I General cube: same Monique Teillaud Contributions by Nico Kruithof I Weighted Delaunay triangulation: adapted number of sheets Applications Flat torus I Iso-cuboid: adapted number of sheets Triangulations Non-orthogonal translations: similar I Covering Spaces
Algorithm
Extensions and Future work
parallelogram: find shortest translation vectors Triangulating the Extensions 3D Periodic Space Manuel Caroli, I General cube: same Monique Teillaud Contributions by Nico Kruithof I Weighted Delaunay triangulation: adapted number of sheets Applications Flat torus I Iso-cuboid: adapted number of sheets Triangulations Non-orthogonal translations: similar I Covering Spaces
Algorithm
Extensions and Future work
parallelogram: find shortest translation vectors Triangulating the Extensions 3D Periodic Space Manuel Caroli, I General cube: same Monique Teillaud Contributions by Nico Kruithof I Weighted Delaunay triangulation: adapted number of sheets Applications Flat torus I Iso-cuboid: adapted number of sheets Triangulations Non-orthogonal translations: similar I Covering Spaces I Other groups? Algorithm Extensions and Future work
reflection: always a self-edge Triangulating the Extensions 3D Periodic Space Manuel Caroli, I General cube: same Monique Teillaud Contributions by Nico Kruithof I Weighted Delaunay triangulation: adapted number of sheets Applications Flat torus I Iso-cuboid: adapted number of sheets Triangulations Non-orthogonal translations: similar I Covering Spaces I Other groups? Algorithm Extensions and Future work
rotation: always self-edges Triangulating the Future work 3D Periodic Space Manuel Caroli, Monique Teillaud Contributions by I Periodic meshes Nico Kruithof
I Periodic alpha shapes Applications Flat torus
I Applications Triangulations Covering Spaces
Algorithm I Other orbifolds 3 Extensions and I of R (e.g. crystallographic space groups) Future work I of the sphere projective plane [Aanjaneya-Teillaud 07] I of the hyperbolic space I general approach
I Preliminary work with M. Aanjaneya Triangulating the 3D Periodic Space
Manuel Caroli, Monique Teillaud Contributions by Nico Kruithof
Applications
Flat torus Demo Triangulations Covering Spaces
(to appear in CGAL 3.5) Algorithm
Extensions and Future work Triangulating the 3D Periodic Space
Manuel Caroli, Monique Teillaud Contributions by Nico Kruithof
Applications
Flat torus
Triangulations
Covering Spaces Thank you! Algorithm Extensions and Future work
Accepted for next release of Research Report 6823 (hal.inria.fr/inria-00356871/) To appear in Proceedings of the 17th European Symposium on Algorithms (ESA)