The document discusses meshing periodic surfaces in CGAL. It describes adapting the CGAL surface meshing algorithm to work with periodic triangulations by modifying point insertion and refinement criteria. Examples meshing various periodic minimal surfaces like the gyroid and schwarz P surface are shown for different criteria values. Future work includes improving the refinement criteria to handle all cases and proving algorithm correctness and termination.

1. . . . . . .
Meshing Periodic Surfaces in CGAL
Vissarion Fisikopoulos
advisor: Monique Teillaud
& )KCKIJ '

2. . . . . . .
Outline
..1 Preliminaries
Periodic Triangulations [CT09]
Surface Meshing [BO05]
..2 Periodic Surface Meshing
Adjust Algorithm
Adjust Renement Criteria
..3 Examples
..4 Conclusion - Future Work

3. . . . . . .
Periodic Triangulations
.
.
v3
v1
v1
v2
Periodic Space
• T!
c := R!
/Z!
Denitions
• domain: cube contains exactly one representative of each element
• compute in original domain (one contains the origin)
• offset (ox, oy, oz): the number of periods far from the original domain
Valid Triangulations
• problem: not all point sets have a triangulation in T!
c
• solutions [CT09]:
• compute in 27-sheeted covering space
• insert 36 dummy points at the beginning

4. . . . . . .
Surface Meshing - Denitions
.
V(P)
S
.
f
c
e
S
Restricted Delaunay triangulation (rDt) 2D complex
in 3D Delaunay triangulation of facets whose dual
Voronoi segments intersect surface
Surface Delaunay ball
ball circumscribing facet with center an intersection
of dual Voronoi edge of facet with the surface
Boundary Facets
facets of restricted Delaunay triangulation

5. . . . . . .
Surface Meshing - Renement Criteria
.
Criteria
• angular bound
minimum angle
• radius bound
radius of the surface delaunay ball
• distance bound
center of a boundary facet and center of its
surface delaunay ball
Bad Facets: boundary facets not meeting criteria

6. . . . . . .
Surface Meshing - The Algorithm
Initialization Step
• nd points on surface (ray
shooting, dichotomic search)
• insert points to triangulation
• construct rDt and queue for bad
facets
Renement Step
• for each bad facet:
• insert the center of its surface
delaunay ball
• update rDt and bad facets'
queue

7. . . . . . .
Periodic Surface Meshing - Algorithm
CGAL 3D Surface Mesher with 3D Periodic
Delaunay Triangulations
Modications
• Point insertion
(Initialization step, Renement step)
translate points inside the domain (use offsets)

8. . . . . . .
Periodic Surface Meshing - Algorithm
CGAL 3D Surface Mesher with 3D Periodic
Delaunay Triangulations
Modications
• Point insertion
(Initialization step, Renement step)
translate points inside the domain (use offsets)
• Star approach
keep track of offsets during insertion

9. . . . . . .
Periodic Surface Meshing - Algorithm
CGAL 3D Surface Mesher with 3D Periodic
Delaunay Triangulations
Modications
• Point insertion
(Initialization step, Renement step)
translate points inside the domain (use offsets)
• Star approach
keep track of offsets during insertion

10. . . . . . .
Periodic Surface Meshing - Algorithm
CGAL 3D Surface Mesher with 3D Periodic
Delaunay Triangulations
Modications
• Point insertion
(Initialization step, Renement step)
translate points inside the domain (use offsets)
• Star approach
keep track of offsets during insertion

11. . . . . . .
Periodic Surface Meshing - Algorithm
CGAL 3D Surface Mesher with 3D Periodic
Delaunay Triangulations
Modications
• Point insertion
(Initialization step, Renement step)
translate points inside the domain (use offsets)
• Star approach
keep track of offsets during insertion
• Renement Criteria
algorithm might compute the wrong quantity

12. . . . . . .
Periodic Surface Meshing - Criteria (1)
Aspect Ratio Criterion (Angular Bound) - Edge Length Criterion
.
v2
v3
v3
domain
criterion: translate all points
v1
Angular Bound: Aspect Ratio Criterion
φφ∗
.
Edge Size Criterion
v2
v3
v3
domain
criterion: translate all points
v1

13. . . . . . .
Periodic Surface Meshing - Criteria (2)
Uniform Size Criterion (Radius Bound)
.
v2
v3
d1
s
d2
v3
domain
criterion1 := min{(s, v1), (s, v2), (s, v3)}
v1
Case 1Radius Bound: Uniform Size Criterion
intersection point
.
v1
domain
criterion2 := min{(s, v1), (s∗
, v1)}
v2
v3
d1
d2
ss∗
Case 2Radius Bound: Uniform Size Criterion
intersection point
criterion: min{(s, v), (s, v ), (s, v!), (s∗
, v)}

14. . . . . . .
Periodic Surface Meshing - Criteria (3)
Curvature Size Criterion (Distance Bound)
.
v2
v3
d1
s
d2
domain
criterion1 := min{(s, c), (s, c∗
)}
v1
Case 1
cc∗
Distance Bound: Curvature Size Criterion
intersection point
.
v1
domain
criterion2 := min{(s, c), (s∗
, c)}
v2
v3
d1
d2
s∗
Case 2
c
Distance Bound: Curvature Size Criterion
s
intersection point
criterion: min{(s, c), (s, c∗
), (s∗
, c)}

15. . . . . . .
Examples
double p
0 = 0.5 · (cosx · cosy + cosy · cosz + cosz · cosx) + 0.2 · (cos2x + cos2y + cos2z)
schwarz p
0 = cosx + cosy + cosz
. .

16. . . . . . .
criteria: angular = 30 radius = 0.1 distance = 0.1
.
cylinder
.
diamond
.
double p
.
D prime
.
G prime
.
gyroid
.
lidinoid
.
schwarz p
.
split p
.
.
.
.
.
.

17. . . . . . .
criteria: angular = 30 radius = 0.05 distance = 0.05
.
cylinder
.
diamond
.
double p
.
D prime
.
G prime
.
gyroid
.
lidinoid
.
schwarz p
.
split p
.
.
.
.
.
.

18. . . . . . .
criteria: angular = 30 radius = 0.01 distance = 0.01
.
cylinder
.
diamond
.
double p
.
D prime
.
G prime
.
gyroid
.
lidinoid
.
schwarz p
.
split p
.
.
.
.
.
.

19. . . . . . .
Bad Examples
schwarz p: 0 = cos(2πx + π
) + cos(2πy + π
) + cos(2πz + π
)
.
angular = ! radius = . distance = .
.
angular = ! radius = .# distance = .#
. .

20. . . . . . .
Examples
double p
0 = 0.5 · (cosx · cosy + cosy · cosz + cosz · cosx) + 0.2 · (cos2x + cos2y + cos2z)
schwarz p
0 = cosx + cosy + cosz
. .

21. . . . . . .
criteria: angular = 30 radius = 0.1 distance = 0.1
.
cylinder
.
diamond
.
double p
.
D prime
.
G prime
.
gyroid
.
lidinoid
.
schwarz p
.
split p
.
.
.
.
.
.

22. . . . . . .
criteria: angular = 30 radius = 0.05 distance = 0.05
.
cylinder
.
diamond
.
double p
.
D prime
.
G prime
.
gyroid
.
lidinoid
.
schwarz p
.
split p
.
.
.
.
.
.

23. . . . . . .
Conclusion - Future Work
Conclusions
• The algorithm terminates for all of the examples
• The algorithm can mesh many periodic minimal surfaces
Future Work
• Improve the renement criteria in order to handle all cases
• Prove the correctness and termination for the new criteria

24. . . . . . .
.
Thank You for Your Attention!
.
cylinder
.
diamond
.
double p
.
D prime
.
G prime
.
gyroid
.
lidinoid
.
schwarz p
.
split p
.
.
.
.
.
.

25. . . . . . .
Appendix
cylinder:
0 = x + y
diamond:
0 = sinx · siny · sinz + sinx · cosy · cosz + cosx · siny · cosz + cosx · cosy · sinz
double p:
0 = 0.5 · (cosx · cosy + cosy · cosz + cosz · cosx) + 0.2 · (cos2x + cos2y + cos2z)
D prime:
0 = 1 · (sinx · siny · sinz) + 1 · (cosx · cosy · cosz) − 1 · (cos2x · cos2y + cos2y ·
cos2z + cos2z · cos2x)) − 0.4
G prime:
0 = 5 · (sin2x · sinz · cosy + sin2y · sinx · cosz + sin2z · siny · cosx) + 1 · (cos2x ·
cos2y + cos2y · cos2z + cos2z · cos2x)

26. . . . . . .
Appendix
gyroid:
0 = cosx · siny + cosy · sinz + cosz · sinx
lidinoid:
0 = 1 · (sin2x · sinz · cosy + sin2y · sinx · cosz + sin2z · siny · cosx) − 1 · (cos2x ·
cos2y + cos2y · cos2z + cos2z · cos2x) + 0.3
schwarz p:
0 = cosx + cosy + cosz
split p:
0 = 1.1 · (sin2x · sinz · cosy + sin2y · sinx · cosz + sin2z · siny · cosx) − 0.2 ·
(cos2x · cos2y + cos2y · cos2z + cos2z · cos2x) − 0.4 · (cosx + cosy + cosz)

27. . . . . . .
References
Jean-Daniel Boissonnat and Steve Oudot.
Provably good sampling and meshing of surfaces.
Graph. Models, 67(5):405–451, 2005.
Manuel Caroli and Monique Teillaud.
Computing 3D Periodic Triangulations.
Research Report RR-6823, INRIA, 2009.