Construction of
Sparse Well-Spaced Point Sets
for Quality Tetrahedralizations
Ravi Jampani
Alper Üngör
Computer and Information Science and Engineering
University of Florida
{rjampani,ungor}@cise.ufl.edu
Sparse Well-Spaced Point Sets
IMR 2007, Seatle, WA
Problem: Quality Steiner Triangulation (2D & 3D)
Given an input domain, compute a small size triangulation such that
all elements (triangles/tetrahedra) are of good quality.
Boeing wing model (airflow simulation)
Applications:
Map of a country
Engineering, Graphics, Visualization, CAD, GIS, ...
1
Sparse Well-Spaced Point Sets
IMR 2007, Seatle, WA
Problem: Quality Steiner Triangulation (2D & 3D)
Given an input domain, compute a small size triangulation such that
all elements (triangles/tetrahedra) are of good quality.
Boeing wing model (airflow simulation)
Applications:
Map of a country
Engineering, Graphics, Visualization, CAD, GIS, ...
1
Sparse Well-Spaced Point Sets
IMR 2007, Seatle, WA
Outline of the Talk
• Review of Delaunay Refinement
BernEppsteinGilbert92, Chew89, Ruppert93, Shewchuk97,
EdelsbrunnerGuoy01, RivaraSimpsonHitschfeld01, Miller04
MillerPavWalkington03, Üngör04, HarPeledÜngör05, ...
• Sparse well-spaced point sets in 2D
ErtenÜngör SGP07
• Sparse well-spaced point sets in 3D
JampaniÜngör IMR07
• Experimental Results
2
Sparse Well-Spaced Point Sets
IMR 2007, Seatle, WA
Delaunay Refinement Algorithm 2D/3D
Input: A domain Ω, quality constraint β
Compute the Delaunay triangulation of Ω
while ∃ a bad triangle/tetrahedron do
Insert a Steiner point
Update the Delaunay triangulation
end while
• Angle guarantee Chew89
• Size-optimal Ruppert93
• Triangle Shewchuk96
• 3D Shewchuk97
3
Sparse Well-Spaced Point Sets
IMR 2007, Seatle, WA
Delaunay Refinement Algorithm 2D/3D
Input: A domain Ω, quality constraint β
Compute the Delaunay triangulation of Ω
while ∃ a bad triangle/tetrahedron do
Insert a Steiner point
Update the Delaunay triangulation
end while
• Angle guarantee Chew89
• Size-optimal Ruppert93
• Triangle Shewchuk96
• 3D Shewchuk97
3
Sparse Well-Spaced Point Sets
IMR 2007, Seatle, WA
Delaunay Refinement Algorithm 2D/3D
Input: A domain Ω, quality constraint β
Compute the Delaunay triangulation of Ω
while ∃ a bad triangle/tetrahedron do
Insert a Steiner point
Update the Delaunay triangulation
end while
• Angle guarantee Chew89
• Size-optimal Ruppert93
• Triangle Shewchuk96
• 3D Shewchuk97
3
Sparse Well-Spaced Point Sets
IMR 2007, Seatle, WA
Delaunay Refinement Algorithm 2D/3D
Input: A domain Ω, quality constraint β
Compute the Delaunay triangulation of Ω
while ∃ a bad triangle/tetrahedron do
Insert a Steiner point
Update the Delaunay triangulation
end while
• Angle guarantee Chew89
• Size-optimal Ruppert93
• Triangle Shewchuk96
• 3D Shewchuk97
3
Sparse Well-Spaced Point Sets
IMR 2007, Seatle, WA
Delaunay Refinement Algorithm 2D/3D
Input: A domain Ω, quality constraint β
Compute the Delaunay triangulation of Ω
while ∃ a bad triangle/tetrahedron do
Insert a Steiner point
Update the Delaunay triangulation
end while
• Angle guarantee Chew89
• Size-optimal Ruppert93
• Triangle Shewchuk96
• 3D Shewchuk97
3
Sparse Well-Spaced Point Sets
IMR 2007, Seatle, WA
Delaunay Refinement Algorithm 2D/3D
Input: A domain Ω, quality constraint β
Compute the Delaunay triangulation of Ω
while ∃ a bad triangle/tetrahedron do
Insert a Steiner point
Update the Delaunay triangulation
end while
• Angle guarantee Chew89
• Size-optimal Ruppert93
• Triangle Shewchuk96
• 3D Shewchuk97
3
Sparse Well-Spaced Point Sets
IMR 2007, Seatle, WA
Delaunay Refinement Algorithm 2D/3D
Input: A domain Ω, quality constraint β
Compute the Delaunay triangulation of Ω
while ∃ a bad triangle/tetrahedron do
Insert a Steiner point
Update the Delaunay triangulation
end while
• Angle guarantee Chew89
• Size-optimal Ruppert93
• Triangle Shewchuk96
• 3D Shewchuk97
3
Sparse Well-Spaced Point Sets
IMR 2007, Seatle, WA
Types of Steiner Points
circumcenter
sink
offcenter
Field, Joe, Chew89
Ruppert93, Shewchuk97
EdelsbrunnerGuoy01
Üngör04
Har-PeledÜngör05
4
Sparse Well-Spaced Point Sets
IMR 2007, Seatle, WA
Sparse Well-Spaced Point Sets in 2D
Careful choice of Steiner points lead to Smaller Meshes
Ruppert93/Shewchuk97 Triangle 1.4
Our Algorithm
5
Sparse Well-Spaced Point Sets
IMR 2007, Seatle, WA
Sparse Well-Spaced Point Sets in 2D
Careful choice of Steiner points lead to Smaller Meshes
Ruppert93/Shewchuk97 Triangle
Our Algorithm
6
Sparse Well-Spaced Point Sets
IMR 2007, Seatle, WA
Sparse, Dense, Denser, Densest
In practice, previous Delaunay refinement algorithms
Chew89, Ruppert93, Shewchuk97, EdelsbrunnerGuoy01, Üngör04
• Terminates for α < 34◦
• No termination for α ≥ 34◦
7
Sparse Well-Spaced Point Sets
IMR 2007, Seatle, WA
Sparse, Dense, Denser, Densest
In practice, previous Delaunay refinement algorithms
Chew89, Ruppert93, Shewchuk97, EdelsbrunnerGuoy01, Üngör04
• Terminates for α < 34◦
• No termination for α ≥ 34◦
7
Sparse Well-Spaced Point Sets
IMR 2007, Seatle, WA
Sparse, Dense, Denser, Densest
In practice, previous Delaunay refinement algorithms
Chew89, Ruppert93, Shewchuk97, EdelsbrunnerGuoy01, Üngör04
• Terminates for α < 34◦
• No termination for α ≥ 34◦
7
Sparse Well-Spaced Point Sets
IMR 2007, Seatle, WA
Sparse, Dense, Denser, Densest
In practice, previous Delaunay refinement algorithms
Chew89, Ruppert93, Shewchuk97, EdelsbrunnerGuoy01, Üngör04
• Terminates for α < 34◦
• No termination for α ≥ 34◦
7
Sparse Well-Spaced Point Sets
IMR 2007, Seatle, WA
Sparse, Dense, Denser, Densest
In practice, previous Delaunay refinement algorithms
Chew89, Ruppert93, Shewchuk97, EdelsbrunnerGuoy01, Üngör04
• Terminates for α < 34◦
• No termination for α ≥ 34◦
7
Sparse Well-Spaced Point Sets
IMR 2007, Seatle, WA
Improving the Termination Bound
ErtenÜngör Symp. on Geometry Processing 2007, Barcelona Spain
• Idea I: A unified Steiner point definition
r
t
x
q
q
q
q
r
r
x
a
a
a
x
p
p
p
r
x
p
s
s
• Idea II: Integration of Steiner point relocation with refinement
q
a
b
p
8
Sparse Well-Spaced Point Sets
IMR 2007, Seatle, WA
Termination upto 42◦
α = 41◦
α = 41◦
α = 42◦
α = 41.5◦
9
Sparse Well-Spaced Point Sets
IMR 2007, Seatle, WA
Performance Comparison
1000 Random Points
Steiner Points (logscale)
106
Triangle 1.4
Triangle 1.6
Algorithm 1
Algorithm 2
105
104
103
30
32
34
36
38
Angle Threshold
40
42
Plots for other data sets are similar. See ErtenUngor07.
10
Sparse Well-Spaced Point Sets
IMR 2007, Seatle, WA
CoreDisk
Definition 1 Let S be the set of spheres that have radius β|pq| and go through
the points p and q. The disk formed by the centers of the spheres in S is called
the core disk of pq, denoted as CoreDisk(pq).
pq|
β|
pq
Note that CoreDisk(pq) is coplanar with V or(pq).
11
Sparse Well-Spaced Point Sets
IMR 2007, Seatle, WA
CoreDisk vs. Voronoi region
Definition 2 A Delaunay edge pq that is not on the boundary is said to be
bad if V or(pq) * CoreDisk(pq). A boundary Delaunay edge is bad if there
exists a vertex of V or(pq) that is not in CoreDisk(pq).
Theorem 1 Delaunay triangulation of a point set has bad tetrahedra if and
only if there is a bad edge in the triangulation.
12
Sparse Well-Spaced Point Sets
IMR 2007, Seatle, WA
CoreDisk vs. Voronoi region
For each bad edge pq, Compute CoreDisk(pq) − V or(pq)
v1
v2
a
Rc
pq
γ
b
v3
v4
13
Sparse Well-Spaced Point Sets
IMR 2007, Seatle, WA
CoreDisk vs. Voronoi region
Sample (multiple) Steiner points in CoreDisk(pq) − V or(pq)
v1
v2
a
s1
v’1
θ
pq
b
v3
s2
v4
14
Sparse Well-Spaced Point Sets
IMR 2007, Seatle, WA
Sample Output
Circumcenters Packing
New Packing
15
Sparse Well-Spaced Point Sets
IMR 2007, Seatle, WA
Sample Output
Circumcenters Packing
New Packing
16
Sparse Well-Spaced Point Sets
IMR 2007, Seatle, WA
Performance Statistics (See the paper for more)
Data Set
Quality
CC algorithm
New algorithm
Imp.
name
v
e
f
R/l
v
t
v
t
%
Cube1
12
12
6
1.5
596
3,420
570
3,280
4.3
2
265
1,440
220
1,196
16.9
4
153
730
79
330
48.3
8
133
607
51
195
61.6
1.5
69,075
431,908
60,861
379,718
11.8
2
39,710
247,031
32,394
200,960
18.4
4
21,396
131,553
17,256
103,380
19.3
8
15,202
89,501
13,328
75,238
12.3
1.5
50,527
310,743
38,965
239,330
22.8
2
26,259
157,163
16,596
99,521
36.7
4
12,763
71,812
5,938
37,738
53.4
8
8,727
49,998
3,923
37,738
55.0
Ellipsoid
Helix
10,008
1,008
12
12
6
6
17
Sparse Well-Spaced Point Sets
IMR 2007, Seatle, WA
Summary and Discussion
• We handle potentially multiple bad tetrahedra.
• We insert multiple Steiner points at each iteration.
• Algorithm is shown to terminate.
• Output point set is well-spaced and sparse.
• We currently explore other benefits of this idea, such as on computing
sliver-free meshing and on termination problem. Preliminary experiments
are promising.
• Send your comments and questions to [email protected]
18