:
Convex Hull construction
via
Star

Shaped
Polyhedron in 3D
Mingcen Gao
Thanh

Tung Cao
Tiow

Seng
Tan
Zhiyong
Huang
Flip

Flop
2
Outline
Flips on 3D polyhedron
Flip

Flop: a novel flip algorithm
ffHull
: 3D convex hull algorithm
Experiments
Flips on 3D polyhedron
3
1
Convex Hull in 3D
Convex Hull
•
Fundamental problem in CG
•
Collision detection
•
Path planning
•
CGAL (CPU)
•
qHull
(CPU)
•
gHull (GPU) [Gao et al. 2012]
Flips on 3D polyhedron
4
1
Flips ?
Compute CH by flips
non

extreme points
reflex edges
3

1 Edge
2

2 Edge
3

1 flip
2

2 flip
Flips on 3D polyhedron
5
1
Difficulty 1: Avoiding
self

intersection
Avoid self

intersection
2D illustration
one flip
Solution:
star

shaped polyhedron
Flips on 3D polyhedron
6
1
Star

shaped polyhedron
s
2D
3D
In star

shaped polyhedron, checking whether a flip creates self

intersection is a
local operation
.
Avoid self

intersection
Flips on 3D polyhedron
7
1
Difficulty 2: Getting
stuck
by Lawson’s flip algorithm
Break the stuck case
Flipping any reflex edges creates self

intersection.
Solution:
Flip

Flop
(for star

shaped polyhedron)
8
Flips on 3D polyhedron
Flip

Flop: a novel flip algorithm
ffHull
: 3D convex hull algorithm
Experiments
Outline
Flip

Flop:
a novel flip algorithm
9
2
An example
a
b
e
d
c
p
f
a
b
e
d
c
p
f
a
b
e
d
c
p
f
a
b
e
d
c
p
f
a
b
e
d
c
p
f
non

extreme point
Flip

Flop:
a novel flip algorithm
10
2
An example
Flips
Volume
Local optimum
Convex hull
Lawson’s flip
Flip

Flop
flip
flop
Flip

Flop:
a novel flip algorithm
11
2
Concepts
Flip

Flop
Increase the volume
Finding non

extreme points
Remove non

extreme points
Star

shaped
Polyhedron
Convex hull
Flip

Flop:
a novel flip algorithm
12
2
Finding non

extreme points
a
b
c
d
s
b (or a)
lies inside the
tetrahedron
sacd
(or
sbcd
)
ab
is a 2

2 edge that is
reflex and unflippable
b (or a)
is non

extreme
Flip

Flop:
a novel flip algorithm
13
2
Flipping criteria
V

Criterion
Increases the
olume of the polyhedron by flipping
reflex edges. (Used in Lawson’s flip algorithm)
D

Criterion
Decreases the
egree of the non

extreme (NE)
vertices by flipping the edges incident to them
degree(v)=5
degree(v)=4
degree(v)=3
degree(v)=0
For any non

extreme point, there is a 2

2 flippable edge incident to it!
Flip

Flop:
a novel flip algorithm
14
2
Flipping criteria
a
b
c
d
Combination of V

and D

criterion
Otherwise flip the edge
if it is reflex (V

criterion).
If a or b is known non

extreme,
flip
the edge (D

criterion).
If c or d is known non

extreme,
keep
the edge (D

criterion).
Use the index of the points to avoid flipping back and forth
Flip

Flop:
a novel flip algorithm
15
2
GPU implementation
while it is not
convex hull
▲
▲
▲
▲
▲
▲
▲
1. identify and label the non

extreme points.
2. If no edge need be flipped, stop the thread.
△
△
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
Flip the edge if the triangles in the flip is
labeled by the thread.
use atomic minimum operation to label the
triangles in the flip
16
Flips on 3D polyhedron
Flip

Flop: a novel flip algorithm
ffHull
: 3D convex hull algorithm
Experiments
Outline
ffHull
:
3D convex hull algorithm
17
3
Algorithm
Point
set
s
s
s
s
Construct
initial tetrahedron
using 4 extreme points,
and use its center as
the kernel
Apply
Flip

Flop
Insert the
furthest point
to grow
the polyhedron and keep the
star

shaped property
.
ffHull
:
3D convex hull algorithm
18
2
GPU implementation
while it is not
convex hull
▲
▲
▲
▲
▲
▲
▲
If the triangle are associated with some points, insert the furthest one.
Update the associated triangle of the point if needed.
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
△
△
▲
▲
●
●
●
●
●
●
●
●
●
●
●
●
●
ffHull
:
3D convex hull algorithm
19
3
Extensions
ffRT
: 2D regular triangulation algorithm
Compute many convex hulls concurrently
Point
set
Lift points by
z=x
2
+y
2

w
Compute
lower hull
Point
set
1
Point
set
n
…
…
Tetrahedron
…
…
…
Apply
ffHull
on all groups
20
Flips on 3D polyhedron
Flip

Flop: a novel flip algorithm
ffHull
: 3D convex hull algorithm
Experiments
Outline
Experiments
21
4
Machine:
•
Intel i7 2600K 3.4GHz CPU
•
16GB of DDR3 RAM
•
Nvidia
GTX 580 Fermi graphics with 3GB of video memory
Experiments
22
4
Convex hull
Ball of radius 0.5
1.4x↑
v.s
. gHull
20x↑
v.s
.
qHull
81x↑
v.s
. CGAL
Cube
2.4x↑
v.s
. gHull
18x↑
v.s
.
qHull
66x↑
v.s
. CGAL
Box (thickness 0.01)
3.2x↑
v.s
. gHull
23x↑
v.s
.
qHull
75x↑
v.s
. CGAL
Sphere (thickness 0.01)
1.3x↑
v.s
. gHull
30x↑
v.s
.
qHull
110x↑
v.s
. CGAL
Experiments
23
4
Convex hull
Thai statue
1.4x↑
v.s
. gHull
5.5x↑
v.s
.
qHull
12x↑
v.s
. CGAL
Lucy
1.1x↑
v.s
. gHull
7.2x↑
v.s
.
qHull
17x↑
v.s
. CGAL
Conclusions
24
5
is provably correct
uses only local operations
works for any star

shaped polyhedron
allows any order of flips
Flip

Flop
GPU friendly
ffHull
GPU friendly
ffRT
GPU friendly
Thank you!
http://www.comp.nus.edu.sg/~tants/flipflop.html
mingcen@comp.nus.edu.sg
Flips on 3D polyhedron
26
1
Convex Edge
outside
inside
a
b
c
d
< 180
°
Reflex Edge
inside
outside
a
b
c
d
> 180
°
one flip
Compute CH by flips
Flips on 3D polyhedron
27
1
Star

shaped polyhedron
The
cone
of
abc
w.r.t
.
s
A polyhedron is
star

shaped
w.r.t
.
s
if the cones of all its triangles do
not overlap.
s
Flips on 3D polyhedron
28
1
Star

shaped polyhedron
Edge
ab
is
reflex
if
d
and
s
lie on
different sides of
abc
.
Otherwise
ab
it is
convex
.
a
b
c
d
a
b
c
d
s
s
Flips on 3D polyhedron
29
1
Star

shaped polyhedron
Edge
ab
is
flippable
w.r.t
.
s
if
1.
Point
s
remains inside after flipping
.
2.
The union of the cones included in the flip is convex.
Flipping a flippable edge results in a star

shaped polyhedron
w.r.t
.
s
.
a
b
c
d
s
ab
is unflippable
a
b
c
d
s
ab
is flippable
a
b
c
d
s
(1)
(2)
Flip

Flop:
a novel flip algorithm
30
2
Algorithm
Initial
polyhedron
Queue of
edges
all edges
Outdated?
pop
ab
with link points {
c
,
d
}
yes
Flippable?
no
yes
no
Apply
V

criterion
Flip

Flop
x
∈
{
a
,
b
,
c
,
d
} is the NE point
with the minimum index
Reflex?
Label
NE point
yes
x
exists?
Apply
D

criterion
no
yes
Push edges
no
Flip

Flop:
a novel flip algorithm
31
2
Correctness
2

2 reflex and unflippable edge
→
a non

extreme point
v is non

extreme
→
v is
locally covered
v is non

extreme with degree > 3
→
an incident edge is 2

2 flippable
v is non

extreme with degree 3
→
3

1 flippable edge
correctness of Flip

Flop
3

1 reflex edge
→
flippable
Flip

Flop:
a novel flip algorithm
32
2
Correctness
3

1 reflex edge
→
flippable
Edge
ab
is a 3

1 edge that is reflex, then the polyhedron
is star

shaped
w.r.t
.
s
after flipping
ab.
a
d
s
b
c
Flip

Flop:
a novel flip algorithm
33
2
Correctness
2

2 reflex and
unflippable
edge
→
a non

extreme point
a
b
c
d
s
b (or a)
lies inside the
tetrahedron
sacd
(or
sbcd
)
ab
is a 2

2 edge that is
reflex and unflippable
b (or a)
is non

extreme
Flip

Flop:
a novel flip algorithm
34
2
Correctness
v
is
locally covered
by
b
,
e
and
f
since
it lies inside the cone of
sb
,
se
and
sf
.
v
is not
locally covered
.
C
s
(
△
扥b
⤠
楳
a
c潶敲
潦 v
Flip

Flop:
a novel flip algorithm
35
2
Correctness
v is non

extreme
→
v is
locally covered
Assume
v
is not locally covered
•
t
1
and
t
2
are two triangles on
C
intersecting with
vs.
(
t
1
is nearer to
s
and
v
than
t
2
)
•
H
passes
s
and is parallel
t
1
.
•
R
T
is the half

space by
H
containing
C
v
and
s
must lie on the same side of
C
by assumption
•
Construct convex hull
C
of the link points of
v.
Any vertex
p
other than
v
must lie
inside
R
T
, otherwise the polyhedron
is not star

shaped
v
is an extreme point,
contradiction!
Flip

Flop:
a novel flip algorithm
36
2
Correctness
v is non

extreme with degree 3
→
3

1 flippable edge
Assume
va
is unflippable
s
is inside the tetrahedron
vabc
v
is not locally covered
v
is an extreme point,
contradiction!
Flip

Flop:
a novel flip algorithm
37
2
Correctness
v is non

extreme with degree > 3
→
an incident edge is 2

2 flippable
Task 1: find 3 special points in the link of
v
Let
C
s
(
△
abc
)
be a cover of
v
such
that
a
,
b
and
c
are link points of
v
While another link point
p
is inside
C
s
(
△
abc
)
, use
p
to replace one of
{
a
,
b
,
c
} and get a smaller cover
Any vertex
q
other than
v
must lie
outside
C
s
(
△
abc
)
, otherwise the
polyhedron is not star

shaped
Flip

Flop:
a novel flip algorithm
38
2
Correctness
v is non

extreme with degree > 3
→
an incident edge is 2

2 flippable
Task 2: find the 2

2 flippable edge incident to
v
•
a
,
b
and
c
partition the link of
v
into three
chains of vertices:
L
ab
,
L
bc
and
L
ca
.
•
W.l.o.g
. let
L
ab
be (
a,p1,p2
,…,
pn,b
),
n
>1.
•
Let
D
sva
be the half

plane through
v
,
s
,
a
defined by line
vs
, similar for
D
svc
and
D
svb
.
•
Let
R
ab
be the region bounded by
D
sva
and
D
svb
and containing the edge
ab
.
Goal: there exists a vertex
p
m
in
L
ab
so that
so that edge
vp
m
is 2

2 flippable
Flip

Flop:
a novel flip algorithm
39
2
Correctness
v is non

extreme with degree > 3
→
an incident edge is 2

2 flippable
Task 2: find the 2

2 flippable edge incident to
v
L
ab
cannot intersect with
D
sva
,
D
svb
and
D
svc
for the star

shaped polyhedron.
L
ab
lies totally
inside
R
ab
L
ab
lies outside
C
s
(
△
abc
)
(task 1).
In the convex hull of {
sv
,
sa
,
sp
1
, …
sp
n
,
s
b
},
there must be a vertex
p
m
on its boundary.
Edge
vp
m
is 2

2 flippable
ffHull
:
3D convex hull algorithm
40
3
Implementation tips
Maintain the orientation of each triangle so that the kernel point
s
is always
beneath it.
Use orientation determinants when finding the furthest point.
Adapt the exact 3D orientation predicate [
Shewchuk
97’] and
SoS
technique
[
Edelsbrunner
90’] to promise the robustness.
In GPU, find an almost furthest point without using atomic operations
In GPU, separate the kernels for fast and exact predicates.
Experiments
41
4
Convex hull
Cube
Ball
Sphere
Box
Running time of
ffHull
compared to gHull, CGAL (on the CPU)
and gHull (on the GPU).
Experiments
42
4
Convex hull
Points on a thin sphere of different thicknesses.
Experiments
43
4
Convex hull
The number of flips performed by
ffHull
on the CPU.
Thank you!
http://www.comp.nus.edu.sg/~tants/flipflop.html
mingcen@comp.nus.edu.sg
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