Triang u Laci on Delaunay

8/18/2019 Triang u Laci on Delaunay

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Algorithmica 1989) 4:9 7-1 08

lgorithmica

9 1989 Springer-Verlag New York Inc.

C o n s t r a i n e d D e l a u n a y T r i a n g u l a t i o n s

L . P a u l C h e w 2

Abs t r a c t Given a set of n vertices in the plane together with a set of noncrossing, straight-line

edges, the

constrained Delaunay triangulation

CD T) is the triangulation of the vertices with the

following properties: 1) the prespecified edges are included in the triangulation, and 2) it is as

close as possible to the Delaunay triangulation. We show that the CDT can be built in optimal

O n

log n) time using a divide-and-conquer technique. Th is matches the time required to build a n

arbitrary unconstrained) Delaunay triangulation an d the time required to build an arbitrary con-

strained non-Delaunay) triagulation. CD Ts, because of their relationship with Delaunay triangula-

tions, have a num ber o f properties that m ake them useful fo r the finite-element method. A pplications

also include m otion planning in the presence of polygonal obstacles and con strained Euclidean

minimum spanning trees, spanning trees subject to the restriction that some edges are prespecified.

Key W ord s. Triangulation, Delaunay triangulation, Constrained triangulation, Algorithm, Vorono i

diagram.

1 . I n t r o d u c t i o n . A s s u m e w e a re g i v e n a n n - v e r t e x , p l a n a r , s t r a ig h t - l i n e g r a p h

G. A constrained triangulation

o f G is a t r i a n g u l a t i o n o f t h e v e r t i c e s o f G t h a t

i n c l u d e s t h e e d g e s o f G a s p a r t o f t h e tr i a n g u l a t i o n . S e e [P S 8 5 ] f o r a n e x p l a n a t i o n

o f h o w s u c h a t r i a n g u l a t i o n c a n b e f o u n d i n O n l o g n ) t i m e . A constrained

De laun ay triangulation

C D T ) o f G c a l le d a n o b s t a c le t r ia n g u l a t i o n i n [ C h 8 6 ]

o r a g e n e r a l i z e d D e l a u n a y t r i a n g u l a t io n i n [L e 7 8 ] a n d [ L L 8 6 ] ) i s a c o n s t r a i n e d

t r i a n g u l a t i o n o f G t h a t a l s o h a s t h e p r o p e r t y t h a t i t is a s c lo s e t o a

Delaunay

triangulation

a s p o s s i b l e .

I n th i s p a p e r w e s h o w t h a t, b y u s i n g a m e t h o d s i m i l a r t o th a t u s e d b y Y a p

[ Y a 8 4 ] f o r b u i l d i n g t h e V o r o n o i d i a g r a m o f a se t o f s i m p l e c u r v e d s e g m e n t s , th e

C D T o f G c a n b e bu i l t in

O n l o g n )

t im e , t h e s a m e t im e b o u n d r e q u i r e d t o b u i l d

t h e u n c o n s t r a i n e d ) D e l a u n a y t r ia n g u l a t io n . T h is a l so m a t c h e s t h e t im e n e e d e d

t o b u i l d a c o n s t r a i n e d n o n - D e l a u n a y ) t r i a n g u l a t i o n . E a r l ie r w o r k h a s s h o w n

t h a t t h e C D T c a n b e b u i l t i n O n 2) t im e [ D F P 8 5 ] , [ L L 8 6 ] . A n

O n logn ) - t ime

a l g o r i t h m f o r t h e s p e c ia l c a s e w h e n t h e e d g e s o f G f o r m a s i m p l e p o l y g o n w a s

p r e s e n t e d i n I L L 8 6 ] .

The Delaunay tr iangulation

i s t h e s t r a i g h t - l i n e d u a l o f t h e

Voronoi diagram

s e e F i g u r e 1 ) . S e e [ P S 8 5 ] f o r d e f i n i t i o n s a n d a n u m b e r o f a p p l i c a t i o n s o f

D e l a u n a y t r i a n g u la t io n s a n d V o r o n o i d i a g ra m s .

A n i m p o r t a n t p r o p e r t y o f t h e D e l a u n a y t r ia n g u l a t i o n is t h a t e d g es c o r r e s p o n d

t o e m p t y c i rc le s . I n d e e d , t h i s p r o p e r t y c a n b e u s e d a s t h e d e f in i t i o n o f D e l a u n a y

An earlier version of the results presented here appeared in the

Proceedings o f the Third Annual

Symposium on Computational Geometry 1987)v

2 Department of M athematics and Com puter Science, Dartmouth College, Hanover, NH 03755, USA .

Received May 23, 1987; revised N ovember 27, 1987. Comm unicated by Chee-Keng Y ap.


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9 8 L P C h e w

Fi g 1 A Vo ronoi d i agram and the cor re spon di ng D e l aunay tr i angul a t ion

t r i a n g u l a t i o n . I t i s s t r a i g h t f o r w a r d t o s h o w t h a t t h i s d e f i n i t i o n i s e q u i v a l e n t t o

t h e m o r e c o m m o n d e f in i ti o n o f t h e D e l a u n a y t r ia n g u l a t io n a s t h e d u a l o f t h e

V o r o n o i d i a g ra m .

D EF IN ITIO N . Le t S b e a s e t o f p o i n t s i n t h e p l an e . A t r i an g u l a t i o n T i s a

elauna y triangulation

o f S i f f o r e a c h e d g e e o f T t h e r e e x i s t s a c ir c l e c w i t h

t h e f o l l o w i n g p r o p e r t i e s :

( 1 ) T h e e n d p o i n t s o f e d g e e a r e o n th e b o u n d a r y o f c.

( 2 ) N o o t h e r v e r t e x o f S i s i n th e i n t e r i o r o f c .

I f n o f o u r p o i n t s o f S a r e c o c i r c u l a r t h e n t h e D e l a u n a y t r i a n g u l a t i o n is u n i q u e .

F o r m o s t c a s e s i n w h i c h t h e r e i s n o t a u n i q u e D e l a u n a y t r i a n g u l a t i o n , a n y o f

t h e m w i l l d o .

T h e f o l l o w i n g d e f i n i t io n , e q u i v a l e n t t o t h e d e f i n i t i o n g i v e n i n [L E T 8 ] , i n d i c a t e s

w h a t w e m e a n w h e n w e s a y a s c l o se as p o s s ib l e to th e D e l a u n a y t r i a n g u la t i o n .

C o m p a r e t h is d e f i n i ti o n w i t h th e d e f i n it io n o f t h e ( u n c o n s t r a i n e d ) D e l a u n a y

t r i a n g u l a t i o n g i v e n a b o v e .

D EF IN ITIO N . Le t G b e a s t r a i g h t - l i n e p l a n a r g r ap h . A t r i an g u l a t i o n T i s a

constrained elaun ay triangulation ( C D T ) o f G i f e a c h e d g e o f G is a n e d g e o f

T a n d f o r e a c h r e m a i n i n g e d g e e o f T t h e r e e x is ts a c ir c le c w i th t h e f o l l o w i n g

p r o p e r t i e s :

( 1) T h e e n d p o i n t s o f e d g e e a r e o n th e b o u n d a r y o f c.

( 2) I f a n y v e r t e x v o f G is i n t h e i n t e r i o r o f c t h e n i t c a n n o t b e s e e n f r o m a t

l e a s t o n e o f th e e n d p o i n t s o f e ( i. e. , i f y o u d r a w t h e l in e s e g m e n t s f r o m v to


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Con strained De launay Triangulations 99

Fig. 2. A graph G and the corresponding constrained Delaunay triangulation

e a c h e n d p o i n t o f e t h e n a t le a s t o n e o f t h e l in e s e g m e n t s c r o s s e s a n e d g e o f

G .

I t f o l lo w s i m m e d i a t e l y f r o m t h e d e f i n i ti o n t h a t i f G h a s n o e d g e s t h e n t h e

c o n s t r a i n e d D e l a u n a y t r i a n g u l a t i o n i s t h e s a m e a s th e ( u n c o n s t r a in e d ) D e l a u n a y

t r i a n g u l a t i o n .

W e d i s ti n g u i sh t w o t y p e s o f e d g e s th a t a p p e a r i n a C D T :

G-edges,

p r e s p e c i f i e d

e d g e s t h a t a r e f o r c e d u p o n u s a s p a r t o f G , a n d Delaunay edges, t h e r e m a i n i n g

e d g e s o f t h e C D T ( s e e F i g u r e 2) .

I n t u it i v e ly , t h e d e f i n it i o n s o f D e l a u n a y t r i a n g u l a t i o n a n d c o n s t r a i n e d D e l a u n a y

t r i a n g u l a t i o n a r e t h e s a m e e x c e p t t h a t, f o r t h e C D T , w e i g n o r e p o r t i o n s o f a

c i rc l e w h e n e v e r t h e c i r cl e p a s s e s t h r o u g h a n e d g e o f G . N o t e t h a t th e C D T i s

not

t h e s a m e a s t h e d u a l o f t h e l in e - s e g m e n t V o r o n o i d i a g r a m . T h e l in e - s e g m e n t

V o r o n o i d i a g r a m c a n h a v e p a r a b o l a s a p p e a r i n g a s p o rt io n s o f V o r o n o i b o u n -

d a r ie s . I ts d u a l w o u l d i n c l u d e e d g e s c o n n e c t i n g o b s t a c le e d g e s , w h i l e t h e C D T

h a s e d g e s o n l y b e t w e e n v e r ti c e s. O n l o g n ) - t i m e a l g o r it h m s f o r c o n s t r u c t in g t h e

l i n e - s e g m e n t V o r o n o i d i a g r a m a p p e a r i n [ K i 7 9 ] a n d [ Y a 8 4 ] .

O n e m e a s u r e o f t h e a p p r o p r i a t e n e s s o f a d e f in i t io n is its u ti li ty . W e d e m o n s t r a t e

t h e u t i li ty o f th e d e f i n it i o n o f a C D T b y p r e s e n t i n g s o m e a p p l i c a t i o n s . J u s t a s

t h e ( u n c o n s t r a i n e d ) D e l a u n a y t r i a n g u l a ti o n o f S c a n b e u s e d t o d e t e r m i n e t he

E u c l i d e a n m i n i m u m s p a n n i n g t r ee ( E M S T ) o f S q u ic k ly , t h e C D T c a n b e u s e d

t o f in d t h e c o n s t r a i n e d E M S T o f S , c o n s t r a i n e d i n t h e s e n s e th a t c e r t a i n e d g e s

o f th e s p a n n i n g t re e a r e p r e s p e c i fi e d a n d m a y n o t b e c r o s se d b y o t h e r e d g e s o f

t h e s p a n n i n g t r e e . A s i m p l e p r o o f b y c o n t r a d i c t i o n s h o w s t h a t th e E M S T is a

s u b g r a p h o f t h e D e l a u n a y t r ia n g u l a t io n f o r b o t h th e c o n s t r a i n e d a n d u n c o n -

s t r a in e d v e r s io n s . S e e [ P S 8 5 ] f o r a v e r s i o n o f t h e p r o o f f o r t h e u n c o n s t r a i n e d c a s e.

A n a d d i t i o n a l a p p l i c a t i o n is p r e s e n t e d i n [ C h 8 6 ] a n d [ C h 8 7 a ] w h e r e v a r i a ti o n s

o f t h e s t a n d a r d C D T a r e u s e d f o r m o t i o n p l a n n i n g in th e p l a n e . T h e s e v a ri a ti o n s

u s e a d i f f e r e n t d i s t a n c e f u n c t i o n , i n e f f e c t u s i n g a c i r c l e t h a t is s h a p e d l ik e a


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100 L .P. Chew

s q u a r e [ C h 8 6 ] o r a tr i a n g l e [ C h 8 7 a ] . W i t h s o m e m o d i f i c a t i o n s , t h e r e s u lt s p r e s e n -

t e d h e r e a r e v a l i d f o r C D T s b a s e d o n t h e s q u a r e - d i s t a n c e t h e L1 m e t r i c ) o r t h e

t r i a n g l e - d i s t a n c e o r b a s e d o n o t h e r

convex d i s tance funct ions .

O n e s u c h

m o d i f i c a t io n is t h a t t h e w o r d triangulation is i n a p p r o p r i a t e - - s o m e r e g io n s c a n n o t

b e d i v i d e d i n t o t ri a n gl e s. I n t h i s c a se , t h e C D T c a n b e d e f i n e d as a p l a n a r g r a p h

w i t h th e e m p t y c i rc le p r o p e r t y a n d w i t h th e m a x i m u m n u m b e r o f e d ge s. ) S e e

[ C D 8 5 ] f o r m o r e i n f o r m a t i o n o n c o n v e x d i s ta n c e f u n c t i o n s a n d t h e i r r e la t i o n to

D e l a u n a y t r ia n g u l a t i o n s .

C D T s a r e a l s o u s e f u l f o r t h e f in i te e l e m e n t m e t h o d , a w i d e l y u s e d t e c h n i q u e

f o r o b t a i n i n g a p p r o x i m a t e n u m e r i c a l s o l u ti o n s to a w i d e v a r i e ty o f e n g in e e r i n g

p r o b l e m s . F o r m a n y a p p l i c a t i o n s o f t h i s m e t h o d , t h e f ir s t s t e p is t o t r i a n g u l a t e

a g i v e n re g i o n i n th e p l a n e . N o t ju s t a n y t r i a n g u l a t i o n w i ll d o ; e r r o r b o u n d s a r e

b e s t i f a ll t h e t r i a n g l e s a r e a s c l o s e a s p o s s i b l e t o e q u i l a t e r a l t r i a n g l e s. P r e s e n t l y ,

e i t h e r t h e s e tr i a n g u l a t i o n s a r e p r o d u c e d b y h a n d o r t h e y a re p r o d u c e d a u t o m a t i -

c a l l y u s i n g o n e o f a n u m b e r o f h e u r i s ti c t e c h n i q u e s . I n [ C h 8 7 b ] w e p r e s e n t a n

e f fi ci en t n e w t e c h n i q u e , b a s e d o n C D T s , f o r p r o d u c i n g d e s i ra b l e t r i a n g u l a ti o n s .

Constrained D e l a u n a y t r i a n g u l a ti o n s a r e n e c e s s a r y b e c a u s e s o m e e d g e s a re p r e -

s p e ci fi e d: b o u n d a r y e d g e s f o r t h e p r o b l e m r e g i o n p lu s o p t i o n a l u s e r - c h o s e n e d g e s

i n th e i n t e r i o r o f t h e r e g i o n . U n l i k e p r e v i o u s t e c h n i q u e s , t h is t e c h n i q u e c o m e s

w i t h a g u a r a n t e e : t h e a n g l e s i n t h e r e s u l t i n g t r i a n g u l a t i o n s a r e a l l b e t w e e n 3 0 ~

a n d 1 20 ~ a n d t h e e d g e l e n g t h s a r e a l l b e t w e e n h / 2 a n d h w h e r e h i s a p a r a m e t e r

c h o s e n b y t h e u s e r .

A n o t h e r a p p l i c a t io n o f C D T s h a s b e e n s u g g e s t e d i n [ D F P 8 5 ] a n d I L L 8 6 ] ,

w h e r e C D T s a r e u s e d f o r s u r f a c e i n t e r p o l a t i o n ~ F o r i n s t a n c e , i n g e o g r a p h i c a l

s u r f a c e i n t e r p o l a t i o n , i n a d d i t i o n t o v a l u e s a t sp e c i f ic p o i n t s , c e r t a i n b o u n d a r i e s

m a y b e k n o w n e ,g ., a l a k e b o u n d a r y o r a m o u n t a i n r i dg e ) . A C D T p r o v i d e s a

n a t u r a l w a y t o r e ta i n t h e b o u n d a r y i n f o r m a t i o n w h i le p r o d u c i n g a g o o d t r ia n g u -

l a t i o n .

2 . T h e A l g o r i th m . W e u s e d i v i d e - a n d - c o n q u e r t o b u i l d t h e C D T . F o r s i m p l i c it y

o f p r e s e n t a t i o n , w e a s s u m e t h a t t h e p l a n a r g r a p h G i s c o n t a i n e d w i t h in a g iv e n

r e c t a n g l e . W e s t a r t b y s o r t i n g t h e v e r ti c e s o f G b y x - c o o r d i n a t e ; t h e n w e u s e th i s

i n f o r m a t i o n t o d i v i d e t h e r e c t a n g l e i n t o v e r t i c a l s t ri p s in s u c h a w a y t h a t t h e r e

i s e x a c t l y o n e v e r t e x in e a c h s t ri p . O f c o u r s e , t h is c a n n o t b e d o n e i f s o m e v e r t e x

i s d ir e c t ly a b o v e a n o t h e r , b u t w e c a n a v o i d t h is p r o b l e m b y r o t a t in g t h e e n t ir e

g r a p h G i f n e c e s s a ry . F o l l o w i n g t h e d i v i d e - a n d - c o n q u e r p a r a d i g m , t h e C D T i s

c a l c u l a t e d f o r e a c h s tr ip , a d j a c e n t s t ri ps a r e p a s t e d t o g e t h e r i n p a ir s t o f o r m n e w

s tr ip s , a n d t h e C D T is c a l c u l a t e d f o r e a c h s u c h n e w l y f o r m e d s t ri p u n t il t h e C D T

f o r t h e e n t i r e G - c o n t a i n i n g r e c t a n g l e h a s b e e n b u i l t . T h i s w h o l e p r o c e s s t a k e s

t i m e O n l o g n ) p r o v i d e d t h e C D T p a s t i n g o p e r a t i o n c a n b e d o n e e f fi ci en t ly .

T h e t r ic k u s e d t o e n s u r e t h a t th i s C D T p a s t i n g o p e r a t i o n i s d o n e i n r e a s o n a b l e

t i m e is to a v o i d k e e p i n g t r a c k o f t o o m u c h e d g e i n f o r m a t i o n . N o t e t h a t i t is n o t

p o s s ib l e t o k e e p t r a c k o f a l l p l a c e s w h e r e G - e d g e s i n t e r se c t s tr ip b o u n d a r i e s ; i f

w e d o s o th e n w e c o u l d h a v e a s m a n y a s O n 2 ) i n t e r s e c t i o n s t o w o r k w i t h . E d g e s

t h a t c r o s s a s t ri p , e d g e s w i t h n o e n d p o i n t s w i t h i n t h e s t r ip , a r e , f o r t h e m o s t p a r t ,


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IP

- . . . .

Fig. 3. The contents of a strip and the contents that we k eep track of.

i g n o r e d . S u c h a n e d g e i s o f i n t e r e s t o n l y i f i t i n t e r a c t s i n s o m e w a y w i t h a v e r t e x

t h a t l i e s w i t h i n t h e s t r i p . Y a p [ Y a 8 4 ] u s e d t h e s a m e t e c h n i q u e t o d e v e l o p a n

a l g o r i th m f o r th e l i n e - s e g m e n t V o r o n o i d i a g r a m .

E a c h s t r i p , t h e n , i s d i v i d e d i n t o r e g i o n s b y

cross edges ,

G - e d g e s t h a t h a v e n o

e n d p o i n t w i t h i n th e s t ri p . W e d o n o t k e e p t r a c k o f al l o f t h e s e r e g io n s t h e r e

c o u l d b e O n 2) o f t h e m ) . I n s t e a d w e k e e p t r a c k o f j u s t th e r e g i o n s t h a t c o n t a i n

o n e o r m o r e v e r ti c e s s e e F i g u r e 3 ).

A t t h is p o i n t , t w o p a r t s o f t h e C D T a l g o r i t h m r e q u i r e f u r t h e r e x p l a n a t i o n : 1 )

h o w t o h a n d l e v e r t e x - c o n t a i n i n g r e g i o n s , i n i t i a l i z i n g t h e m a n d k e e p i n g t r a c k o f

t h e m a s a d j a c e n t st ri p s a r e c o m b i n e d , a n d 2 ) h o w t o st it c h t o g e t h e r C D T s a s

a d j a c e n t s t r i p s a r e c o m b i n e d .

3 . Control o f Vertex Containing Regions. E a c h i n i t i a l s t r i p h a s a s i n g l e r e g i o n

c o n t a i n i n g i t s s i n g l e v e r t e x . T o c r e a t e t h e s e i n i t i a l r e g i o n s w e n e e d t o k n o w t h e

e d g e i m m e d i a t e ly a b o v e a n d t h e e d g e i m m e d i a t e l y b e l o w e a c h v e rt ex . N o t e th a t

t h e e d g e i m m e d i a t e l y a b o v e a v e r t e x m a y b e t h e t o p e d g e o f t h e r e c t a n g l e t h a t

c o n t a in s t h e e n t ir e g r a p h G a n d t h e e d g e i m m e d i a t e l y b e l o w m a y b e th e b o t t o m

e d g e o f th e r e c t a n g l e .) T h i s i n f o r m a t i o n c a n b e f o u n d f o r a ll v e rt ic e s in O n l o g n )

t i m e b y u s i n g a v e r t ic a l l in e a s a s w e e p - l in e . S e e [ P S 8 5 ] f o r a n e x p l a n a t i o n o f

t h is te c h n i q u e a n d a n u m b e r o f a p p l i c a t io n s .

I t is n o t d i f fi c ul t t o d e t e r m i n e a p p r o p r i a t e r e g i o n s w h e n t w o a d j a c e n t s t ri p s

a r e s t it c h e d t o g e t h e r . B a s i c a ll y , w e m e r g e r e g i o n s b y r u n n i n g t h r o u g h t h e r e g i o n s

i n t h e i r o r d e r a l o n g th e s t r ip s s e e F i g u r e 4 ). A s w e m o v e f r o m b o t t o m t o t o p i n

t h e c o m b i n e d s t ri p , w e s t a r t a n e w r e g i o n w h e n e v e r e i t h e r st r ip s t a rt s a r e g i o n ,

w e c o n t i n u e t h e r e g i o n a s l o n g a s a r e g i o n c o n t i n u e s i n e i t h e r s t r i p , w e s t o p a

r e g i o n o n l y w h e n w e r e a c h a p o i n t w h e r e n e i t h e r s tr ip h a s a v e r t e x - c o n t a i n i n g


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102 L .P . Chew

o ~

Fig. 4. M erging he reg ions of two adjac ent strips.

r e gi o n. A s w e d o t h is m e r g e p r o c e s s, w e c r e a t e a n o r d e r e d l is t o f th e i m p o r t a n t

G - e d g e s t h a t c r o s s t h e b o u n d a r y b e t w e e n t h e t w o s t r i p s . R e c a l l t h a t c r o s s e d g e s

G - e d g e s w i t h e n d p o i n t s o u t s i d e e i t h e r s tr i p ) a r e ig n o r e d . I t i s e a s y t o s e e t h a t

t h i s m e r g e o p e r a t i o n c a n b e d o n e i n t i m e

O v + h )

w h e r e v is t h e n u m b e r o f

G - v e r t i c e s w i t h i n th e t w o s tr ip s a n d h i s t h e n u m b e r o f half-edges w i t h i n t h e t w o

s t r i ps .

A half-edge

is o n e h a l f o f a G - e d g e ; i t s t a rt s a t a v e r t e x w h i c h m u s t b e w i t h i n

t h e s tr ip , b u t i ts o t h e r e n d m a y b e a n y w h e r e . A s t a n d a r d G - e d g e i s m a d e o f t w o

h a l f - e d g e s .

4 . I n f in i te Ve r t ic e s H o w c a n w e e f fi ci en t l y c o m b i n e t h e C D T s o f a d j a c e n t s t r ip s ?

E a c h s t r ip c o n t a i n s o n e o r m o r e v e r t e x - c o n t a i n i n g r e g io n s ; e a c h s u c h r e g i o n

c o n t a in s a C D T . I f w e m u s t s e a r c h e a c h C D T t o c o m b i n e it w i th a C D T f r o m

a n a d j a c e n t s t r ip , a n d i f s u c h a s e a r c h t a k e s m o r e t h a n c o n s t a n t t i m e f o r e a c h

v e r t e x - c o n t a i n i n g r e g i o n , t h e n w e l o s e t h e

O n

l og n ) - t im e b o u n d f o r b u i l d in g

t h e c o m p l e t e C D T . T h u s , w e n e e d a n e f fi ci en t w a y t o a c c e s s t h e C D T s w i t h in a

s t r i p .

T o d o t h is , w e l i n k s u c h C D T s b y i n t r o d u c i n g f a l s e v e rt ic e s . A f a l s e v e r te x is

i n t r o d u c e d w h e r e v e r a n e d g e c r o s s es a s t ri p b o u n d a r y . N o t e t h a t, a s b e f o r e , t h is

is n o t d o n e f o r a ll e d g e s ; i n st e a d , w e u s e o n l y t h o s e G - e d g e s w h i c h e i t h e r 1 )

f o r m a r e g i o n b o u n d a r y f o r a v e r t e x - c o n t a i n i n g r e g i o n , o r 2 ) h a v e e x a c t l y o n e

e n d p o i n t w i t h i n t h e s t r i p . T h e t o t a l n u m b e r o f f a l s e v e r t i c e s i s t h u s

O v + h ) ,

w h e r e v is t h e n u m b e r o f G - v e r t ic e s w i t h i n th e s t ri p a n d h is t h e n u m b e r o f

h a l f - e d g e s w i t h i n t h e s t r i p .

C D T s o f a d j a c e n t re g i o n s a re c o m b i n e d b y e x e c u t i n g t h e fo l lo w i n g st ep s : 1 )

e l i m i n a t e fa l s e v e r ti c e s a n d D e l a u n a y - e d g e s th a t u s e t h e s e v e rt ic e s ) a l o n g t h e


7/12

Con strained Delaun ay Triangu lations 103

b o u n d a r y b e t w e e n t h e s t r i p s , l e a v i n g a p a r t i a l C D T i n e a c h h a l f ; a n d ( 2 ) s t i t c h

t h e p a r ti a l C D T s t o g e t h e r to c r e a t e a c o m p l e t e C D T f o r t h e n e w l y c o m b i n e d s tr ip .

T h i s m e t h o d l e a d s t o a n e w p r o b l e m . W h e n a f a l se v e r t e x is e l i m i n a t e d i t s e e m s

t h a t f u r t h e r c h a n g e s s h o u l d o c c u r i n th e C D T t h a t u s e s th a t v e r t ex . I n f a c t, i f a

f a l s e v e r t e x i s a n y w h e r e i n t h e p l a n e t h e n i t s e e m s t h a t e l i m i n a t i n g i t c a n c a u s e

t h e C D T t o c h a n g e. W e u s e a t r ic k t o k e e p t h es e c h a n g e s m a n a g e a b l e : w e t re a t

t h e s e f a l s e v e rt i c e s as i f t h e y a r e l o c a t e d a t in f in i ty . W e a r e t a k i n g a d v a n t a g e o f

t h e f a c t t h a t e l i m i n a t i n g a p o i n t a t i n f in i ty d o e s n o t a f f e c t D e l a u n a y e d g e s t h a t

a r e f i n it e . I n a s e n s e , e a c h f a l s e v e r t e x h a s t w o l o c a t i o n s ; i t i s c r e a t e d ( a n d i s

s h o w n i n p i c t u r e s) w h e r e a G - e d g e i n t er s e c ts t h e b o u n d a r y o f a s t ri p , b u t i n al l

o t h e r r e s p e c t s i t i s t r e a t e d a s i f i s l o c a t e d a t in f i n it y . T h e C D T w i t h i n a s tr i p is

c o n s t r u c t e d u s i n g t h e s e i n fi n it e v e r t ic e s as a d d i t i o n a l d a t a p o i n t s f o r th e C D T .

W e s p e c i f y t h a t t w o d i f f e r e n t i n f i n it e v e r t ic e s d o n o t i n t e r a c t .

O n c e w e h a v e i n c l u d e d t h e i nf in i te v e r ti c e s, t h e C D T o f a s t r i p is w e l l - d ef i n e d .

W i t h o u t i n f i n i t e v e r t i c e s , w e h a v e t o a l l o w f o r G - e d g e s t h a t l e a v e t h e s t r i p , a n d

i t i s n o t i m m e d i a t e l y c l e a r h o w s u c h e d g e s s h o u l d b e t r e a t e d a s p a r t o f a C D T .

W i t h i n f i n i t e v e r t i c e s , t h e r e a r e n o e d g e s t h a t l e a v e t h e s t r i p .

T h e i n f in i te v e r t i ce s w e r e i n t r o d u c e d s o t h a t w e a l w a y s h a v e a s t a r t in g p l a c e

f o r a C D T . T h e f o l l o w i n g l e m m a s h o w s t h a t t h e i n fi n it e v e r t ic e s d o n o t o t h e r w i s e

a f f ec t t h e C D T s .

LEMMA 1. Let A and B be vertices of a straight-line planar graph G where edge

AB is a Delaunay edge of the CDT of G. For any strip S that contains both A and

B edge AB appears as a Delaunay edge in the CDT of S.

PROOF. Since edge AB is a D e l a u n a y e d g e o f t h e C D T o f G , w e k n o w , b y

d e f i n it i o n , th a t t h e r e i s a c i rc l e c t h r o u g h A a n d B s u c h t h a t a n y G - v e r t e x w i th i n

c c a n n o t b e s e e n f r o m e i t h e r A o r B o r b o t h . T h e s a m e c i rc le c a n b e u s e d w i t h

t h e s t r i p S . C e r t a i n l y , a n y G - v e r t e x t h a t i s w i t h i n c i s s ti ll h i d d e n f r o m A o r B .

I n f i n i t e v e r t i c e s a r e l o c a t e d a t i n f i n i t y a n d c a n n o t b e w i t h i n a n y f i n i t e c i r c l e .

T h u s , e d g e AB is a D e l a u n a y e d g e o f t h e C D T o f S . [ ]

A n i n fi n it e v e r t e x is t r e a t e d a s i f i t i s a t o n e o f f o u r p o s i t i o n s : ( - o % - c o ) ,

( + o % - ~ ) , ( - o o , + o @ , o r ( + o o , + c o ) . A n i n f i n it e v e r t e x o n t h e l e f t s i d e o f a st r i p

is a l w a y s t r e a t e d a s i f i t h a s x - c o o r d i n a t e - o o . S i m i l a r ly , a n i n f in i te v e r t e x o n t h e

r i g h t is tr e a t e d a s i f i t h a s x - c o o r d i n a t e + c ~. T h e y - c o o r d i n a t e o f a n i n f in i te v e r t e x

d e p e n d s o n w h e t h e r w e a r e l o o k i n g a t it f r o m a b o v e o r b e l o w i ts G - e d g e . ( R e c a l l

t h a t a n i n fi n it e v e r t e x c a n e x i s t o n l y w h e r e a G - e d g e i n t e r s e ct s a s t ri p b o u n d a r y . )

I f w e a p p r o a c h t h e v e r t e x f r o m a b o v e i ts G - e d g e t h e n i t is t r e a t e d a s i f i ts

y - c o o r d i n a t e is - o o ; f r o m b e l o w , + o o.

( A l t e r n a t e l y , e a c h s i n g l e f a l s e v e r t e x a s u s e d h e r e c a n b e c o n s i d e r e d t w o

v e r t ic e s , o n e w i t h + o o a s t h e y - c o o r d i n a t e a n d o n e w i th - ~ a s t h e y - c o o r d i n a t e .

T h i s d o e s n o t a f f e c t t h e r u n n i n g t i m e o f t h e a l g o r i t h m , b u t i t i s h a r d e r t o s h o w

i n p i c t u r e s . )

I n t h e f o l l o w i n g a l g o r i t h m , w e c r e a t e c i r c l e s t h a t g o t h r o u g h i n f i n i t e v e r t i c e s .

T o d e t e r m i n e t h e p r o p e r f o r m f o r s u c h a c ir c le , f o r i n s t a n c e a c i rc l e t h r o u g h


8/12

104 L.P . Chew

(cO , o 0), t h e r e a d e r s h o u l d f ir st c o n s i d e r a c i r c l e u s i n g t h e p o i n t ( w , w ) , w h e r e w

is a l a r g e n u m b e r . U s e t h e l im i t c i r c le a s w a p p r o a c h e s o o. T h e l i m i t c i r c l e w i ll

b e a h a l f - p l a n e .

5 . C o m b i n i n g C D T s of Adjacent Str ips T h e m e t h o d w e u s e f o r c o m b i n i n g C D T s

is si m i l a r t o o n e o f t h e m e t h o d s u s e d b y L e e a n d S c h a c h t e r [ L S 8 0 ] t o b u il d

( u n c o n s t r a i n e d ) D e l a u n a y t r i a n g u la t i o n s . T h i s m e t h o d i s r o u g h l y e q u i v a l e n t to

t h e p r o c e s s o f c o n s t r u c t i n g th e d i v i d in g c h a i n i n t h e d i v i d e - a n d - c o n q u e r a l g o r i th m

f o r b u i l d i n g t h e s t a n d a r d V o r o n o i d i a g r a m . S e e [P S 8 5 ] f o r a n e x p l a n a t i o n o f t h e

V o r o n o i - d i a g r a m a l g or it h m .

W e a s s u m e t h e f o l l o w i n g o p e r a t i o n s c a n b e d o n e i n u n i t ti m e : ( 1 ) g iv e n p o i n t

P a n d c i r c l e c, t e s t w h e t h e r P i s i n t h e i n t e r i o r o f c ; ( 2) g iv e n p o i n t s P , Q , a n d

R , r e t u r n t h e c i r c le th r o u g h t h e s e p o i n ts . T h e s e f u n c t i o n s c l e a r ly t a k e c o n s t a n t

t im e f o r a n y r e a s o n a b l e m o d e l o f c o m p u t a t i o n , e v e n if w e u s e n o n s t a n d a r d

c i r c l e s a s i n [ C D 8 5 ] .

L e t A a n d B b e v e r t ic e s o f G . A s s u m e e d g e

AB

c r os s es t h e b o u n d a r y b e t w e e n

t h e t w o s t r i p s a n d t h a t e d g e

AB

is k n o w n t o b e p a r t o f t h e d e s ir e d C D T

AB

m a y b e e i t h e r a G - e d g e o r a j u s t - c r e a t e d D e l a u n a y e d g e ) . C o n s i d e r a c ir c le w i t h

p o i n ts A a n d B o n i ts b o u n d a r y a n d w i th c e n t e r w e l l b e l o w e d g e

AB.

C h a n g e

t h e c i rc le b y m o v i n g t h e c e n t e r u p w a r d t o w a r d

AB

a lw a y s k e e p i n g A a n d B o n

t h e b o u n d a r y . C o n t i n u e m o v i n g t h e c e n t e r u p w a r d u n t i l t h e c i r c l e i n t e r s e c t s t h e

f i r s t v e r t e x a b o v e e d g e

AB

t h a t c a n b e s e e n f r o m b o t h A a n d B . C a l l t h is v e r t e x

X ( s e e F i g u r e 5 ) . I t is e a s y t o s e e th a t , b y d e f i n i t i o n , t h e e d g e s

AX

a n d

BX

a r e

D e l a u n a y e d g es o f t h e C D T .

T h i s p r o c e s s , a s o u t l i n e d i n t h e p r e v i o u s p a r a g r a p h , g iv e s a n i c e in t u it iv e w a y

t o f i nd X , b u t w e n e e d a n e f f ic i e n t a l g o r i t h m f o r d e t e r m i n i n g t h i s v e r t e x . V e r t e x

X is e it h e r i n B ' s s t ri p o r in A ' s s t r ip ; w e a s s u m e f o r th e m o m e n t t h a t X is i n

A ' s s t rip . B y L e m m a 1, w e k n o w

AX

e x i st s in t h e C D T o f th e l e f t - h a n d s tr ip .

W e c o n c l u d e f r o m t h i s th a t f o r e d g e

AB,

t h e n e x t b o u n d a r y - c r o s s i n g D e l a u n a y -

t

i

x

B

I

i

l

i

Fig. 5. Vertex X is the first vertex that can be see n from A and B.


9/12


e d g e a b o v e

AB

m u s t g o f r o m B t o s o m e v e r t e x Y a l r e a d y c o n n e c t e d to A , o r

f r o m A t o s o m e v e r t e x Z a l r e a d y c o n n e c t e d to B . O f c o u rs e , w e c a n n o t t el l a h e a d

o f ti m e w h e t h e r t h e v e r t e x w e a r e l o o k i n g f o r is o n t h e l e ft o r t h e r ig h t , s o w e

c h o o s e t h e b e s t c a n d i d a t e v e r t e x o n e a c h s i d e, t h e n w e c h o o s e b e t w e e n t h e m .

W e s t il l n e e d t o d e v e l o p a n e f fi c ie n t w a y t o d e t e r m i n e w h i c h o f t h e v e r ti c e s

c o n n e c t e d t o A i s t h e b e s t c a n d id a t e . W e k n o w f r o m t h e a r g u m e n t a b o v e t h a t

f o r X ( th e v e r te x w e a r e l o o k in g f o r) e d g e A X h a s to b e a D e l a u n a y e d g e ;

f u r t h e r , c i r c l e

ABX

is a n e m p t y c i r c le f o r e d g e A X . I n o t h e r w o r d s , a v e r t e x Y ,

a d j a c e n t t o A , c a n b e e l i m i n a t e d a s a p o s si b l e c a n d i d a t e i f c i rc l e

A B Y

c o n t a i n s

a v e r t e x t h a t c a n b e s e e n f r o m A a n d Y .

L e t

AC

b e t h e n e x t e d g e c o u n t e r c lo c k w i s e a r o u n d A f r o m

AB.

W e t e s t i f

AC

i s a g o o d c a n d i d a t e b y e x a m i n i n g t r i a n g l e ADC ( i f i t e x i s t s ) w h e r e AD i s t h e

n e x t e d g e a r o u n d A f r o m

AC.

I f n o s u c h t r i a n g l e e x i s ts

AC

i s o n t h e e d g e o f

t h e p a r t ia l C D T ) o r i f

AC

is a G - e d g e t h e n e d g e

AC

is a u t o m a t i c a l l y c o n s i d e r e d

a g o o d c a n d i d a t e . I f t r ia n g l e

ADC

d o e s e x i s t a n d

AC

i s a D e l a u n a y e d g e ( o f

A s s t r ip ) t h e n w e t e s t t o s e e i f D i s w i t h i n c i r c l e ABC; i f i t i s t h en AC i s n o t

a g o o d c a n d i d a t e ( s e e F i g u r e 6 ). F u r t h e r , w e k n o w n , b y th e d e f i n i ti o n o f a

D e l a u n a y e d g e , t h a t

AC

i s n o t a v a l i d e d g e i n t h e C D T a t a l l . I n t h i s c a s e , e d g e

AC c a n b e e l i m i n a t e d ; AD w o u l d t h e n b e t e s t e d to s e e i f AD i s a g o o d c a n d i d a t e .

T h e s e r e s u l ts i n d i c a t e t h a t t w o s tr ip s c a n b e e f fi c ie n tl y c o m b i n e d u s i n g t h e

f o l l o w i n g 2 - s t e p a l g o r i t h m .

Step 1.

( S e e F i g u r e 7 . ) E l i m i n a t e t h e i n f in i te v e r ti c e s a l o n g t h e b o u n d a r y b e t w e e n

t h e t w o s t r i p s . A l s o , e l i m i n a t e a n y e d g e s t h a t u s e t h e s e v e r t i c e s a s e n d p o i n t s .

A d d a n e w i n fi n it e v e r t e x f o r e a c h G - e d g e t h a t s ta r ts i n o n e s t ri p a n d g o e s t h r o u g h

t h e o t h e r s tr ip . A p a r t i a l C D T i s l e f t i n e a c h s t r ip .

Step

2 . B y L e m m a 1 , t h e n e w e dg e s t h a t w e n e e d m u s t c ro s s t h e b o u n d a r y

b e t w e e n s tr ip s. T h e s e n e w D e l a u n a y e d g e s t h a t c r os s t h e b o u n d a r y c a n b e f o u n d

b y e x e c u t i n g th e f o l l o w i n g p r o c e d u r e i n th e r e g i o n a b o v e e a c h G - e d g e t h a t c r o s se s

t h e b o u n d a r y b e t w e e n t h e tw o s t ri ps . ( T h e s e G - e d g e s c a n b e f o u n d e f f ic i en t ly

b y u s i n g t h e o r d e r e d l i st c r e a t e d a s v e r t e x - c o n t a i n i n g re g i o n s w e r e m e r g e d . ) L e t

A a n d B b e t h e e n d p o i n t s o f t h e G - e d g e ( e i t h e r A o r B m a y b e a n in f in i te v e r t e x ) .

A B

i

F i g 6

C

i s n o t a g o o d c a n d i d a t e a n d i s e l i m i n a t e d


10/12

1 06 L P C h e w

F i g 7 S t e p 1 :

e l i m i n a t i n g i n f in i t e v e r t ic e s a s st r i p s a r e c o m b i n e d O p e n c i r cl e s r e p r e s e n t i n f in i t e

v e r t i c e s

loop

E l i m i n a t e A - e d g e s t h a t c a n b e s h o w n t o b e i ll eg a l b e c a u s e o f t h e i r

i n t e r a c t i o n w i t h B ;

E l i m i n a t e B - e d g e s t h a t c a n b e s h o w n t o b e il le g a l b e c a u s e o f t h e i r

i n t e r a c t i o n w i t h A ;

L e t C b e a c a n d i d a t e w h e r e

AC

i s t h e n e x t e d g e c o u n t e r c l o c k w i s e

a r o u n d A f r o m AB i f s u c h a n e d g e e x i s t s) ;

L e t D b e a c a n d i d a t e w h e r e BD i s t h e n e x t e d g e c l o c k w i s e a r o u n d B

f r o m BA i f s u c h a n e d g e e x i s ts ) ;

e x i t lo o p i f n o c a n d i d a t e s e x i s t;

L e t X b e t h e c a n d i d a t e t h a t c o r r e s p o n d s t o t h e l o w e r o f c ir c le s ABC

a n d ABD;

A d d e d g e A X o r BX a s a p p r o p r i a t e a n d c a l l t h i s n e w e d g e AB;

end loop

U s i n g th e p r o c e s s o u t l i n e d a b o v e , t h e ti m e t o c o m b i n e th e C D T s o f a d j a c e n t

s t r i ps i s O v + h) w h e r e v i s t h e t o ta l n u m b e r o f v e rt ic e s i n t h e n e w l y c o m b i n e d

s t ri p a n d h i s t h e n u m b e r o f h a l f -e d g e s f r o m e a c h o f t h e t w o s tr ip s . T o s e e th i s

n o t e t h a t i n e a c h p a r t o f s t e p 2 w e e i t h e r e l i m i n a t e a n e d g e f r o m a p a r ti a l C D T

o r w e a d d a n e d g e f o r t h e c o m b i n e d C D T . I t is e a sy to s e e t h a t th e t o t a l n u m b e r

o f e d g e s i n v o l v e d i s O v + h).

W e c l a im t h a t t h e t i m e to b u i l d t h e c o m p l e t e C D T f o r a s t ra i g h t- l in e p l a n a r

g r a p h G i s O n l o g n ) w h e r e n i s t h e n u m b e r o f v e r t ic e s o f G . T h e p r o b l e m is

i n i t ia l l y b r o k e n d o w n i n t o n s t ri p s e a c h c o n t a i n i n g a s i n g le v e r te x . S t ri p s a r e

c o m b i n e d i n p a i r s t o m a k e s t ri p s c o n t a i n i n g t w o v e r t ic e s , t h e n f o u r v e r t ic e s , e t c.

T h e s e s u b p r o b l e m s c a n b e r e p r e s e n t e d i n a t r e e o f d e p t h O l o g n ) w i th t h e

c o m p l e t e C D T a t t h e r o o t a n d w i t h s in g l e - v e r te x s tr ip s a p p e a r i n g a s th e l e a v e s.

R e c a l l t h a t t h e t i m e t o s o l v e o n e s u b p r o b l e m i s O v + h) w h e r e v is t h e n u m b e r

o f v e r ti c e s w i t h i n t h e s t r ip a n d h i s t h e n u m b e r o f h a l f- e d g e s f o r t h a t s tr ip . F o r


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C o n s t r a i n e d D e l a u n a y T r i a n g u l a t i o n s 1 07

a single level of the tree, each vertex of G appears in exactly one subproblem

and each half-edge of G (a G-edge is always made of two half-edges) appears

in exactly one subproblem. Thus, the total time spent solving all the subproblems

on one level of the tree is O n + e ) where e is the number of edges in G. Since

G is planar, e is O n ) . This give us the following theorem.

THEOREM. Le t G be a st ra igh t - line p lanar graph. The C D T o f G can be bu il t in

O n log n) t ime where n i s the num ber o f vert ices o f G .

6. Conclusions. We have shown how a divide-and-conquer algori thm can be

used to produce the CDT in O n log n) time. This time bound is optimal since

it is easy to show that a CDT can be used to sort. There are two ideas that have

been particularly important in reaching this time bound: (1) as in [Ya84], the

only cross edges that we keep track of are those that bound vertex-containing

regions; (2) infinite vertices are used so that partial CDTs are linked for efficient

access. A useful property of these infinite vertices is that such a vertex can be

eliminated with minimal effect on edges between noninfinite vertices.

It may be possible to develop a sweep-line algorithm for constructing the CDT.

Such an algorithm would probably run faster than the divide-and-conquer

algorithm presented here, although it would, of course, have the same asymptotic

time bound. Not surprisingly, the CDT, a type of Delaunay triangulation, has a

dual that is a type of Voronoi diagram. However, this dual graph can overlap

itself, with different portions of the graph sharing the same portion of the

Euclidean plane (see Figure 8). Thus, although we suspect Fortune s sweep-line

F i g 8 T h e d u a l o f a C D T : a V o r o n o i d i a g r a m t h a t o v e r la p s i ts e l f


12/12

1 08 L . P . C h e w

technique [Fo87] developed to build the standard Voronoi diagram can be

adapted to build the CDT it is not immediately clear how it should be done.

e f e r e n c e s

[ C D 8 5 ]

[Ch86]

[Ch87a ]

[Ch87b]

[ D F P 8 5 ]

[Fo87]

[Ki79]

[Le78]

[LL86]

[LS80]

[PS85]

[Ya84]

L . P . Che w and R . L . Drysda l e , Voro noi d i agrams based on con vex d i s t ance func t i ons ,

Proceedings of the First Symposium on Computational Geometry,

Bal t imo re 1985) , pp.

235-244 . Revi sed ve rs i on subm i t ted t o Discrete and Computational Geometry.)

L. P . Chew, The re i s a p l ana r g raph a l mos t a s good a s t he compl e t e g raph , Proceedings

of the Second Annual Symposium on Computational Geometry,

Yorkt ow n He i ght s 1986) ,

pp. 169-177.

L . P . Chew , P l ana r g raph s and spa rse g raphs fo r ef fi ci en t mot i on p l anni ng i n t he p l ane ,

in prepara t ion.

L . P . Chew, Gua ran t eed-qua l i t y t r i angul a r meshes , i n p repa ra t i on .

L . De F l of i an i , B. Fa l c id i eno , and C . P i enovi , De l aunay-based repre sen t a t i on of sur face s

de f i ned ove r a rb i tr a r i ly sh aped dom a i ns , Computer Vision, Graphics, and Image Processing,

32 1985), 127-140.

S . For t une , A S weepl i ne a l gor i thm for Voron oi d i agrams , Algorithmica,2 1987), 153-174.

D. G. Ki rkpat r ick, Eff ic ient com puta t ion o f con t inuo us ske le tons , Proceedings of the 20th

Annual Symposium on the Foundations of Computer Science,

I E E E C o m p u t e r S o c i e t y

1979), pp. 18-27.

D. T . Lee , P rox i mi t y and t eachabi l i ty i n t he p l ane , Techn i ca l Repor t R -831 , Co ord i n a t ed

Sc i ence Labora t ory , U ni ve rs i t y o f I l l ino i s 1978) .

D. T . Lee and A. K. L i n , Gene ra l i zed De l au nay t r i angul a t i on for p l ana r g raphs ,

Discrete

and Computational Geometry, 1 1986), 201-217.

D. T . Lee and B . Schacht e r , Two a l gor i t hms for cons t ruc t i ng De l aunay t r i angul a t i ons ,

International Journal of Computer and Information Sciences,

9 1980), 219-242.

F. P. Prepara ta and M. I . Shamos,

Computational Geometry,

Spr i nge r -Ver l ag , New York

1985).

C . K. Yap , An

O n

l o g n ) a l g o r i t h m f o r th e V o r o n o i d i a g ra m o f a s e t o f s i m p l e c u rv e

segments . Techni ca l Repor t , Couran t Ins t i t u t e, N ew York Uni ve rs i t y Oc t ob e r 1984) .

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