Voronoi Diagram –
Fortune’s Algorithm
Reporter: GI1 11號 蔡逸凡
Date:
2004/10/28
Reference: Computational Geometry ch7
ISBN: 3-540-65620-0
Fortune’s Algorithm (summary)
HandleSiteEvent(pi)
1. Search T for arc directly
above pi; delete circle
events associated with
this arc
2. Add arc of pi into T (with
two new break points
<pi,pj>,<pj,pi>)
3. [DCEL] edge between
<pi,pj>
4. Detect circle events
where the new arc is the
left arc in a triple of
consecutive arcs
HandleCircleEvent(pl)
1. Delete disappearing arc
from T and its
associated circle events
from Q
2. [DCEL] Center of circle
as vertex in V.diagram
3. Check new triples for
circle events (only if the
circle intersects sweep
line and not yet in Q)
4. [DCEL] create edges due
to disappearing arc
From the Textbook
These can be ignored if
using simplified T
From the Textbook
Case 1
Initially:
p3
p1
Q={p1,p2,p3}
T ={}
D={}
p2
Case 1(cont)
SiteEvent(p1)
Q={p2,p3}
T ={p1}
D={}
Entries in T:
the arcs
induced by
the sites
Case 1 (Cont)
SiteEvent(p2)
Q={p3}
Arc above: p1
T ={p1,p2,p1}
D={e1=<p2,p1>}
The arcs in T
(bottom to
top)
Here two p1
represents
different
segments of the
arc!
Case 1 (Cont)
SiteEvent (p3)
Q={}
Arc above: p1
T ={p1,p2,p1,p3,p1}
D={e1,e2=<p1,p3>}
Detect circle event:
c1:<p2,p1,p3>
→ add c1 in Q
Case 1 (Cont)
CircleEvent (c1)
Q={}
Disappearing arc: p1

T ={p1,p2,p1,p3,p1}
D={e1,e2,v=<p2,p1,p3>,
e3=<p2,p3>}
Case 1 (Cont)
Finally,
Q={}
T ={p1,p2,p3,p1}
D={e1,e2,v,e3}
Case 2
Initially:
Q={p1,p2,p3,p4}
T ={}
D={}
Case 2(cont)
SiteEvent(p1)
p2
Q={p2,p3,p4}
p1
p4
p3
T ={p1}
D={}
Case 2 (Cont)
SiteEvent(p2)
Q={p3,p4}
Arc above: p1
T ={p1,p2,p1}
D={e1=<p1,p2>}
Case 2 (Cont)
SiteEvent(p3)
Q={p4}
Arc above: p1
T ={p1,p3,p1,p2,p1}
D={e1,e2=<p3,p1>}
Detect circle event:
c1:<p3,p1,p2>
→ add c1 in Q={p4,c1}
Make a note
that c1 is due
to <p3,p1,p2>
Case 2 (Cont)
SiteEvent(p4)
Q={c1}
Arc above: p1
Delete c1 from Q
T ={p1,p3,p1,p4,p1,p2,p1}
D={e1,e2,e3=<p1,p4>}
Detect circle event:
<p3p1p4>c2,<p4p1p2>c3
Case 2 (Cont)
CircleEvent(c2)
Q={c3}
Disappearing arc: p1

T ={p1,p3,p1,p4,p1,p2,p1}
D={e1,e2,e3,v1=<p3p1p4>,
e4=<p3,p4>}
Case 2 (Cont)
CircleEvent(c3)
Q={}
Disappearing arc: p1

T ={p1,p3,p4,p1,p2,p1}
D={e1,e2,e3,v1,e4,v2=<p4p1p2
>, e5=<p4,p2> }
Case 2 (Cont)
Finally,
Q={}
T ={p1,p3,p4,p2,p1}
D={e1,e2,e3,v1,e4,v2,e5}
Case 3
Initially:
Q={p1,p2,p3,p4}
T ={}
Case 3 (cont)
SiteEvent (p1)
Q={p2,p3,p4}
T ={p1}
Case 3 (cont)
SiteEvent (p2)
Q={p3,p4}
Arc above: p1
T ={p1,p2,p1}
D={e1=<p2,p1>}
Case 3 (cont)
SiteEvent (p3)
Q={p4}
Arc above: p1
T ={p1,p2,p1,p3,p1}
D={e1,e2=<p1,p3>}
Circle event <p2,p1,p3>c1
Q={p4,c1}
Case 3 (cont)
SiteEvent (p4)
Q={c1}
Arc above:p3; remove c1
T ={p1,p2,p1,p3,p4,p3,p1}
D={e1,e2,e3=<p3,p4>}
Circle event <p1,p3,p4>c2
Q={c2}
Case 3 (cont)
CircleEvent (c2)
Q={}
Disappearing arc p3
T ={p1,p2,p1,p3,p4,p3,p1}
D={e1,e2,e3,v1=<p1,p3,p4>,e4=<p1,p4
>}
Circle event <p2,p1,p4>c3
Q={c3}

Case 3 (cont)
CircleEvent (c3)
Q={}
Disappearing arc p1
T ={p1,p2,p1,p4,p3,p1}
<p2,p4,p3> circle before sweep line
D={e1,e2,e3,v1,e4,v2=<p2,p1,p4>,
e5=<p2,p4>}

Case 3 (cont)
Finally:
D={e1,e2,e3,v1, e4,v2, e5}
Case 4
Initial:
SiteEvent(p1)
Q={p1,p2,p3,p4}
Q={p2,p3,p4}
B={}
T ={p1}
D={}
D={}
Case 4 (Cont)
SiteEvent(p2)
Q={p3,p4}
Arc above: p1
T ={p1,p2,p1}
D={e1=<p2,p1>}
Case 4 (Cont)
SiteEvent(p3)
Q={p4}
Arc above: p1
T ={p1,p2,p1,p3,p1}
D={e1,e2=<p1,p3>}
Detect circle event:
c1:<p2,p1,p3>
→ add c1 in Q
Case 4 (Cont)
CircleEvent(c1)
Q={p4}
Disappearing arc: p1

T ={p1,p2,p1,p3,p1}
D={e1,e2,v1=<p2,p1,p3>,e
3=<p2,p3>}
Case 4 (Cont)
SiteEvent(p4)
Q={}
Arc directly above: p2
T ={p1,p2,p4,p2,p3,p1}
Circle event <p4,p2,p3>c2
D={e1,e2,v1,e3,e4=<p2,p4>}
P4 is equal distance
to p2 & p3
Case 4 (Cont)
CircleEvent(c2)
Q={}
Disappearing arc: p2
T ={p1,p2,p4,p2,p3,p1}
D={e1,e2,v1,e3,e4,v2=<p4p2p3
>, e5=<p4,p3>}

Case 4 (Cont)
Finally,
D={e1,e2,v1,e3,v2,e3,e4,e5}
Case 5
SiteEvent(p1)
Q={p2,p3,p4}
T ={p1}
D={}
Case 5 (cont)
SiteEvent(p2)
Q={p3,p4}
Arc above: p1
T ={p1,p2,p1}
D={e1=<p2,p1>}
Case 5 (cont)
SiteEvent(p3)
Q={p4}
Arc above: p1
T ={p1,p3,p1,p2,p1}
D={e1,e2=<p1,p3>}
Detect circle event:
c1:<p3,p1,p2>
→ add c1 in Q
Case 5 (cont)
CircleEvent(c1)
Q={p4}
Disappearing arc: p1

T ={p1,p3,p1,p2,p1}
D={e1,e2,v1=<p3,p1,p2>,e3
=<p3,p2>}
Case 5 (cont)
SiteEvent(p4)
Q={}
Arc above: p2
T ={p1,p3,p2,p4,p2,p1}
D={e1,e2,v1,e3,e4=<p2,p4>}
Detect circle event:
c2:<p3,p2,p4>
→ add c2 in Q
Case 5 (cont)
CircleEvent(c2)
Q={}
Disappearing arc: p2

T ={p1,p3,p2,p4,p2,p1}
D={e1,e2,v1,e3,e4,v2=<p3,p
2,p4>,e5=<p3,p4>}
Case 5 (cont)
Finally,
D={e1,e2,v1,e3,e4,v2, e5}
Questions
 How do I know which one is the arc above?
 Location above
 How to know which arc is disappearing
 Location above?!
 Link between circle event and arc
 Things in
site)
T should be arcs (with a corresponding
 Degeneracy: 4points on a circle
 Does it matter?