<s>
A	O
kinetic	B-Algorithm
convex	I-Algorithm
hull	I-Algorithm
data	O
structure	O
is	O
a	O
kinetic	B-Data_Structure
data	I-Data_Structure
structure	I-Data_Structure
that	O
maintains	O
the	O
convex	O
hull	O
of	O
a	O
set	O
of	O
continuously	O
moving	O
points	O
.	O
</s>
<s>
It	O
should	O
be	O
distinguished	O
from	O
dynamic	B-Algorithm
convex	I-Algorithm
hull	I-Algorithm
data	O
structures	O
,	O
which	O
handle	O
points	O
undergoing	O
discrete	O
changes	O
such	O
as	O
insertions	O
or	O
deletions	O
of	O
points	O
rather	O
than	O
continuous	O
motion	O
.	O
</s>
<s>
The	O
best	O
known	O
data	O
structure	O
for	O
the	O
2-dimensional	O
kinetic	B-Algorithm
convex	I-Algorithm
hull	I-Algorithm
problem	O
is	O
by	O
Basch	O
,	O
Guibas	O
,	O
and	O
Hershberger	O
.	O
</s>
<s>
The	O
dual	O
of	O
a	O
convex	O
hull	O
of	O
a	O
set	O
of	O
points	O
is	O
the	O
upper	B-Algorithm
and	I-Algorithm
lower	I-Algorithm
envelopes	I-Algorithm
of	O
the	O
dual	O
set	O
of	O
lines	O
.	O
</s>
<s>
Therefore	O
,	O
maintaining	O
the	O
upper	B-Algorithm
and	I-Algorithm
lower	I-Algorithm
envelopes	I-Algorithm
of	O
a	O
set	O
of	O
moving	O
lines	O
is	O
equivalent	O
to	O
maintaining	O
the	O
convex	O
hull	O
of	O
a	O
set	O
of	O
moving	O
points	O
.	O
</s>
<s>
Computing	O
upper	B-Algorithm
and	I-Algorithm
lower	I-Algorithm
envelopes	I-Algorithm
are	O
equivalent	O
problems	O
,	O
so	O
computing	O
the	O
upper	B-Algorithm
envelope	I-Algorithm
of	O
a	O
set	O
of	O
lines	O
is	O
equivalent	O
to	O
computing	O
the	O
convex	O
hull	O
of	O
a	O
set	O
of	O
moving	O
points	O
.	O
</s>
<s>
The	O
upper	B-Algorithm
envelope	I-Algorithm
of	O
a	O
set	O
of	O
static	O
lines	O
can	O
be	O
computed	O
using	O
a	O
divide	B-Algorithm
and	I-Algorithm
conquer	I-Algorithm
algorithm	I-Algorithm
which	O
partitions	O
the	O
lines	O
into	O
two	O
sets	O
of	O
equal	O
size	O
,	O
calls	O
itself	O
recursively	O
on	O
the	O
two	O
sets	O
to	O
find	O
their	O
upper	B-Algorithm
envelopes	I-Algorithm
,	O
and	O
then	O
merges	O
the	O
two	O
resulting	O
upper	B-Algorithm
envelopes	I-Algorithm
.	O
</s>
<s>
The	O
merge	O
step	O
is	O
performed	O
using	O
a	O
vertical	B-Algorithm
line	I-Algorithm
sweep	I-Algorithm
.	O
</s>
<s>
The	O
standard	O
line	B-Algorithm
sweep	I-Algorithm
algorithm	I-Algorithm
for	O
merging	O
upper	B-Algorithm
envelopes	I-Algorithm
sweeps	O
though	O
all	O
of	O
vertices	O
of	O
the	O
red	O
and	O
blue	O
upper	B-Algorithm
envelopes	I-Algorithm
,	O
from	O
left	O
to	O
right	O
.	O
</s>
<s>
If	O
it	O
is	O
above	O
,	O
that	O
point	O
is	O
added	O
to	O
the	O
merged	O
upper	B-Algorithm
envelope	I-Algorithm
.	O
</s>
<s>
If	O
it	O
is	O
of	O
a	O
different	O
color	O
than	O
the	O
last	O
added	O
point	O
,	O
the	O
red	O
and	O
blue	O
envelopes	O
have	O
crossed	O
,	O
so	O
the	O
intersection	O
point	O
is	O
also	O
added	O
to	O
the	O
merged	O
upper	B-Algorithm
envelope	I-Algorithm
.	O
</s>
<s>
They	O
are	O
the	O
certificates	O
for	O
a	O
kinetic	B-Data_Structure
sorted	I-Data_Structure
list	I-Data_Structure
on	O
the	O
set	O
of	O
vertices	O
.	O
</s>
<s>
As	O
long	O
as	O
all	O
of	O
these	O
certificates	O
hold	O
,	O
the	O
merge	O
steps	O
will	O
be	O
executed	O
identically	O
,	O
so	O
the	O
resulting	O
upper	B-Algorithm
envelope	I-Algorithm
will	O
be	O
the	O
same	O
.	O
</s>
<s>
A	O
kinetic	B-Data_Structure
data	I-Data_Structure
structure	I-Data_Structure
for	O
upper	B-Algorithm
envelopes	I-Algorithm
can	O
be	O
created	O
by	O
using	O
these	O
certificates	O
to	O
certify	O
the	O
static	O
upper	B-Algorithm
envelope	I-Algorithm
algorithm	O
.	O
</s>
<s>
The	O
remaining	O
5	O
certificates	O
,	O
,	O
,	O
,	O
,	O
and	O
place	O
edges	O
that	O
are	O
not	O
in	O
the	O
upper	B-Algorithm
envelopes	I-Algorithm
in	O
the	O
sequence	O
of	O
slopes	O
of	O
the	O
edges	O
that	O
are	O
above	O
it	O
.	O
</s>
<s>
The	O
kinetic	B-Data_Structure
data	I-Data_Structure
structure	I-Data_Structure
for	O
convex	O
hull	O
is	O
constructed	O
by	O
using	O
these	O
certificates	O
to	O
certify	O
the	O
recursive	O
merge	O
of	O
the	O
upper	B-Algorithm
envelopes	I-Algorithm
.	O
</s>
<s>
This	O
is	O
due	O
to	O
the	O
logarithmic	O
depth	O
of	O
the	O
merges	O
used	O
to	O
certify	O
the	O
upper	B-Algorithm
envelope	I-Algorithm
.	O
</s>
<s>
Certifying	O
an	O
upper	B-Algorithm
envelope	I-Algorithm
of	O
n	O
lines	O
requires	O
certificates	O
for	O
the	O
merge	O
upper	B-Algorithm
envelope	I-Algorithm
of	O
the	O
two	O
subsets	O
of	O
n/2	O
lines	O
,	O
and	O
certificates	O
for	O
the	O
merge	O
of	O
the	O
envelopes	O
.	O
</s>
<s>
Thus	O
the	O
number	O
of	O
certificates	O
,	O
C(n )	O
,	O
required	O
to	O
certify	O
the	O
upper	B-Algorithm
envelope	I-Algorithm
of	O
n	O
lines	O
satisfies	O
the	O
recurrence	O
C(n )	O
=	O
2C( 	O
n/2	O
)	O
+	O
O(n )	O
,	O
with	O
C(1 )	O
=O	O
( 1	O
)	O
.	O
</s>
<s>
By	O
the	O
master	B-Algorithm
theorem	I-Algorithm
for	I-Algorithm
divide-and-conquer	I-Algorithm
recurrences	I-Algorithm
,	O
C(n )	O
=O	O
( n	O
log	O
n	O
)	O
.	O
</s>
<s>
Finding	O
an	O
efficient	O
kinetic	B-Data_Structure
data	I-Data_Structure
structure	I-Data_Structure
for	O
maintaining	O
the	O
convex	O
hull	O
of	O
moving	O
points	O
in	O
dimensions	O
higher	O
than	O
2	O
is	O
an	O
open	O
problem	O
.	O
</s>
<s>
Kinetic	B-Algorithm
convex	I-Algorithm
hull	I-Algorithm
can	O
be	O
used	O
to	O
solve	O
the	O
following	O
related	O
problems	O
:	O
</s>
