<s>
A	O
kinetic	B-Data_Structure
data	I-Data_Structure
structure	I-Data_Structure
is	O
a	O
data	B-General_Concept
structure	I-General_Concept
used	O
to	O
track	O
an	O
attribute	O
of	O
a	O
geometric	O
system	O
that	O
is	O
moving	O
continuously	O
.	O
</s>
<s>
For	O
example	O
,	O
a	O
kinetic	B-Algorithm
convex	I-Algorithm
hull	I-Algorithm
data	B-General_Concept
structure	I-General_Concept
maintains	O
the	O
convex	O
hull	O
of	O
a	O
group	O
of	O
moving	O
points	O
.	O
</s>
<s>
The	O
development	O
of	O
kinetic	B-Data_Structure
data	I-Data_Structure
structures	I-Data_Structure
was	O
motivated	O
by	O
computational	O
geometry	O
problems	O
involving	O
physical	O
objects	O
in	O
continuous	O
motion	O
,	O
such	O
as	O
collision	O
or	O
visibility	O
detection	O
in	O
robotics	O
,	O
animation	O
or	O
computer	O
graphics	O
.	O
</s>
<s>
Kinetic	B-Data_Structure
data	I-Data_Structure
structures	I-Data_Structure
are	O
used	O
on	O
systems	O
where	O
there	O
is	O
a	O
set	O
of	O
values	O
that	O
are	O
changing	O
as	O
a	O
function	O
of	O
time	O
,	O
in	O
a	O
known	O
fashion	O
.	O
</s>
<s>
Kinetic	B-Data_Structure
data	I-Data_Structure
structures	I-Data_Structure
allow	O
queries	O
on	O
a	O
system	O
at	O
the	O
current	O
virtual	O
time	O
,	O
and	O
two	O
additional	O
operations	O
:	O
</s>
<s>
For	O
example	O
,	O
kinetic	B-Data_Structure
data	I-Data_Structure
structures	I-Data_Structure
are	O
often	O
used	O
with	O
a	O
set	O
of	O
points	O
.	O
</s>
<s>
A	O
kinetic	B-Data_Structure
data	I-Data_Structure
structure	I-Data_Structure
allows	O
the	O
values	O
stored	O
in	O
it	O
to	O
change	O
continuously	O
with	O
time	O
.	O
</s>
<s>
In	O
principle	O
,	O
this	O
can	O
be	O
approximated	O
by	O
sampling	O
the	O
position	O
of	O
the	O
points	O
at	O
fixed	O
intervals	O
of	O
time	O
,	O
and	O
deleting	O
and	O
re-inserting	O
each	O
point	O
into	O
a	O
"	O
static	O
"	O
(	O
traditional	O
)	O
data	B-General_Concept
structure	I-General_Concept
.	O
</s>
<s>
The	O
following	O
general	O
approach	O
can	O
be	O
used	O
to	O
construct	O
kinetic	B-Data_Structure
data	I-Data_Structure
structures	I-Data_Structure
:	O
</s>
<s>
Store	O
a	O
data	B-General_Concept
structure	I-General_Concept
on	O
the	O
system	O
at	O
the	O
current	O
time	O
.	O
</s>
<s>
This	O
data	B-General_Concept
structure	I-General_Concept
allows	O
queries	O
on	O
the	O
system	O
at	O
the	O
current	O
virtual	O
time	O
.	O
</s>
<s>
Augment	O
the	O
data	B-General_Concept
structure	I-General_Concept
with	O
certificates	O
.	O
</s>
<s>
Certificates	O
are	O
conditions	O
under	O
which	O
the	O
data	B-General_Concept
structure	I-General_Concept
is	O
accurate	O
.	O
</s>
<s>
The	O
certificates	O
are	O
all	O
true	O
now	O
,	O
and	O
the	O
data	B-General_Concept
structure	I-General_Concept
will	O
only	O
cease	O
to	O
be	O
accurate	O
when	O
one	O
of	O
the	O
certificates	O
is	O
no	O
longer	O
true	O
.	O
</s>
<s>
If	O
the	O
certificate	O
fails	O
before	O
time	O
,	O
pop	O
it	O
from	O
the	O
queue	O
and	O
fix	O
the	O
data	B-General_Concept
structure	I-General_Concept
so	O
it	O
is	O
accurate	O
at	O
the	O
time	O
of	O
failure	O
,	O
and	O
update	O
the	O
certificates	O
.	O
</s>
<s>
If	O
the	O
certificate	O
with	O
the	O
lowest	O
failure	O
time	O
in	O
the	O
priority	O
queue	O
fails	O
after	O
time	O
,	O
then	O
all	O
certificates	O
are	O
true	O
at	O
time	O
so	O
the	O
data	B-General_Concept
structure	I-General_Concept
can	O
correctly	O
answer	O
queries	O
at	O
time	O
.	O
</s>
<s>
An	O
event	O
is	O
considered	O
internal	O
if	O
the	O
property	O
maintained	O
by	O
the	O
kinetic	B-Data_Structure
data	I-Data_Structure
structure	I-Data_Structure
does	O
not	O
change	O
when	O
the	O
event	O
occurs	O
.	O
</s>
<s>
An	O
event	O
is	O
considered	O
external	O
if	O
the	O
property	O
maintained	O
by	O
the	O
data	B-General_Concept
structure	I-General_Concept
changes	O
when	O
the	O
event	O
occurs	O
.	O
</s>
<s>
Responsiveness	O
is	O
the	O
worst	O
case	O
amount	O
of	O
time	O
required	O
to	O
fix	O
the	O
data	B-General_Concept
structure	I-General_Concept
and	O
augmenting	O
certificates	O
when	O
a	O
certificate	O
fails	O
.	O
</s>
<s>
A	O
kinetic	B-Data_Structure
data	I-Data_Structure
structure	I-Data_Structure
is	O
responsive	O
if	O
the	O
worst	O
case	O
amount	O
of	O
time	O
required	O
for	O
an	O
update	O
is	O
small	O
.	O
</s>
<s>
A	O
kinetic	B-Data_Structure
data	I-Data_Structure
structure	I-Data_Structure
is	O
local	O
if	O
the	O
maximum	O
number	O
of	O
certificates	O
any	O
one	O
value	O
is	O
involved	O
with	O
is	O
small	O
.	O
</s>
<s>
The	O
maximum	O
number	O
of	O
certificates	O
used	O
to	O
augment	O
the	O
data	B-General_Concept
structure	I-General_Concept
at	O
any	O
time	O
.	O
</s>
<s>
A	O
kinetic	B-Data_Structure
data	I-Data_Structure
structure	I-Data_Structure
is	O
compact	O
if	O
the	O
number	O
of	O
certificates	O
it	O
uses	O
is	O
or	O
for	O
arbitrarily	O
small	O
.	O
</s>
<s>
The	O
ratio	O
of	O
the	O
worst	O
case	O
number	O
of	O
events	O
that	O
can	O
occur	O
when	O
the	O
structure	O
is	O
advanced	O
to	O
to	O
the	O
worst	O
case	O
number	O
of	O
"	O
necessary	O
changes	O
"	O
to	O
the	O
data	B-General_Concept
structure	I-General_Concept
.	O
</s>
<s>
For	O
example	O
,	O
in	O
the	O
case	O
of	O
a	O
kinetic	B-Data_Structure
data	I-Data_Structure
structure	I-Data_Structure
maintaining	O
the	O
kinetic	O
hull	O
of	O
a	O
set	O
of	O
moving	O
points	O
,	O
the	O
number	O
of	O
necessary	O
changes	O
would	O
be	O
the	O
number	O
of	O
times	O
the	O
kinetic	O
hull	O
changes	O
as	O
time	O
is	O
advanced	O
to	O
.	O
</s>
<s>
A	O
kinetic	B-Data_Structure
data	I-Data_Structure
structure	I-Data_Structure
is	O
said	O
to	O
be	O
efficient	O
if	O
this	O
ratio	O
is	O
small	O
.	O
</s>
<s>
The	O
performance	O
of	O
a	O
certain	O
kinetic	B-Data_Structure
data	I-Data_Structure
structure	I-Data_Structure
may	O
be	O
analyzed	O
for	O
certain	O
types	O
of	O
trajectories	O
.	O
</s>
