<s>
In	O
mathematics	O
,	O
a	O
spline	B-Algorithm
is	O
a	O
special	O
function	O
defined	O
piecewise	B-Algorithm
by	O
polynomials	O
.	O
</s>
<s>
In	O
interpolating	B-Algorithm
problems	O
,	O
spline	B-Algorithm
interpolation	I-Algorithm
is	O
often	O
preferred	O
to	O
polynomial	O
interpolation	B-Algorithm
because	O
it	O
yields	O
similar	O
results	O
,	O
even	O
when	O
using	O
low	O
degree	O
polynomials	O
,	O
while	O
avoiding	O
Runge	O
's	O
phenomenon	O
for	O
higher	O
degrees	O
.	O
</s>
<s>
In	O
the	O
computer	B-General_Concept
science	I-General_Concept
subfields	O
of	O
computer-aided	B-Application
design	I-Application
and	O
computer	O
graphics	O
,	O
the	O
term	O
spline	B-Algorithm
more	O
frequently	O
refers	O
to	O
a	O
piecewise	B-Algorithm
polynomial	I-Algorithm
(	O
parametric	O
)	O
curve	O
.	O
</s>
<s>
Splines	B-Algorithm
are	O
popular	O
curves	O
in	O
these	O
subfields	O
because	O
of	O
the	O
simplicity	O
of	O
their	O
construction	O
,	O
their	O
ease	O
and	O
accuracy	O
of	O
evaluation	O
,	O
and	O
their	O
capacity	O
to	O
approximate	O
complex	O
shapes	O
through	O
curve	B-Algorithm
fitting	I-Algorithm
and	O
interactive	O
curve	O
design	O
.	O
</s>
<s>
The	O
term	O
spline	B-Algorithm
comes	O
from	O
the	O
flexible	O
spline	B-Algorithm
devices	O
used	O
by	O
shipbuilders	O
and	O
draftsmen	B-Application
to	O
draw	O
smooth	O
shapes	O
.	O
</s>
<s>
The	O
term	O
"	O
spline	B-Algorithm
"	O
is	O
used	O
to	O
refer	O
to	O
a	O
wide	O
class	O
of	O
functions	O
that	O
are	O
used	O
in	O
applications	O
requiring	O
data	O
interpolation	B-Algorithm
and/or	O
smoothing	O
.	O
</s>
<s>
Spline	B-Algorithm
functions	I-Algorithm
for	O
interpolation	B-Algorithm
are	O
normally	O
determined	O
as	O
the	O
minimizers	O
of	O
suitable	O
measures	O
of	O
roughness	O
(	O
for	O
example	O
integral	O
squared	O
curvature	O
)	O
subject	O
to	O
the	O
interpolation	B-Algorithm
constraints	O
.	O
</s>
<s>
Smoothing	O
splines	B-Algorithm
may	O
be	O
viewed	O
as	O
generalizations	O
of	O
interpolation	B-Algorithm
splines	B-Algorithm
where	O
the	O
functions	O
are	O
determined	O
to	O
minimize	O
a	O
weighted	O
combination	O
of	O
the	O
average	O
squared	O
approximation	O
error	O
over	O
observed	O
data	O
and	O
the	O
roughness	O
measure	O
.	O
</s>
<s>
For	O
a	O
number	O
of	O
meaningful	O
definitions	O
of	O
the	O
roughness	O
measure	O
,	O
the	O
spline	B-Algorithm
functions	I-Algorithm
are	O
found	O
to	O
be	O
finite	O
dimensional	O
in	O
nature	O
,	O
which	O
is	O
the	O
primary	O
reason	O
for	O
their	O
utility	O
in	O
computations	O
and	O
representation	O
.	O
</s>
<s>
For	O
the	O
rest	O
of	O
this	O
section	O
,	O
we	O
focus	O
entirely	O
on	O
one-dimensional	O
,	O
polynomial	O
splines	B-Algorithm
and	O
use	O
the	O
term	O
"	O
spline	B-Algorithm
"	O
in	O
this	O
restricted	O
sense	O
.	O
</s>
<s>
We	O
begin	O
by	O
limiting	O
our	O
discussion	O
to	O
polynomials	B-General_Concept
in	I-General_Concept
one	I-General_Concept
variable	I-General_Concept
.	O
</s>
<s>
In	O
this	O
case	O
,	O
a	O
spline	B-Algorithm
is	O
a	O
piecewise	B-Algorithm
polynomial	I-Algorithm
function	O
.	O
</s>
<s>
We	O
want	O
S	O
to	O
be	O
piecewise	B-Algorithm
defined	O
.	O
</s>
<s>
To	O
accomplish	O
this	O
,	O
let	O
the	O
interval	O
 [ a , b ] 	O
be	O
covered	O
by	O
k	O
ordered	O
,	O
disjoint	B-Algorithm
subintervals	O
,	O
</s>
<s>
The	O
given	O
k+1	O
points	O
ti	O
are	O
called	O
knots	B-Algorithm
.	O
</s>
<s>
is	O
called	O
a	O
knot	B-Algorithm
vector	I-Algorithm
for	O
the	O
spline	B-Algorithm
.	O
</s>
<s>
If	O
the	O
knots	B-Algorithm
are	O
equidistantly	O
distributed	O
in	O
the	O
interval	O
 [ a , b ] 	O
we	O
say	O
the	O
spline	B-Algorithm
is	O
uniform	O
,	O
otherwise	O
we	O
say	O
it	O
is	O
non-uniform	O
.	O
</s>
<s>
such	O
that	O
the	O
spline	B-Algorithm
has	O
smoothness	O
at	O
ti	O
for	O
is	O
called	O
a	O
smoothness	O
vector	O
for	O
the	O
spline	B-Algorithm
.	O
</s>
<s>
This	O
spline	B-Algorithm
space	O
is	O
commonly	O
denoted	O
by	O
.	O
</s>
<s>
In	O
the	O
mathematical	O
study	O
of	O
polynomial	O
splines	B-Algorithm
the	O
question	O
of	O
what	O
happens	O
when	O
two	O
knots	B-Algorithm
,	O
</s>
<s>
This	O
leads	O
to	O
a	O
more	O
general	O
understanding	O
of	O
a	O
knot	B-Algorithm
vector	I-Algorithm
.	O
</s>
<s>
of	O
the	O
same	O
degree	O
with	O
the	O
same	O
extended	O
knot	B-Algorithm
vectors	I-Algorithm
on	O
that	O
interval	O
.	O
</s>
<s>
(	O
Note	O
:	O
while	O
the	O
polynomial	O
piece	O
2t	O
is	O
not	O
quadratic	O
,	O
the	O
result	O
is	O
still	O
called	O
a	O
quadratic	B-Algorithm
spline	I-Algorithm
.	O
</s>
<s>
This	O
demonstrates	O
that	O
the	O
degree	O
of	O
a	O
spline	B-Algorithm
is	O
the	O
maximum	O
degree	O
of	O
its	O
polynomial	O
parts	O
.	O
)	O
</s>
<s>
The	O
extended	O
knot	B-Algorithm
vector	I-Algorithm
for	O
this	O
type	O
of	O
spline	B-Algorithm
would	O
be	O
(	O
0	O
,	O
1	O
,	O
2	O
,	O
2	O
,	O
3	O
)	O
.	O
</s>
<s>
The	O
simplest	O
spline	B-Algorithm
has	O
degree	O
0	O
.	O
</s>
<s>
The	O
next	O
most	O
simple	O
spline	B-Algorithm
has	O
degree	O
1	O
.	O
</s>
<s>
It	O
is	O
also	O
called	O
a	O
linear	B-Algorithm
spline	I-Algorithm
.	O
</s>
<s>
A	O
closed	O
linear	B-Algorithm
spline	I-Algorithm
(	O
i.e	O
,	O
the	O
first	O
knot	B-Algorithm
and	O
the	O
last	O
are	O
the	O
same	O
)	O
in	O
the	O
plane	O
is	O
just	O
a	O
polygon	B-General_Concept
.	O
</s>
<s>
A	O
common	O
spline	B-Algorithm
is	O
the	O
natural	B-Algorithm
cubic	I-Algorithm
spline	I-Algorithm
of	O
degree	O
3	O
with	O
continuity	O
C2	O
.	O
</s>
<s>
This	O
forces	O
the	O
spline	B-Algorithm
to	O
be	O
a	O
straight	O
line	O
outside	O
of	O
the	O
interval	O
,	O
while	O
not	O
disrupting	O
its	O
smoothness	O
.	O
</s>
<s>
Cubic	B-Algorithm
splines	I-Algorithm
are	O
of	O
the	O
form	O
.	O
</s>
<s>
Algorithm	O
for	O
computing	O
Natural	B-Algorithm
Cubic	I-Algorithm
Splines	I-Algorithm
:	O
</s>
<s>
Output	O
:	O
set	O
splines	B-Algorithm
which	O
is	O
composed	O
of	O
n	O
5-tuples	O
.	O
</s>
<s>
Create	O
new	O
set	O
Splines	B-Algorithm
and	O
call	O
it	O
output_set	O
.	O
</s>
<s>
Populate	O
it	O
with	O
n	O
splines	B-Algorithm
S	O
.	O
</s>
