<s>
In	O
the	O
mathematical	O
field	O
of	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
spline	B-Algorithm
interpolation	I-Algorithm
is	O
a	O
form	O
of	O
interpolation	B-Algorithm
where	O
the	O
interpolant	O
is	O
a	O
special	O
type	O
of	O
piecewise	B-Algorithm
polynomial	I-Algorithm
called	O
a	O
spline	B-Algorithm
.	O
</s>
<s>
That	O
is	O
,	O
instead	O
of	O
fitting	O
a	O
single	O
,	O
high-degree	O
polynomial	O
to	O
all	O
of	O
the	O
values	O
at	O
once	O
,	O
spline	B-Algorithm
interpolation	I-Algorithm
fits	O
low-degree	O
polynomials	O
to	O
small	O
subsets	O
of	O
the	O
values	O
,	O
for	O
example	O
,	O
fitting	O
nine	O
cubic	O
polynomials	O
between	O
each	O
of	O
the	O
pairs	O
of	O
ten	O
points	O
,	O
instead	O
of	O
fitting	O
a	O
single	O
degree-ten	O
polynomial	O
to	O
all	O
of	O
them	O
.	O
</s>
<s>
Spline	B-Algorithm
interpolation	I-Algorithm
is	O
often	O
preferred	O
over	O
polynomial	O
interpolation	B-Algorithm
because	O
the	O
interpolation	B-Algorithm
error	O
can	O
be	O
made	O
small	O
even	O
when	O
using	O
low-degree	O
polynomials	O
for	O
the	O
spline	B-Algorithm
.	O
</s>
<s>
Spline	B-Algorithm
interpolation	I-Algorithm
also	O
avoids	O
the	O
problem	O
of	O
Runge	O
's	O
phenomenon	O
,	O
in	O
which	O
oscillation	O
can	O
occur	O
between	O
points	O
when	O
interpolating	O
using	O
high-degree	O
polynomials	O
.	O
</s>
<s>
Originally	O
,	O
spline	B-Algorithm
was	O
a	O
term	O
for	O
elastic	O
rulers	O
that	O
were	O
bent	O
to	O
pass	O
through	O
a	O
number	O
of	O
predefined	O
points	O
,	O
or	O
knots	B-Algorithm
.	O
</s>
<s>
These	O
were	O
used	O
to	O
make	O
technical	B-Application
drawings	I-Application
for	O
shipbuilding	O
and	O
construction	O
by	O
hand	O
,	O
as	O
illustrated	O
in	O
the	O
figure	O
.	O
</s>
<s>
Assume	O
we	O
have	O
a	O
sequence	O
of	O
knots	B-Algorithm
,	O
through	O
.	O
</s>
<s>
There	O
will	O
be	O
a	O
cubic	O
polynomial	O
between	O
each	O
successive	O
pair	O
of	O
knots	B-Algorithm
and	O
connecting	O
to	O
both	O
of	O
them	O
,	O
where	O
.	O
</s>
<s>
To	O
make	O
the	O
spline	B-Algorithm
take	O
a	O
shape	O
that	O
minimizes	O
the	O
bending	O
(	O
under	O
the	O
constraint	O
of	O
passing	O
through	O
all	O
knots	B-Algorithm
)	O
,	O
we	O
will	O
define	O
both	O
and	O
to	O
be	O
continuous	O
everywhere	O
,	O
including	O
at	O
the	O
knots	B-Algorithm
.	O
</s>
<s>
The	O
classical	O
approach	O
is	O
to	O
use	O
polynomials	O
of	O
exactly	O
degree	O
3	O
—	O
cubic	B-Algorithm
splines	I-Algorithm
.	O
</s>
<s>
In	O
addition	O
to	O
the	O
three	O
conditions	O
above	O
,	O
a	O
'	O
natural	B-Algorithm
cubic	I-Algorithm
spline	I-Algorithm
 '	O
has	O
the	O
condition	O
that	O
.	O
</s>
<s>
In	O
addition	O
to	O
the	O
three	O
main	O
conditions	O
above	O
,	O
a	O
'	O
clamped	O
cubic	B-Algorithm
spline	I-Algorithm
 '	O
has	O
the	O
conditions	O
that	O
and	O
where	O
is	O
the	O
derivative	O
of	O
the	O
interpolated	O
function	O
.	O
</s>
<s>
In	O
addition	O
to	O
the	O
three	O
main	O
conditions	O
above	O
,	O
a	O
'	O
not-a-knot	O
spline	B-Algorithm
 '	O
has	O
the	O
conditions	O
that	O
and	O
.	O
</s>
<s>
For	O
the	O
elastic	O
rulers	O
being	O
the	O
model	O
for	O
the	O
spline	B-Algorithm
interpolation	I-Algorithm
,	O
one	O
has	O
that	O
to	O
the	O
left	O
of	O
the	O
left-most	O
"	O
knot	B-Algorithm
"	O
and	O
to	O
the	O
right	O
of	O
the	O
right-most	O
"	O
knot	B-Algorithm
"	O
the	O
ruler	O
can	O
move	O
freely	O
and	O
will	O
therefore	O
take	O
the	O
form	O
of	O
a	O
straight	O
line	O
with	O
.	O
</s>
<s>
There	O
exist	O
other	O
end	O
conditions	O
,	O
"	O
clamped	O
spline	B-Algorithm
"	O
,	O
which	O
specifies	O
the	O
slope	O
at	O
the	O
ends	O
of	O
the	O
spline	B-Algorithm
,	O
and	O
the	O
popular	O
"	O
not-a-knot	O
spline	B-Algorithm
"	O
,	O
which	O
requires	O
that	O
the	O
third	O
derivative	O
is	O
also	O
continuous	O
at	O
the	O
and	O
points	O
.	O
</s>
<s>
For	O
the	O
"	O
not-a-knot	O
"	O
spline	B-Algorithm
,	O
the	O
additional	O
equations	O
will	O
read	O
:	O
</s>
<s>
In	O
the	O
figure	O
,	O
the	O
spline	B-Algorithm
function	I-Algorithm
consisting	O
of	O
the	O
two	O
cubic	O
polynomials	O
and	O
given	O
by	O
(	O
)	O
is	O
displayed	O
.	O
</s>
