<s>
In	O
the	O
mathematical	O
subfield	O
of	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
a	O
B-spline	B-Algorithm
or	O
basis	O
spline	B-Algorithm
is	O
a	O
spline	B-Algorithm
function	I-Algorithm
that	O
has	O
minimal	O
support	O
with	O
respect	O
to	O
a	O
given	O
degree	O
,	O
smoothness	O
,	O
and	O
domain	B-Algorithm
partition	O
.	O
</s>
<s>
Any	O
spline	B-Algorithm
function	I-Algorithm
of	O
given	O
degree	O
can	O
be	O
expressed	O
as	O
a	O
linear	O
combination	O
of	O
B-splines	B-Algorithm
of	O
that	O
degree	O
.	O
</s>
<s>
Cardinal	O
B-splines	B-Algorithm
have	O
knots	B-Algorithm
that	O
are	O
equidistant	O
from	O
each	O
other	O
.	O
</s>
<s>
B-splines	B-Algorithm
can	O
be	O
used	O
for	O
curve-fitting	B-Algorithm
and	O
numerical	B-Algorithm
differentiation	I-Algorithm
of	O
experimental	O
data	O
.	O
</s>
<s>
In	O
computer-aided	B-Application
design	I-Application
and	O
computer	O
graphics	O
,	O
spline	B-Algorithm
functions	I-Algorithm
are	O
constructed	O
as	O
linear	O
combinations	O
of	O
B-splines	B-Algorithm
with	O
a	O
set	O
of	O
control	O
points	O
.	O
</s>
<s>
The	O
term	O
"	O
B-spline	B-Algorithm
"	O
was	O
coined	O
by	O
Isaac	O
Jacob	O
Schoenberg	O
and	O
is	O
short	O
for	O
basis	O
spline	B-Algorithm
.	O
</s>
<s>
A	O
spline	B-Algorithm
function	I-Algorithm
of	O
order	O
is	O
a	O
piecewise	B-Algorithm
polynomial	I-Algorithm
function	O
of	O
degree	O
in	O
a	O
variable	O
.	O
</s>
<s>
The	O
places	O
where	O
the	O
pieces	O
meet	O
are	O
known	O
as	O
knots	B-Algorithm
.	O
</s>
<s>
The	O
key	O
property	O
of	O
spline	B-Algorithm
functions	I-Algorithm
is	O
that	O
they	O
and	O
their	O
derivatives	O
may	O
be	O
continuous	O
,	O
depending	O
on	O
the	O
multiplicities	O
of	O
the	O
knots	B-Algorithm
.	O
</s>
<s>
B-splines	B-Algorithm
of	O
order	O
are	O
basis	O
functions	O
for	O
spline	B-Algorithm
functions	I-Algorithm
of	O
the	O
same	O
order	O
defined	O
over	O
the	O
same	O
knots	B-Algorithm
,	O
meaning	O
that	O
all	O
possible	O
spline	B-Algorithm
functions	I-Algorithm
can	O
be	O
built	O
from	O
a	O
linear	O
combination	O
of	O
B-splines	B-Algorithm
,	O
and	O
there	O
is	O
only	O
one	O
unique	O
combination	O
for	O
each	O
spline	B-Algorithm
function	I-Algorithm
.	O
</s>
<s>
A	O
spline	B-Algorithm
of	O
order	O
is	O
a	O
piecewise	B-Algorithm
polynomial	I-Algorithm
function	O
of	O
degree	O
in	O
a	O
variable	O
.	O
</s>
<s>
The	O
values	O
of	O
where	O
the	O
pieces	O
of	O
polynomial	O
meet	O
are	O
known	O
as	O
knots	B-Algorithm
,	O
denoted	O
and	O
sorted	O
into	O
nondecreasing	O
order	O
.	O
</s>
<s>
When	O
the	O
knots	B-Algorithm
are	O
distinct	O
,	O
the	O
first	O
derivatives	O
of	O
the	O
polynomial	O
pieces	O
are	O
continuous	O
across	O
each	O
knot	B-Algorithm
.	O
</s>
<s>
When	O
knots	B-Algorithm
are	O
coincident	O
,	O
then	O
only	O
the	O
first	O
derivatives	O
of	O
the	O
spline	B-Algorithm
are	O
continuous	O
across	O
that	O
knot	B-Algorithm
.	O
</s>
<s>
If	O
we	O
add	O
the	O
additional	O
constraint	O
that	O
for	O
all	O
between	O
the	O
first	O
and	O
last	O
knot	B-Algorithm
,	O
then	O
the	O
scaling	O
factor	O
of	O
becomes	O
fixed	O
.	O
</s>
<s>
The	O
resulting	O
spline	B-Algorithm
functions	I-Algorithm
are	O
called	O
B-splines	B-Algorithm
.	O
</s>
<s>
Alternatively	O
,	O
B-splines	B-Algorithm
can	O
be	O
defined	O
by	O
construction	O
by	O
means	O
of	O
the	O
Cox	O
–	O
de	O
Boor	O
recursion	O
formula	O
.	O
</s>
<s>
A	O
B-spline	B-Algorithm
function	O
is	O
a	O
combination	O
of	O
flexible	O
bands	O
that	O
is	O
controlled	O
by	O
a	O
number	O
of	O
points	O
that	O
are	O
called	O
control	O
points	O
,	O
creating	O
smooth	O
curves	O
.	O
</s>
<s>
B-spline	B-Algorithm
function	O
and	O
Bézier	O
functions	O
are	O
applied	O
extensively	O
in	O
shape	O
optimization	O
methods	O
.	O
</s>
<s>
A	O
B-spline	B-Algorithm
of	O
order	O
is	O
a	O
piecewise	B-Algorithm
polynomial	I-Algorithm
function	O
of	O
degree	O
in	O
a	O
variable	O
.	O
</s>
<s>
It	O
is	O
defined	O
over	O
locations	O
,	O
called	O
knots	B-Algorithm
or	O
breakpoints	O
,	O
which	O
must	O
be	O
in	O
non-descending	O
order	O
.	O
</s>
<s>
The	O
B-spline	B-Algorithm
contributes	O
only	O
in	O
the	O
range	O
between	O
the	O
first	O
and	O
last	O
of	O
these	O
knots	B-Algorithm
and	O
is	O
zero	O
elsewhere	O
.	O
</s>
<s>
If	O
each	O
knot	B-Algorithm
is	O
separated	O
by	O
the	O
same	O
distance	O
(	O
where	O
)	O
from	O
its	O
predecessor	O
,	O
the	O
knot	B-Algorithm
vector	I-Algorithm
and	O
the	O
corresponding	O
B-splines	B-Algorithm
are	O
called	O
"	O
uniform	O
"	O
(	O
see	O
cardinal	O
B-spline	B-Algorithm
below	O
)	O
.	O
</s>
<s>
For	O
each	O
finite	O
knot	B-Algorithm
interval	O
where	O
it	O
is	O
non-zero	O
,	O
a	O
B-spline	B-Algorithm
is	O
a	O
polynomial	O
of	O
degree	O
.	O
</s>
<s>
A	O
B-spline	B-Algorithm
is	O
a	O
continuous	O
function	O
at	O
the	O
knots	B-Algorithm
.	O
</s>
<s>
When	O
all	O
knots	B-Algorithm
belonging	O
to	O
the	O
B-spline	B-Algorithm
are	O
distinct	O
,	O
its	O
derivatives	O
are	O
also	O
continuous	O
up	O
to	O
the	O
derivative	O
of	O
degree	O
.	O
</s>
<s>
If	O
the	O
knots	B-Algorithm
are	O
coincident	O
at	O
a	O
given	O
value	O
of	O
,	O
the	O
continuity	O
of	O
derivative	O
order	O
is	O
reduced	O
by	O
1	O
for	O
each	O
additional	O
coincident	O
knot	B-Algorithm
.	O
</s>
<s>
B-splines	B-Algorithm
may	O
share	O
a	O
subset	O
of	O
their	O
knots	B-Algorithm
,	O
but	O
two	O
B-splines	B-Algorithm
defined	O
over	O
exactly	O
the	O
same	O
knots	B-Algorithm
are	O
identical	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
a	O
B-spline	B-Algorithm
is	O
uniquely	O
defined	O
by	O
its	O
knots	B-Algorithm
.	O
</s>
<s>
One	O
distinguishes	O
internal	O
knots	B-Algorithm
and	O
end	O
points	O
.	O
</s>
<s>
Internal	O
knots	B-Algorithm
cover	O
the	O
-domain	O
one	O
is	O
interested	O
in	O
.	O
</s>
<s>
Since	O
a	O
single	O
B-spline	B-Algorithm
already	O
extends	O
over	O
knots	B-Algorithm
,	O
it	O
follows	O
that	O
the	O
internal	O
knots	B-Algorithm
need	O
to	O
be	O
extended	O
with	O
endpoints	O
on	O
each	O
side	O
,	O
to	O
give	O
full	O
support	O
to	O
the	O
first	O
and	O
last	O
B-spline	B-Algorithm
,	O
which	O
affect	O
the	O
internal	O
knot	B-Algorithm
intervals	O
.	O
</s>
<s>
The	O
values	O
of	O
the	O
endpoints	O
do	O
not	O
matter	O
,	O
usually	O
the	O
first	O
or	O
last	O
internal	O
knot	B-Algorithm
is	O
just	O
repeated	O
.	O
</s>
<s>
The	O
usefulness	O
of	O
B-splines	B-Algorithm
lies	O
in	O
the	O
fact	O
that	O
any	O
spline	B-Algorithm
function	I-Algorithm
of	O
order	O
on	O
a	O
given	O
set	O
of	O
knots	B-Algorithm
can	O
be	O
expressed	O
as	O
a	O
linear	O
combination	O
of	O
B-splines	B-Algorithm
:	O
</s>
<s>
B-splines	B-Algorithm
play	O
the	O
role	O
of	O
basis	O
functions	O
for	O
the	O
spline	B-Algorithm
function	I-Algorithm
space	O
,	O
hence	O
the	O
name	O
.	O
</s>
<s>
This	O
property	O
follows	O
from	O
the	O
fact	O
that	O
all	O
pieces	O
have	O
the	O
same	O
continuity	O
properties	O
,	O
within	O
their	O
individual	O
range	O
of	O
support	O
,	O
at	O
the	O
knots	B-Algorithm
.	O
</s>
<s>
That	O
is	O
,	O
is	O
piecewise	B-Algorithm
constant	O
one	O
or	O
zero	O
indicating	O
which	O
knot	B-Algorithm
span	O
x	O
is	O
in	O
(	O
zero	O
if	O
knot	B-Algorithm
span	O
j	O
is	O
repeated	O
)	O
.	O
</s>
<s>
However	O
,	O
because	O
B-spline	B-Algorithm
basis	O
functions	O
have	O
local	O
support	O
,	O
B-splines	B-Algorithm
are	O
typically	O
computed	O
by	O
algorithms	O
that	O
do	O
not	O
need	O
to	O
evaluate	O
basis	O
functions	O
where	O
they	O
are	O
zero	O
,	O
such	O
as	O
de	B-Algorithm
Boor	I-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
This	O
relation	O
leads	O
directly	O
to	O
the	O
FORTRAN-coded	O
algorithm	O
BSPLV	O
,	O
which	O
generates	O
values	O
of	O
the	O
B-splines	B-Algorithm
of	O
order	O
n	O
at	O
x	O
.	O
</s>
<s>
The	O
following	O
scheme	O
illustrates	O
how	O
each	O
piece	O
of	O
order	O
n	O
is	O
a	O
linear	O
combination	O
of	O
the	O
pieces	O
of	O
B-splines	B-Algorithm
of	O
order	O
n−1	O
to	O
its	O
left	O
.	O
</s>
<s>
The	O
second	O
derivative	O
of	O
a	O
B-spline	B-Algorithm
of	O
degree	O
2	O
is	O
discontinuous	O
at	O
the	O
knots	B-Algorithm
:	O
</s>
<s>
Faster	O
variants	O
of	O
the	O
de	B-Algorithm
Boor	I-Algorithm
algorithm	I-Algorithm
have	O
been	O
proposed	O
,	O
but	O
they	O
suffer	O
from	O
comparatively	O
lower	O
stability	O
.	O
</s>
<s>
A	O
cardinal	O
B-spline	B-Algorithm
has	O
a	O
constant	O
separation	O
h	O
between	O
knots	B-Algorithm
.	O
</s>
<s>
The	O
cardinal	O
B-splines	B-Algorithm
for	O
a	O
given	O
order	O
n	O
are	O
just	O
shifted	O
copies	O
of	O
each	O
other	O
.	O
</s>
<s>
The	O
"	O
placeholder	O
"	O
notation	O
is	O
used	O
to	O
indicate	O
that	O
the	O
n-th	O
divided	B-Algorithm
difference	I-Algorithm
of	O
the	O
function	O
of	O
the	O
two	O
variables	O
t	O
and	O
x	O
is	O
to	O
be	O
taken	O
by	O
fixing	O
x	O
and	O
considering	O
as	O
a	O
function	O
of	O
t	O
alone	O
.	O
</s>
<s>
A	O
cardinal	O
B-spline	B-Algorithm
has	O
uniformly	O
spaced	O
knots	B-Algorithm
,	O
therefore	O
interpolation	O
between	O
the	O
knots	B-Algorithm
equals	O
convolution	O
with	O
a	O
smoothing	O
kernel	O
.	O
</s>
<s>
Convolution	O
of	O
the	O
signal	O
with	O
a	O
rectangle	O
function	O
gives	O
first	O
order	O
interpolated	O
B-spline	B-Algorithm
values	O
.	O
</s>
<s>
Second-order	O
B-spline	B-Algorithm
interpolation	O
is	O
convolution	O
with	O
a	O
rectangle	O
function	O
twice	O
;	O
by	O
iterative	O
filtering	O
with	O
a	O
rectangle	O
function	O
,	O
higher-order	O
interpolation	O
is	O
obtained	O
.	O
</s>
<s>
Fast	O
B-spline	B-Algorithm
interpolation	O
on	O
a	O
uniform	O
sample	O
domain	B-Algorithm
can	O
be	O
done	O
by	O
iterative	O
mean-filtering	O
.	O
</s>
<s>
Alternatively	O
,	O
a	O
rectangle	O
function	O
equals	O
sinc	B-Algorithm
in	O
Fourier	O
domain	B-Algorithm
.	O
</s>
<s>
Therefore	O
,	O
cubic	O
spline	B-Algorithm
interpolation	O
equals	O
multiplying	O
the	O
signal	O
in	O
Fourier	O
domain	B-Algorithm
with	O
sinc4	O
.	O
</s>
<s>
See	O
Irwin	O
–	O
Hall	O
distribution	O
#Special	O
cases	O
for	O
algebraic	O
expressions	O
for	O
the	O
cardinal	O
B-splines	B-Algorithm
of	O
degree	O
1	O
–	O
4	O
.	O
</s>
<s>
The	O
term	O
P-spline	O
stands	O
for	O
"	O
penalized	O
B-spline	B-Algorithm
"	O
.	O
</s>
<s>
It	O
refers	O
to	O
using	O
the	O
B-spline	B-Algorithm
representation	O
where	O
the	O
coefficients	O
are	O
determined	O
partly	O
by	O
the	O
data	O
to	O
be	O
fitted	B-Algorithm
,	O
and	O
partly	O
by	O
an	O
additional	O
penalty	B-Algorithm
function	I-Algorithm
that	O
aims	O
to	O
impose	O
smoothness	O
to	O
avoid	O
overfitting	B-Error_Name
.	O
</s>
<s>
Two	O
-	O
and	O
multidimensional	O
P-spline	O
approximations	O
of	O
data	O
can	O
use	O
the	O
face-splitting	O
product	O
of	O
matrices	O
to	O
the	O
minimization	O
of	O
calculation	O
operations	O
.	O
</s>
<s>
The	O
derivative	O
of	O
a	O
B-spline	B-Algorithm
of	O
degree	O
k	O
is	O
simply	O
a	O
function	O
of	O
B-splines	B-Algorithm
of	O
degree	O
k−1	O
:	O
</s>
<s>
which	O
shows	O
that	O
there	O
is	O
a	O
simple	O
relationship	O
between	O
the	O
derivative	O
of	O
a	O
spline	B-Algorithm
function	I-Algorithm
and	O
the	O
B-splines	B-Algorithm
of	O
degree	O
one	O
less	O
.	O
</s>
<s>
Univariate	O
B-splines	B-Algorithm
,	O
i.e.	O
</s>
<s>
B-splines	B-Algorithm
where	O
the	O
knot	B-Algorithm
positions	O
lie	O
in	O
a	O
single	O
dimension	O
,	O
can	O
be	O
used	O
to	O
represent	O
1-d	O
probability	O
density	O
functions	O
.	O
</s>
<s>
An	O
example	O
is	O
a	O
weighted	O
sum	O
of	O
B-spline	B-Algorithm
basis	O
functions	O
of	O
order	O
,	O
which	O
each	O
are	O
area-normalized	O
to	O
unity	O
(	O
i.e.	O
</s>
<s>
Here	O
,	O
represents	O
a	O
vector	O
with	O
the	O
knot	B-Algorithm
positions	O
and	O
a	O
vector	O
with	O
the	O
respective	O
knot	B-Algorithm
multiplicities	O
.	O
</s>
<s>
One	O
can	O
therefore	O
calculate	O
any	O
moment	O
of	O
a	O
probability	O
density	O
function	O
represented	O
by	O
a	O
sum	O
of	O
B-spline	B-Algorithm
basis	O
functions	O
exactly	O
,	O
without	O
resorting	O
to	O
numerical	O
techniques	O
.	O
</s>
<s>
A	O
Bézier	O
curve	O
is	O
also	O
a	O
polynomial	O
curve	O
definable	O
using	O
a	O
recursion	O
from	O
lower-degree	O
curves	O
of	O
the	O
same	O
class	O
and	O
encoded	O
in	O
terms	O
of	O
control	O
points	O
,	O
but	O
a	O
key	O
difference	O
is	O
that	O
all	O
terms	O
in	O
the	O
recursion	O
for	O
a	O
Bézier	O
curve	O
segment	O
have	O
the	O
same	O
domain	B-Algorithm
of	O
definition	O
(	O
usually	O
)	O
,	O
whereas	O
the	O
supports	O
of	O
the	O
two	O
terms	O
in	O
the	O
B-spline	B-Algorithm
recursion	O
are	O
different	O
(	O
the	O
outermost	O
subintervals	O
are	O
not	O
common	O
)	O
.	O
</s>
<s>
This	O
means	O
that	O
a	O
Bézier	O
curve	O
of	O
degree	O
given	O
by	O
control	O
points	O
consists	O
of	O
about	O
mostly	O
independent	O
segments	O
,	O
whereas	O
the	O
B-spline	B-Algorithm
with	O
the	O
same	O
parameters	O
smoothly	O
transitions	O
from	O
subinterval	O
to	O
subinterval	O
.	O
</s>
<s>
To	O
get	O
something	O
comparable	O
from	O
a	O
Bézier	O
curve	O
,	O
one	O
would	O
need	O
to	O
impose	O
a	O
smoothness	O
condition	O
on	O
transitions	O
between	O
segments	O
,	O
resulting	O
in	O
some	O
manner	O
of	O
Bézier	O
spline	B-Algorithm
(	O
for	O
which	O
many	O
control	O
points	O
would	O
be	O
determined	O
by	O
the	O
smoothness	O
requirement	O
)	O
.	O
</s>
<s>
A	O
piecewise/composite	B-Algorithm
Bézier	I-Algorithm
curve	I-Algorithm
is	O
a	O
series	O
of	O
Bézier	O
curves	O
joined	O
with	O
at	O
least	O
C0	O
continuity	O
(	O
the	O
last	O
point	O
of	O
one	O
curve	O
coincides	O
with	O
the	O
starting	O
point	O
of	O
the	O
next	O
curve	O
)	O
.	O
</s>
<s>
Usually	O
in	O
curve	B-Algorithm
fitting	I-Algorithm
,	O
a	O
set	O
of	O
data	O
points	O
is	O
fitted	B-Algorithm
with	O
a	O
curve	O
defined	O
by	O
some	O
mathematical	O
function	O
.	O
</s>
<s>
For	O
example	O
,	O
common	O
types	O
of	O
curve	B-Algorithm
fitting	I-Algorithm
use	O
a	O
polynomial	O
or	O
a	O
set	O
of	O
exponential	O
functions	O
.	O
</s>
<s>
When	O
there	O
is	O
no	O
theoretical	O
basis	O
for	O
choosing	O
a	O
fitting	O
function	O
,	O
the	O
curve	O
may	O
be	O
fitted	B-Algorithm
with	O
a	O
spline	B-Algorithm
function	I-Algorithm
composed	O
of	O
a	O
sum	O
of	O
B-splines	B-Algorithm
,	O
using	O
the	O
method	B-Algorithm
of	I-Algorithm
least	I-Algorithm
squares	I-Algorithm
.	O
</s>
<s>
Thus	O
,	O
the	O
objective	O
function	O
for	O
least-squares	B-Algorithm
minimization	O
is	O
,	O
for	O
a	O
spline	B-Algorithm
function	I-Algorithm
of	O
degree	O
k	O
,	O
</s>
<s>
The	O
knot	B-Algorithm
values	I-Algorithm
may	O
be	O
fixed	O
or	O
treated	O
as	O
parameters	O
.	O
</s>
<s>
The	O
main	O
difficulty	O
in	O
applying	O
this	O
process	O
is	O
in	O
determining	O
the	O
number	O
of	O
knots	B-Algorithm
to	O
use	O
and	O
where	O
they	O
should	O
be	O
placed	O
.	O
</s>
<s>
For	O
instance	O
,	O
the	O
spacing	O
between	O
knots	B-Algorithm
is	O
decreased	O
in	O
proportion	O
to	O
the	O
curvature	O
(	O
2nd	O
derivative	O
)	O
of	O
the	O
data	O
.	O
</s>
<s>
For	O
instance	O
,	O
the	O
use	O
of	O
B-splines	B-Algorithm
for	O
fitting	O
single	O
Lorentzian	O
and	O
Gaussian	O
curves	O
has	O
been	O
investigated	O
.	O
</s>
<s>
Optimal	O
spline	B-Algorithm
functions	I-Algorithm
of	O
degrees	O
3	O
–	O
7	O
inclusive	O
,	O
based	O
on	O
symmetric	O
arrangements	O
of	O
5	O
,	O
6	O
,	O
and	O
7	O
knots	B-Algorithm
,	O
have	O
been	O
computed	O
and	O
the	O
method	O
was	O
applied	O
for	O
smoothing	O
and	O
differentiation	O
of	O
spectroscopic	O
curves	O
.	O
</s>
<s>
In	O
a	O
comparable	O
study	O
,	O
the	O
two-dimensional	O
version	O
of	O
the	O
Savitzky	O
–	O
Golay	O
filtering	O
and	O
the	O
spline	B-Algorithm
method	O
produced	O
better	O
results	O
than	O
moving	O
average	O
or	O
Chebyshev	O
filtering	O
.	O
</s>
<s>
In	O
computer-aided	B-Application
design	I-Application
and	O
computer	O
graphics	O
applications	O
,	O
a	O
spline	B-Algorithm
curve	I-Algorithm
is	O
sometimes	O
represented	O
as	O
,	O
a	O
parametric	O
curve	O
of	O
some	O
real	O
parameter	O
.	O
</s>
<s>
The	O
coordinate	O
functions	O
,	O
and	O
are	O
each	O
spline	B-Algorithm
functions	I-Algorithm
,	O
with	O
a	O
common	O
set	O
of	O
knot	B-Algorithm
values	I-Algorithm
.	O
</s>
<s>
Working	O
in	O
reverse	O
,	O
a	O
sequence	O
of	O
control	O
points	O
,	O
knot	B-Algorithm
values	I-Algorithm
,	O
and	O
order	O
of	O
the	O
B-spline	B-Algorithm
define	O
a	O
parametric	O
curve	O
.	O
</s>
<s>
Because	O
the	O
B-splines	B-Algorithm
are	O
non-zero	O
for	O
just	O
a	O
finite	O
number	O
of	O
knot	B-Algorithm
intervals	O
,	O
if	O
a	O
single	O
control	O
point	O
is	O
moved	O
,	O
the	O
corresponding	O
change	O
to	O
the	O
parametric	O
curve	O
is	O
just	O
over	O
the	O
parameter	O
range	O
of	O
a	O
small	O
number	O
knot	B-Algorithm
intervals	O
.	O
</s>
<s>
In	O
computer	B-Application
aided	I-Application
design	I-Application
,	O
computer	O
aided	O
manufacturing	O
,	O
and	O
computer	O
graphics	O
,	O
a	O
powerful	O
extension	O
of	O
B-splines	B-Algorithm
is	O
non-uniform	O
rational	O
B-splines	B-Algorithm
(	O
NURBS	O
)	O
.	O
</s>
<s>
NURBS	O
are	O
essentially	O
B-splines	B-Algorithm
in	O
homogeneous	O
coordinates	O
.	O
</s>
<s>
Like	O
B-splines	B-Algorithm
,	O
they	O
are	O
defined	O
by	O
their	O
order	O
,	O
and	O
a	O
knot	B-Algorithm
vector	I-Algorithm
,	O
and	O
a	O
set	O
of	O
control	O
points	O
,	O
but	O
unlike	O
simple	O
B-splines	B-Algorithm
,	O
the	O
control	O
points	O
each	O
have	O
a	O
weight	O
.	O
</s>
<s>
When	O
the	O
weight	O
is	O
equal	O
to	O
1	O
,	O
a	O
NURBS	O
is	O
simply	O
a	O
B-spline	B-Algorithm
and	O
as	O
such	O
NURBS	O
generalizes	O
both	O
B-splines	B-Algorithm
and	O
Bézier	O
curves	O
and	O
surfaces	O
,	O
the	O
primary	O
difference	O
being	O
the	O
weighting	O
of	O
the	O
control	O
points	O
which	O
makes	O
NURBS	O
curves	O
"	O
rational	O
"	O
.	O
</s>
<s>
Like	O
B-splines	B-Algorithm
,	O
NURBS	O
control	O
points	O
determine	O
the	O
shape	O
of	O
the	O
curve	O
.	O
</s>
<s>
For	O
a	O
curve	O
of	O
degree	O
d	O
,	O
the	O
influence	O
of	O
any	O
control	O
point	O
is	O
only	O
nonzero	O
in	O
d+1	O
intervals	O
(	O
knot	B-Algorithm
spans	O
)	O
of	O
the	O
parameter	O
space	O
.	O
</s>
<s>
The	O
knot	B-Algorithm
vector	I-Algorithm
is	O
a	O
sequence	O
of	O
parameter	O
values	O
that	O
determines	O
where	O
and	O
how	O
the	O
control	O
points	O
affect	O
the	O
NURBS	O
curve	O
.	O
</s>
<s>
The	O
number	O
of	O
knots	B-Algorithm
is	O
always	O
equal	O
to	O
the	O
number	O
of	O
control	O
points	O
plus	O
curve	O
degree	O
plus	O
one	O
.	O
</s>
<s>
Each	O
time	O
the	O
parameter	O
value	O
enters	O
a	O
new	O
knot	B-Algorithm
span	O
,	O
a	O
new	O
control	O
point	O
becomes	O
active	O
,	O
while	O
an	O
old	O
control	O
point	O
is	O
discarded	O
.	O
</s>
<s>
u	O
is	O
the	O
independent	O
variable	O
(	O
instead	O
of	O
x	O
)	O
,	O
k	O
is	O
the	O
number	O
of	O
control	O
points	O
,	O
N	O
is	O
a	O
B-spline	B-Algorithm
(	O
used	O
instead	O
of	O
B	O
)	O
,	O
n	O
is	O
the	O
polynomial	O
degree	O
,	O
P	O
is	O
a	O
control	O
point	O
and	O
w	O
is	O
a	O
weight	O
.	O
</s>
