<s>
Non-uniform	O
rational	O
basis	O
spline	B-Algorithm
(	O
NURBS	B-Algorithm
)	O
is	O
a	O
mathematical	O
model	O
using	O
basis	B-Algorithm
splines	I-Algorithm
(	O
B-splines	B-Algorithm
)	O
that	O
is	O
commonly	O
used	O
in	O
computer	O
graphics	O
for	O
representing	O
curves	O
and	O
surfaces	O
.	O
</s>
<s>
NURBS	B-Algorithm
curves	I-Algorithm
are	O
commonly	O
used	O
in	O
computer-aided	B-Application
design	I-Application
(	O
CAD	B-Application
)	O
,	O
manufacturing	O
(	O
CAM	O
)	O
,	O
and	O
engineering	O
(	O
CAE	O
)	O
.	O
</s>
<s>
They	O
are	O
part	O
of	O
numerous	O
industry-wide	O
standards	O
,	O
such	O
as	O
IGES	O
,	O
STEP	O
,	O
ACIS	O
,	O
and	O
PHIGS	B-Library
.	O
</s>
<s>
Tools	O
for	O
creating	O
and	O
editing	O
NURBS	B-Algorithm
surfaces	I-Algorithm
are	O
found	O
in	O
various	O
3D	O
graphics	O
and	O
animation	O
software	O
packages	O
.	O
</s>
<s>
NURBS	B-Algorithm
surfaces	I-Algorithm
are	O
functions	O
of	O
two	O
parameters	O
mapping	O
to	O
a	O
surface	O
in	O
three-dimensional	O
space	O
.	O
</s>
<s>
The	O
shape	O
of	O
the	O
surface	O
is	O
determined	O
by	O
control	B-Algorithm
points	I-Algorithm
.	O
</s>
<s>
In	O
a	O
compact	O
form	O
,	O
NURBS	B-Algorithm
surfaces	I-Algorithm
can	O
represent	O
simple	O
geometrical	O
shapes	O
.	O
</s>
<s>
For	O
complex	O
organic	O
shapes	O
,	O
T-splines	B-Algorithm
and	O
subdivision	O
surfaces	O
are	O
more	O
suitable	O
because	O
they	O
halve	O
the	O
number	O
of	O
control	B-Algorithm
points	I-Algorithm
in	O
comparison	O
with	O
the	O
NURBS	B-Algorithm
surfaces	I-Algorithm
.	O
</s>
<s>
In	O
general	O
,	O
editing	O
NURBS	B-Algorithm
curves	I-Algorithm
and	O
surfaces	O
is	O
intuitive	O
and	O
predictable	O
.	O
</s>
<s>
Control	B-Algorithm
points	I-Algorithm
are	O
always	O
either	O
connected	O
directly	O
to	O
the	O
curve	O
or	O
surface	O
,	O
or	O
else	O
act	O
as	O
if	O
they	O
were	O
connected	O
by	O
a	O
rubber	O
band	O
.	O
</s>
<s>
Depending	O
on	O
the	O
type	O
of	O
user	O
interface	O
,	O
the	O
editing	O
of	O
NURBS	B-Algorithm
curves	I-Algorithm
and	O
surfaces	O
can	O
be	O
via	O
their	O
control	B-Algorithm
points	I-Algorithm
(	O
similar	O
to	O
Bézier	O
curves	O
)	O
or	O
via	O
higher	O
level	O
tools	O
such	O
as	O
spline	B-Algorithm
modeling	O
and	O
hierarchical	O
editing	O
.	O
</s>
<s>
Before	O
computers	O
,	O
designs	O
were	O
drawn	O
by	O
hand	O
on	O
paper	O
with	O
various	O
drafting	B-Application
tools	I-Application
.	O
</s>
<s>
Such	O
large	O
drawings	O
were	O
done	O
with	O
the	O
help	O
of	O
flexible	O
strips	O
of	O
wood	O
,	O
called	O
splines	B-Algorithm
.	O
</s>
<s>
The	O
splines	B-Algorithm
were	O
held	O
in	O
place	O
at	O
a	O
number	O
of	O
predetermined	O
points	O
,	O
called	O
"	O
ducks	O
"	O
(	O
which	O
were	O
made	O
of	O
lead	O
&	O
3-ish	O
inches	O
long	O
.	O
</s>
<s>
The	O
"	O
beak	O
"	O
of	O
the	O
"	O
duck	O
"	O
pushed	O
against	O
the	O
spline	B-Algorithm
:	O
the	O
old	O
yacht-design	O
books	O
assumed	O
these	O
methods	O
)	O
;	O
between	O
the	O
ducks	O
,	O
the	O
elasticity	O
of	O
the	O
spline	B-Algorithm
material	O
caused	O
the	O
strip	O
to	O
take	O
the	O
shape	O
that	O
minimized	O
the	O
energy	O
of	O
bending	O
,	O
thus	O
creating	O
the	O
smoothest	O
possible	O
shape	O
that	O
fit	O
the	O
constraints	O
.	O
</s>
<s>
In	O
1946	O
,	O
mathematicians	O
started	O
studying	O
the	O
spline	B-Algorithm
shape	O
,	O
and	O
derived	O
the	O
piecewise	B-Algorithm
polynomial	I-Algorithm
formula	O
known	O
as	O
the	O
spline	B-Algorithm
curve	I-Algorithm
or	O
spline	B-Algorithm
function	I-Algorithm
.	O
</s>
<s>
I	O
.	O
J	O
.	O
Schoenberg	O
gave	O
the	O
spline	B-Algorithm
function	I-Algorithm
its	O
name	O
after	O
its	O
resemblance	O
to	O
the	O
mechanical	O
spline	B-Algorithm
used	O
by	O
draftsmen	O
.	O
</s>
<s>
As	O
computers	O
were	O
introduced	O
into	O
the	O
design	O
process	O
,	O
the	O
physical	O
properties	O
of	O
such	O
splines	B-Algorithm
were	O
investigated	O
so	O
that	O
they	O
could	O
be	O
modelled	O
with	O
mathematical	O
precision	O
and	O
reproduced	O
where	O
needed	O
.	O
</s>
<s>
At	O
first	O
NURBS	B-Algorithm
were	O
only	O
used	O
in	O
the	O
proprietary	O
CAD	B-Application
packages	O
of	O
car	O
companies	O
.	O
</s>
<s>
Real-time	O
,	O
interactive	O
rendering	O
of	O
NURBS	B-Algorithm
curves	I-Algorithm
and	O
surfaces	O
was	O
first	O
made	O
commercially	O
available	O
on	O
Silicon	O
Graphics	O
workstations	O
in	O
1989	O
.	O
</s>
<s>
In	O
1993	O
,	O
the	O
first	O
interactive	O
NURBS	B-Algorithm
modeller	O
for	O
PCs	O
,	O
called	O
NöRBS	O
,	O
was	O
developed	O
by	O
CAS	O
Berlin	O
,	O
a	O
small	O
startup	O
company	O
cooperating	O
with	O
the	O
Technical	O
University	O
of	O
Berlin	O
.	O
</s>
<s>
the	O
hull	O
of	O
a	O
motor	O
yacht	O
,	O
is	O
usually	O
composed	O
of	O
several	O
NURBS	B-Algorithm
surfaces	I-Algorithm
known	O
as	O
NURBS	B-Algorithm
patches	I-Algorithm
(	O
or	O
just	O
patches	O
)	O
.	O
</s>
<s>
Higher-level	O
tools	O
exist	O
that	O
benefit	O
from	O
the	O
ability	O
of	O
NURBS	B-Algorithm
to	O
create	O
and	O
establish	O
geometric	O
continuity	O
of	O
different	O
levels	O
:	O
</s>
<s>
This	O
level	O
of	O
continuity	O
is	O
very	O
useful	O
in	O
the	O
creation	O
of	O
models	O
that	O
require	O
many	O
bi-cubic	B-Algorithm
patches	O
composing	O
one	O
continuous	O
surface	O
.	O
</s>
<s>
Geometric	O
continuity	O
mainly	O
refers	O
to	O
the	O
shape	O
of	O
the	O
resulting	O
surface	O
;	O
since	O
NURBS	B-Algorithm
surfaces	I-Algorithm
are	O
functions	O
,	O
it	O
is	O
also	O
possible	O
to	O
discuss	O
the	O
derivatives	O
of	O
the	O
surface	O
with	O
respect	O
to	O
the	O
parameters	O
.	O
</s>
<s>
In	O
practice	O
,	O
C²	O
continuity	O
is	O
easier	O
to	O
achieve	O
if	O
uniform	O
B-splines	B-Algorithm
are	O
used	O
.	O
</s>
<s>
Highlights	O
and	O
reflections	O
can	O
reveal	O
the	O
perfect	O
smoothing	O
,	O
which	O
is	O
otherwise	O
practically	O
impossible	O
to	O
achieve	O
without	O
NURBS	B-Algorithm
surfaces	I-Algorithm
that	O
have	O
at	O
least	O
G²	O
continuity	O
.	O
</s>
<s>
This	O
same	O
principle	O
is	O
used	O
as	O
one	O
of	O
the	O
surface	O
evaluation	O
methods	O
whereby	O
a	O
ray-traced	B-Algorithm
or	O
reflection-mapped	O
image	O
of	O
a	O
surface	O
with	O
white	O
stripes	O
reflecting	O
on	O
it	O
will	O
show	O
even	O
the	O
smallest	O
deviations	O
on	O
a	O
surface	O
or	O
set	O
of	O
surfaces	O
.	O
</s>
<s>
A	O
NURBS	B-Algorithm
curve	I-Algorithm
is	O
defined	O
by	O
its	O
order	O
,	O
a	O
set	O
of	O
weighted	B-Algorithm
control	B-Algorithm
points	I-Algorithm
,	O
and	O
a	O
knot	B-Algorithm
vector	I-Algorithm
.	O
</s>
<s>
NURBS	B-Algorithm
curves	I-Algorithm
and	O
surfaces	O
are	O
generalizations	O
of	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	B-Algorithm
points	I-Algorithm
,	O
which	O
makes	O
NURBS	B-Algorithm
curves	I-Algorithm
rational	O
.	O
</s>
<s>
(	O
Non-rational	O
,	O
aka	O
simple	O
,	O
B-splines	B-Algorithm
are	O
a	O
special	O
case/subset	O
of	O
rational	O
B-splines	B-Algorithm
,	O
where	O
each	O
control	B-Algorithm
point	I-Algorithm
is	O
a	O
regular	O
non-homogenous	O
coordinate	O
[	O
no	O
'	O
w' ]	O
rather	O
than	O
a	O
homogeneous	O
coordinate	O
.	O
</s>
<s>
That	O
is	O
equivalent	O
to	O
having	O
weight	O
"	O
1	O
"	O
at	O
each	O
control	B-Algorithm
point	I-Algorithm
;	O
Rational	O
B-splines	B-Algorithm
use	O
the	O
'	O
w	O
 '	O
of	O
each	O
control	B-Algorithm
point	I-Algorithm
as	O
a	O
weight	O
.	O
)	O
</s>
<s>
By	O
using	O
a	O
two-dimensional	O
grid	O
of	O
control	B-Algorithm
points	I-Algorithm
,	O
NURBS	B-Algorithm
surfaces	I-Algorithm
including	O
planar	O
patches	O
and	O
sections	O
of	O
spheres	O
can	O
be	O
created	O
.	O
</s>
<s>
This	O
can	O
be	O
extended	O
to	O
arbitrary	O
dimensions	O
to	O
create	O
NURBS	B-Algorithm
mapping	O
.	O
</s>
<s>
NURBS	B-Algorithm
curves	I-Algorithm
and	O
surfaces	O
are	O
useful	O
for	O
a	O
number	O
of	O
reasons	O
:	O
</s>
<s>
The	O
set	O
of	O
NURBS	B-Algorithm
for	O
a	O
given	O
order	O
is	O
invariant	O
under	O
affine	B-Algorithm
transformations	I-Algorithm
:	O
operations	O
like	O
rotations	O
and	O
translations	O
can	O
be	O
applied	O
to	O
NURBS	B-Algorithm
curves	I-Algorithm
and	O
surfaces	O
by	O
applying	O
them	O
to	O
their	O
control	B-Algorithm
points	I-Algorithm
.	O
</s>
<s>
They	O
can	O
be	O
evaluated	O
reasonably	O
quickly	O
by	O
numerically	B-Algorithm
stable	I-Algorithm
and	O
accurate	O
algorithms	O
.	O
</s>
<s>
Here	O
,	O
NURBS	B-Algorithm
is	O
mostly	O
discussed	O
in	O
one	O
dimension	O
(	O
curves	O
)	O
;	O
it	O
can	O
be	O
generalized	O
to	O
two	O
(	O
surfaces	O
)	O
or	O
even	O
more	O
dimensions	O
.	O
</s>
<s>
The	O
order	O
of	O
a	O
NURBS	B-Algorithm
curve	I-Algorithm
defines	O
the	O
number	O
of	O
nearby	O
control	B-Algorithm
points	I-Algorithm
that	O
influence	O
any	O
given	O
point	O
on	O
the	O
curve	O
.	O
</s>
<s>
The	O
number	O
of	O
control	B-Algorithm
points	I-Algorithm
must	O
be	O
greater	O
than	O
or	O
equal	O
to	O
the	O
order	O
of	O
the	O
curve	O
.	O
</s>
<s>
The	O
control	B-Algorithm
points	I-Algorithm
determine	O
the	O
shape	O
of	O
the	O
curve	O
.	O
</s>
<s>
Typically	O
,	O
each	O
point	O
of	O
the	O
curve	O
is	O
computed	O
by	O
taking	O
a	O
weighted	B-Algorithm
sum	I-Algorithm
of	O
a	O
number	O
of	O
control	B-Algorithm
points	I-Algorithm
.	O
</s>
<s>
For	O
a	O
curve	O
of	O
degree	O
d	O
,	O
the	O
weight	O
of	O
any	O
control	B-Algorithm
point	I-Algorithm
is	O
only	O
nonzero	O
in	O
d+1	O
intervals	O
of	O
the	O
parameter	O
space	O
.	O
</s>
<s>
While	O
it	O
rises	O
,	O
the	O
basis	O
function	O
of	O
the	O
previous	O
control	B-Algorithm
point	I-Algorithm
falls	O
.	O
</s>
<s>
In	O
that	O
way	O
,	O
the	O
curve	O
interpolates	O
between	O
the	O
two	O
points	O
,	O
and	O
the	O
resulting	O
curve	O
is	O
a	O
polygon	O
,	O
which	O
is	O
continuous	O
,	O
but	O
not	O
differentiable	O
at	O
the	O
interval	O
boundaries	O
,	O
or	O
knots	B-Algorithm
.	O
</s>
<s>
Note	O
that	O
within	O
the	O
interval	O
the	O
polynomial	O
nature	O
of	O
the	O
basis	O
functions	O
and	O
the	O
linearity	O
of	O
the	O
construction	O
make	O
the	O
curve	O
perfectly	O
smooth	O
,	O
so	O
it	O
is	O
only	O
at	O
the	O
knots	B-Algorithm
that	O
discontinuity	O
can	O
arise	O
.	O
</s>
<s>
In	O
many	O
applications	O
the	O
fact	O
that	O
a	O
single	O
control	B-Algorithm
point	I-Algorithm
only	O
influences	O
those	O
intervals	O
where	O
it	O
is	O
active	O
is	O
a	O
highly	O
desirable	O
property	O
,	O
known	O
as	O
local	O
support	O
.	O
</s>
<s>
Adding	O
more	O
control	B-Algorithm
points	I-Algorithm
allows	O
better	O
approximation	O
to	O
a	O
given	O
curve	O
,	O
although	O
only	O
a	O
certain	O
class	O
of	O
curves	O
can	O
be	O
represented	O
exactly	O
with	O
a	O
finite	O
number	O
of	O
control	B-Algorithm
points	I-Algorithm
.	O
</s>
<s>
NURBS	B-Algorithm
curves	I-Algorithm
also	O
feature	O
a	O
scalar	O
weight	O
for	O
each	O
control	B-Algorithm
point	I-Algorithm
.	O
</s>
<s>
This	O
allows	O
for	O
more	O
control	O
over	O
the	O
shape	O
of	O
the	O
curve	O
without	O
unduly	O
raising	O
the	O
number	O
of	O
control	B-Algorithm
points	I-Algorithm
.	O
</s>
<s>
The	O
term	O
rational	O
in	O
NURBS	B-Algorithm
refers	O
to	O
these	O
weights	O
.	O
</s>
<s>
The	O
control	B-Algorithm
points	I-Algorithm
can	O
have	O
any	O
dimensionality	O
.	O
</s>
<s>
Three-dimensional	O
control	B-Algorithm
points	I-Algorithm
are	O
used	O
abundantly	O
in	O
3D	O
modeling	O
,	O
where	O
they	O
are	O
used	O
in	O
the	O
everyday	O
meaning	O
of	O
the	O
word	O
'	O
point	O
 '	O
,	O
a	O
location	O
in	O
3D	O
space	O
.	O
</s>
<s>
NURBS	B-Algorithm
surfaces	I-Algorithm
are	O
just	O
an	O
application	O
of	O
this	O
.	O
</s>
<s>
Each	O
control	O
'	O
point	O
 '	O
is	O
actually	O
a	O
full	O
vector	O
of	O
control	B-Algorithm
points	I-Algorithm
,	O
defining	O
a	O
curve	O
.	O
</s>
<s>
These	O
curves	O
share	O
their	O
degree	O
and	O
the	O
number	O
of	O
control	B-Algorithm
points	I-Algorithm
,	O
and	O
span	O
one	O
dimension	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	B-Algorithm
points	I-Algorithm
affect	O
the	O
NURBS	B-Algorithm
curve	I-Algorithm
.	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	B-Algorithm
points	I-Algorithm
plus	O
curve	O
degree	O
plus	O
one	O
(	O
i.e.	O
</s>
<s>
number	O
of	O
control	B-Algorithm
points	I-Algorithm
plus	O
curve	O
order	O
)	O
.	O
</s>
<s>
The	O
knot	B-Algorithm
vector	I-Algorithm
divides	O
the	O
parametric	O
space	O
in	O
the	O
intervals	O
mentioned	O
before	O
,	O
usually	O
referred	O
to	O
as	O
knot	B-Algorithm
spans	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	B-Algorithm
point	I-Algorithm
becomes	O
active	O
,	O
while	O
an	O
old	O
control	B-Algorithm
point	I-Algorithm
is	O
discarded	O
.	O
</s>
<s>
It	O
follows	O
that	O
the	O
values	O
in	O
the	O
knot	B-Algorithm
vector	I-Algorithm
should	O
be	O
in	O
nondecreasing	O
order	O
,	O
so	O
(	O
0	O
,	O
0	O
,	O
1	O
,	O
2	O
,	O
3	O
,	O
3	O
)	O
is	O
valid	O
while	O
(	O
0	O
,	O
0	O
,	O
2	O
,	O
1	O
,	O
3	O
,	O
3	O
)	O
is	O
not	O
.	O
</s>
<s>
Consecutive	O
knots	B-Algorithm
can	O
have	O
the	O
same	O
value	O
.	O
</s>
<s>
This	O
then	O
defines	O
a	O
knot	B-Algorithm
span	O
of	O
zero	O
length	O
,	O
which	O
implies	O
that	O
two	O
control	B-Algorithm
points	I-Algorithm
are	O
activated	O
at	O
the	O
same	O
time	O
(	O
and	O
of	O
course	O
two	O
control	B-Algorithm
points	I-Algorithm
become	O
deactivated	O
)	O
.	O
</s>
<s>
This	O
has	O
impact	O
on	O
continuity	O
of	O
the	O
resulting	O
curve	O
or	O
its	O
higher	O
derivatives	O
;	O
for	O
instance	O
,	O
it	O
allows	O
the	O
creation	O
of	O
corners	O
in	O
an	O
otherwise	O
smooth	O
NURBS	B-Algorithm
curve	I-Algorithm
.	O
</s>
<s>
A	O
number	O
of	O
coinciding	O
knots	B-Algorithm
is	O
sometimes	O
referred	O
to	O
as	O
a	O
knot	B-Algorithm
with	O
a	O
certain	O
multiplicity	O
.	O
</s>
<s>
Knots	B-Algorithm
with	O
multiplicity	O
two	O
or	O
three	O
are	O
known	O
as	O
double	O
or	O
triple	O
knots	B-Algorithm
.	O
</s>
<s>
The	O
multiplicity	O
of	O
a	O
knot	B-Algorithm
is	O
limited	O
to	O
the	O
degree	O
of	O
the	O
curve	O
;	O
since	O
a	O
higher	O
multiplicity	O
would	O
split	O
the	O
curve	O
into	O
disjoint	O
parts	O
and	O
it	O
would	O
leave	O
control	B-Algorithm
points	I-Algorithm
unused	O
.	O
</s>
<s>
For	O
first-degree	O
NURBS	B-Algorithm
,	O
each	O
knot	B-Algorithm
is	O
paired	O
with	O
a	O
control	B-Algorithm
point	I-Algorithm
.	O
</s>
<s>
The	O
knot	B-Algorithm
vector	I-Algorithm
usually	O
starts	O
with	O
a	O
knot	B-Algorithm
that	O
has	O
multiplicity	O
equal	O
to	O
the	O
order	O
.	O
</s>
<s>
This	O
makes	O
sense	O
,	O
since	O
this	O
activates	O
the	O
control	B-Algorithm
points	I-Algorithm
that	O
have	O
influence	O
on	O
the	O
first	O
knot	B-Algorithm
span	O
.	O
</s>
<s>
Similarly	O
,	O
the	O
knot	B-Algorithm
vector	I-Algorithm
usually	O
ends	O
with	O
a	O
knot	B-Algorithm
of	O
that	O
multiplicity	O
.	O
</s>
<s>
Curves	O
with	O
such	O
knot	B-Algorithm
vectors	I-Algorithm
start	O
and	O
end	O
in	O
a	O
control	B-Algorithm
point	I-Algorithm
.	O
</s>
<s>
The	O
values	O
of	O
the	O
knots	B-Algorithm
control	O
the	O
mapping	O
between	O
the	O
input	O
parameter	O
and	O
the	O
corresponding	O
NURBS	B-Algorithm
value	O
.	O
</s>
<s>
For	O
example	O
,	O
if	O
a	O
NURBS	B-Algorithm
describes	O
a	O
path	O
through	O
space	O
over	O
time	O
,	O
the	O
knots	B-Algorithm
control	O
the	O
time	O
that	O
the	O
function	O
proceeds	O
past	O
the	O
control	B-Algorithm
points	I-Algorithm
.	O
</s>
<s>
For	O
the	O
purposes	O
of	O
representing	O
shapes	O
,	O
however	O
,	O
only	O
the	O
ratios	O
of	O
the	O
difference	O
between	O
the	O
knot	B-Algorithm
values	I-Algorithm
matter	O
;	O
in	O
that	O
case	O
,	O
the	O
knot	B-Algorithm
vectors	I-Algorithm
(	O
0	O
,	O
0	O
,	O
1	O
,	O
2	O
,	O
3	O
,	O
3	O
)	O
and	O
(	O
0	O
,	O
0	O
,	O
2	O
,	O
4	O
,	O
6	O
,	O
6	O
)	O
produce	O
the	O
same	O
curve	O
.	O
</s>
<s>
The	O
positions	O
of	O
the	O
knot	B-Algorithm
values	I-Algorithm
influences	O
the	O
mapping	O
of	O
parameter	O
space	O
to	O
curve	O
space	O
.	O
</s>
<s>
Rendering	O
a	O
NURBS	B-Algorithm
curve	I-Algorithm
is	O
usually	O
done	O
by	O
stepping	O
with	O
a	O
fixed	O
stride	O
through	O
the	O
parameter	O
range	O
.	O
</s>
<s>
By	O
changing	O
the	O
knot	B-Algorithm
span	O
lengths	O
,	O
more	O
sample	O
points	O
can	O
be	O
used	O
in	O
regions	O
where	O
the	O
curvature	O
is	O
high	O
.	O
</s>
<s>
The	O
knot	B-Algorithm
span	O
lengths	O
then	O
translate	O
into	O
velocity	O
and	O
acceleration	O
,	O
which	O
are	O
essential	O
to	O
get	O
right	O
to	O
prevent	O
damage	O
to	O
the	O
robot	O
arm	O
or	O
its	O
environment	O
.	O
</s>
<s>
This	O
flexibility	O
in	O
the	O
mapping	O
is	O
what	O
the	O
phrase	O
non	O
uniform	O
in	O
NURBS	B-Algorithm
refers	O
to	O
.	O
</s>
<s>
Necessary	O
only	O
for	O
internal	O
calculations	O
,	O
knots	B-Algorithm
are	O
usually	O
not	O
helpful	O
to	O
the	O
users	O
of	O
modeling	O
software	O
.	O
</s>
<s>
Therefore	O
,	O
many	O
modeling	O
applications	O
do	O
not	O
make	O
the	O
knots	B-Algorithm
editable	O
or	O
even	O
visible	O
.	O
</s>
<s>
It	O
's	O
usually	O
possible	O
to	O
establish	O
reasonable	O
knot	B-Algorithm
vectors	I-Algorithm
by	O
looking	O
at	O
the	O
variation	O
in	O
the	O
control	B-Algorithm
points	I-Algorithm
.	O
</s>
<s>
More	O
recent	O
versions	O
of	O
NURBS	B-Algorithm
software	O
(	O
e.g.	O
,	O
Autodesk	B-Language
Maya	I-Language
and	O
Rhinoceros	B-Algorithm
3D	I-Algorithm
)	O
allow	O
for	O
interactive	O
editing	O
of	O
knot	B-Algorithm
positions	O
,	O
but	O
this	O
is	O
significantly	O
less	O
intuitive	O
than	O
the	O
editing	O
of	O
control	B-Algorithm
points	I-Algorithm
.	O
</s>
<s>
control	B-Algorithm
point	I-Algorithm
,	O
and	O
corresponds	O
with	O
the	O
degree	O
of	O
the	O
basis	O
function	O
.	O
</s>
<s>
The	O
degree-0	O
functions	O
are	O
piecewise	B-Algorithm
constant	O
functions	O
.	O
</s>
<s>
They	O
are	O
one	O
on	O
the	O
corresponding	O
knot	B-Algorithm
span	O
and	O
zero	O
everywhere	O
else	O
.	O
</s>
<s>
knot	B-Algorithm
spans	O
,	O
overlapping	O
for	O
knot	B-Algorithm
spans	O
.	O
</s>
<s>
As	O
mentioned	O
before	O
,	O
is	O
a	O
triangular	O
function	O
,	O
nonzero	O
over	O
two	O
knot	B-Algorithm
spans	O
rising	O
from	O
zero	O
to	O
one	O
on	O
the	O
first	O
,	O
and	O
falling	O
to	O
zero	O
on	O
the	O
second	O
knot	B-Algorithm
span	O
.	O
</s>
<s>
Higher	O
order	O
basis	O
functions	O
are	O
non-zero	O
over	O
corresponding	O
more	O
knot	B-Algorithm
spans	O
and	O
have	O
correspondingly	O
higher	O
degree	O
.	O
</s>
<s>
By	B-Algorithm
induction	I-Algorithm
on	O
n	O
it	O
follows	O
that	O
the	O
basis	O
functions	O
are	O
non-negative	O
for	O
all	O
values	O
of	O
and	O
.	O
</s>
<s>
This	O
makes	O
the	O
computation	O
of	O
the	O
basis	O
functions	O
numerically	B-Algorithm
stable	I-Algorithm
.	O
</s>
<s>
Again	O
by	B-Algorithm
induction	I-Algorithm
,	O
it	O
can	O
be	O
proved	O
that	O
the	O
sum	O
of	O
the	O
basis	O
functions	O
for	O
a	O
particular	O
value	O
of	O
the	O
parameter	O
is	O
unity	O
.	O
</s>
<s>
The	O
figures	O
show	O
the	O
linear	O
and	O
the	O
quadratic	O
basis	O
functions	O
for	O
the	O
knots	B-Algorithm
{	O
...	O
,	O
0	O
,	O
1	O
,	O
2	O
,	O
3	O
,	O
4	O
,	O
4.1	O
,	O
5.1	O
,	O
6.1	O
,	O
7.1	O
,	O
...	O
}	O
</s>
<s>
One	O
knot	B-Algorithm
span	O
is	O
considerably	O
shorter	O
than	O
the	O
others	O
.	O
</s>
<s>
On	O
that	O
knot	B-Algorithm
span	O
,	O
the	O
peak	O
in	O
the	O
quadratic	O
basis	O
function	O
is	O
more	O
distinct	O
,	O
reaching	O
almost	O
one	O
.	O
</s>
<s>
In	O
the	O
geometrical	O
interpretation	O
,	O
this	O
means	O
that	O
the	O
curve	O
approaches	O
the	O
corresponding	O
control	B-Algorithm
point	I-Algorithm
closely	O
.	O
</s>
<s>
In	O
case	O
of	O
a	O
double	O
knot	B-Algorithm
,	O
the	O
length	O
of	O
the	O
knot	B-Algorithm
span	O
becomes	O
zero	O
and	O
the	O
peak	O
reaches	O
one	O
exactly	O
.	O
</s>
<s>
The	O
curve	O
will	O
have	O
a	O
sharp	O
corner	O
if	O
the	O
neighbour	O
control	B-Algorithm
points	I-Algorithm
are	O
not	O
collinear	O
.	O
</s>
<s>
Using	O
the	O
definitions	O
of	O
the	O
basis	O
functions	O
from	O
the	O
previous	O
paragraph	O
,	O
a	O
NURBS	B-Algorithm
curve	I-Algorithm
takes	O
the	O
following	O
form	O
:	O
</s>
<s>
In	O
this	O
,	O
is	O
the	O
number	O
of	O
control	B-Algorithm
points	I-Algorithm
and	O
are	O
the	O
corresponding	O
weights	O
.	O
</s>
<s>
A	O
NURBS	B-Algorithm
surface	I-Algorithm
is	O
obtained	O
as	O
the	O
tensor	O
product	O
of	O
two	O
NURBS	B-Algorithm
curves	I-Algorithm
,	O
thus	O
using	O
two	O
independent	O
parameters	O
and	O
(	O
with	O
indices	O
and	O
respectively	O
)	O
:	O
</s>
<s>
A	O
number	O
of	O
transformations	O
can	O
be	O
applied	O
to	O
a	O
NURBS	B-Algorithm
object	O
.	O
</s>
<s>
For	O
instance	O
,	O
if	O
some	O
curve	O
is	O
defined	O
using	O
a	O
certain	O
degree	O
and	O
N	O
control	B-Algorithm
points	I-Algorithm
,	O
the	O
same	O
curve	O
can	O
be	O
expressed	O
using	O
the	O
same	O
degree	O
and	O
N+1	O
control	B-Algorithm
points	I-Algorithm
.	O
</s>
<s>
In	O
the	O
process	O
a	O
number	O
of	O
control	B-Algorithm
points	I-Algorithm
change	O
position	O
and	O
a	O
knot	B-Algorithm
is	O
inserted	O
in	O
the	O
knot	B-Algorithm
vector	I-Algorithm
.	O
</s>
<s>
When	O
adding	O
a	O
control	B-Algorithm
point	I-Algorithm
,	O
the	O
shape	O
of	O
the	O
curve	O
should	O
stay	O
the	O
same	O
,	O
forming	O
the	O
starting	O
point	O
for	O
further	O
adjustments	O
.	O
</s>
<s>
As	O
the	O
term	O
suggests	O
,	O
knot	B-Algorithm
insertion	O
inserts	O
a	O
knot	B-Algorithm
into	O
the	O
knot	B-Algorithm
vector	I-Algorithm
.	O
</s>
<s>
If	O
the	O
degree	O
of	O
the	O
curve	O
is	O
,	O
then	O
control	B-Algorithm
points	I-Algorithm
are	O
replaced	O
by	O
new	O
ones	O
.	O
</s>
<s>
A	O
knot	B-Algorithm
can	O
be	O
inserted	O
multiple	O
times	O
,	O
up	O
to	O
the	O
maximum	O
multiplicity	O
of	O
the	O
knot	B-Algorithm
.	O
</s>
<s>
This	O
is	O
sometimes	O
referred	O
to	O
as	O
knot	B-Algorithm
refinement	O
and	O
can	O
be	O
achieved	O
by	O
an	O
algorithm	O
that	O
is	O
more	O
efficient	O
than	O
repeated	O
knot	B-Algorithm
insertion	O
.	O
</s>
<s>
Knot	B-Algorithm
removal	O
is	O
the	O
reverse	O
of	O
knot	B-Algorithm
insertion	O
.	O
</s>
<s>
Its	O
purpose	O
is	O
to	O
remove	O
knots	B-Algorithm
and	O
the	O
associated	O
control	B-Algorithm
points	I-Algorithm
in	O
order	O
to	O
get	O
a	O
more	O
compact	O
representation	O
.	O
</s>
<s>
In	O
practice	O
,	O
a	O
tolerance	O
in	O
the	O
accuracy	O
is	O
used	O
to	O
determine	O
whether	O
a	O
knot	B-Algorithm
can	O
be	O
removed	O
.	O
</s>
<s>
a	O
curve	O
from	O
a	O
different	O
representation	O
,	O
where	O
a	O
straightforward	O
conversion	O
process	O
leads	O
to	O
redundant	O
control	B-Algorithm
points	I-Algorithm
.	O
</s>
<s>
A	O
NURBS	B-Algorithm
curve	I-Algorithm
of	O
a	O
particular	O
degree	O
can	O
always	O
be	O
represented	O
by	O
a	O
NURBS	B-Algorithm
curve	I-Algorithm
of	O
higher	O
degree	O
.	O
</s>
<s>
This	O
is	O
frequently	O
used	O
when	O
combining	O
separate	O
NURBS	B-Algorithm
curves	I-Algorithm
,	O
</s>
<s>
e.g.	O
,	O
when	O
creating	O
a	O
NURBS	B-Algorithm
surface	I-Algorithm
interpolating	O
between	O
a	O
set	O
of	O
NURBS	B-Algorithm
curves	I-Algorithm
or	O
when	O
unifying	O
adjacent	O
curves	O
.	O
</s>
<s>
The	O
most	O
important	O
property	O
in	O
differential	B-Language
geometry	I-Language
is	O
the	O
curvature	O
.	O
</s>
<s>
Non-rational	O
splines	B-Algorithm
or	O
Bézier	O
curves	O
may	O
approximate	O
a	O
circle	O
,	O
but	O
they	O
cannot	O
represent	O
it	O
exactly	O
.	O
</s>
<s>
Rational	O
splines	B-Algorithm
can	O
represent	O
any	O
conic	O
section	O
,	O
including	O
the	O
circle	O
,	O
exactly	O
.	O
</s>
<s>
The	O
order	O
is	O
three	O
,	O
since	O
a	O
circle	O
is	O
a	O
quadratic	O
curve	O
and	O
the	O
spline	B-Algorithm
's	O
order	O
is	O
one	O
more	O
than	O
the	O
degree	O
of	O
its	O
piecewise	B-Algorithm
polynomial	I-Algorithm
segments	O
.	O
</s>
<s>
The	O
knot	B-Algorithm
vector	I-Algorithm
is	O
.	O
</s>
<s>
The	O
circle	O
is	O
composed	O
of	O
four	O
quarter	O
circles	O
,	O
tied	O
together	O
with	O
double	O
knots	B-Algorithm
.	O
</s>
<s>
Although	O
double	O
knots	B-Algorithm
in	O
a	O
third	O
order	O
NURBS	B-Algorithm
curve	I-Algorithm
would	O
normally	O
result	O
in	O
loss	O
of	O
continuity	O
in	O
the	O
first	O
derivative	O
,	O
the	O
control	B-Algorithm
points	I-Algorithm
are	O
positioned	O
in	O
such	O
a	O
way	O
that	O
the	O
first	O
derivative	O
is	O
continuous	O
.	O
</s>
<s>
The	O
circle	O
does	O
make	O
one	O
full	O
revolution	O
as	O
its	O
parameter	O
goes	O
from	O
0	O
to	O
,	O
but	O
this	O
is	O
only	O
because	O
the	O
knot	B-Algorithm
vector	I-Algorithm
was	O
arbitrarily	O
chosen	O
as	O
multiples	O
of	O
.	O
</s>
