<s>
In	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
a	O
cubic	B-Algorithm
Hermite	I-Algorithm
spline	I-Algorithm
or	O
cubic	O
Hermite	O
interpolator	O
is	O
a	O
spline	B-Algorithm
where	O
each	O
piece	O
is	O
a	O
third-degree	O
polynomial	O
specified	O
in	O
Hermite	O
form	O
,	O
that	O
is	O
,	O
by	O
its	O
values	O
and	O
first	B-Algorithm
derivatives	I-Algorithm
at	O
the	O
end	O
points	O
of	O
the	O
corresponding	O
domain	B-Algorithm
interval	O
.	O
</s>
<s>
Cubic	B-Algorithm
Hermite	I-Algorithm
splines	I-Algorithm
are	O
typically	O
used	O
for	O
interpolation	B-Algorithm
of	O
numeric	O
data	O
specified	O
at	O
given	O
argument	O
values	O
,	O
to	O
obtain	O
a	O
continuous	O
function	O
.	O
</s>
<s>
The	O
data	O
should	O
consist	O
of	O
the	O
desired	O
function	O
value	O
and	O
derivative	B-Algorithm
at	O
each	O
.	O
</s>
<s>
(	O
If	O
only	O
the	O
values	O
are	O
provided	O
,	O
the	O
derivatives	B-Algorithm
must	O
be	O
estimated	O
from	O
them	O
.	O
)	O
</s>
<s>
The	O
resulting	O
spline	B-Algorithm
will	O
be	O
continuous	O
and	O
will	O
have	O
continuous	O
first	B-Algorithm
derivative	I-Algorithm
.	O
</s>
<s>
Cubic	O
polynomial	O
splines	B-Algorithm
can	O
be	O
specified	O
in	O
other	O
ways	O
,	O
the	O
Bezier	O
cubic	O
being	O
the	O
most	O
common	O
.	O
</s>
<s>
However	O
,	O
these	O
two	O
methods	O
provide	O
the	O
same	O
set	O
of	O
splines	B-Algorithm
,	O
and	O
data	O
can	O
be	O
easily	O
converted	O
between	O
the	O
Bézier	O
and	O
Hermite	O
forms	O
;	O
so	O
the	O
names	O
are	O
often	O
used	O
as	O
if	O
they	O
were	O
synonymous	O
.	O
</s>
<s>
Cubic	O
polynomial	O
splines	B-Algorithm
are	O
extensively	O
used	O
in	O
computer	O
graphics	O
and	O
geometric	B-Application
modeling	I-Application
to	O
obtain	O
curves	O
or	O
motion	O
trajectories	B-Application
that	O
pass	O
through	O
specified	O
points	O
of	O
the	O
plane	O
or	O
three-dimensional	O
space	O
.	O
</s>
<s>
In	O
these	O
applications	O
,	O
each	O
coordinate	O
of	O
the	O
plane	O
or	O
space	O
is	O
separately	O
interpolated	O
by	O
a	O
cubic	B-Algorithm
spline	I-Algorithm
function	O
of	O
a	O
separate	O
parametert	O
.	O
</s>
<s>
Cubic	O
polynomial	O
splines	B-Algorithm
are	O
also	O
used	O
extensively	O
in	O
structural	O
analysis	O
applications	O
,	O
such	O
as	O
Euler	O
–	O
Bernoulli	O
beam	O
theory	O
.	O
</s>
<s>
Cubic	O
polynomial	O
splines	B-Algorithm
have	O
also	O
been	O
applied	O
to	O
mortality	O
analysis	O
and	O
mortality	O
forecasting	O
.	O
</s>
<s>
Cubic	B-Algorithm
splines	I-Algorithm
can	O
be	O
extended	O
to	O
functions	O
of	O
two	O
or	O
more	O
parameters	O
,	O
in	O
several	O
ways	O
.	O
</s>
<s>
Bicubic	B-Algorithm
splines	B-Algorithm
(	O
Bicubic	B-Algorithm
interpolation	I-Algorithm
)	O
are	O
often	O
used	O
to	O
interpolate	O
data	O
on	O
a	O
regular	O
rectangular	O
grid	O
,	O
such	O
as	O
pixel	B-Algorithm
values	O
in	O
a	O
digital	B-Algorithm
image	I-Algorithm
or	O
altitude	B-Application
data	O
on	O
a	O
terrain	O
.	O
</s>
<s>
Bicubic	B-Algorithm
surface	O
patches	O
,	O
defined	O
by	O
three	O
bicubic	B-Algorithm
splines	B-Algorithm
,	O
are	O
an	O
essential	O
tool	O
in	O
computer	O
graphics	O
.	O
</s>
<s>
Cubic	B-Algorithm
splines	I-Algorithm
are	O
often	O
called	O
csplines	B-Algorithm
,	O
especially	O
in	O
computer	O
graphics	O
.	O
</s>
<s>
Hermite	O
splines	B-Algorithm
are	O
named	O
after	O
Charles	O
Hermite	O
.	O
</s>
<s>
Interpolating	O
in	O
an	O
arbitrary	O
interval	O
is	O
done	O
by	O
mapping	O
the	O
latter	O
to	O
through	O
an	O
affine	B-Algorithm
(	O
degree-1	O
)	O
change	O
of	O
variable	O
.	O
</s>
<s>
The	O
"	O
Bernstein	O
"	O
column	O
shows	O
the	O
decomposition	O
of	O
the	O
Hermite	O
basis	O
functions	O
into	O
Bernstein	B-Algorithm
polynomials	I-Algorithm
of	O
order3	O
:	O
</s>
<s>
Using	O
this	O
connection	O
you	O
can	O
express	O
cubic	O
Hermite	O
interpolation	B-Algorithm
in	O
terms	O
of	O
cubic	O
Bézier	O
curves	O
with	O
respect	O
to	O
the	O
four	O
values	O
and	O
do	O
Hermite	O
interpolation	B-Algorithm
using	O
the	O
de	B-Algorithm
Casteljau	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
It	O
shows	O
that	O
in	O
a	O
cubic	O
Bézier	O
patch	O
the	O
two	O
control	O
points	O
in	O
the	O
middle	O
determine	O
the	O
tangents	O
of	O
the	O
interpolation	B-Algorithm
curve	O
at	O
the	O
respective	O
outer	O
points	O
.	O
</s>
<s>
The	O
interpolated	O
curve	O
then	O
consists	O
of	O
piecewise	O
cubic	B-Algorithm
Hermite	I-Algorithm
splines	I-Algorithm
and	O
is	O
globally	O
continuously	O
differentiable	O
in	O
.	O
</s>
<s>
Choosing	O
yields	O
all	O
zero	O
tangents	O
,	O
and	O
choosing	O
yields	O
a	O
Catmull	O
–	O
Rom	O
spline	B-Algorithm
in	O
the	O
uniform	O
parameterization	O
case	O
.	O
</s>
<s>
a	O
Catmull	O
–	O
Rom	O
spline	B-Algorithm
is	O
obtained	O
,	O
being	O
a	O
special	O
case	O
of	O
a	O
cardinal	O
spline	B-Algorithm
.	O
</s>
<s>
The	O
principal	O
advantage	O
of	O
this	O
technique	O
is	O
that	O
the	O
points	O
along	O
the	O
original	O
set	O
of	O
points	O
also	O
make	O
up	O
the	O
control	O
points	O
for	O
the	O
spline	B-Algorithm
curve	I-Algorithm
.	O
</s>
<s>
The	O
chordal	O
and	O
centripetal	B-Algorithm
Catmull	I-Algorithm
–	I-Algorithm
Rom	I-Algorithm
implementations	O
solve	O
this	O
problem	O
,	O
but	O
use	O
a	O
slightly	O
different	O
calculation	O
.	O
</s>
<s>
In	O
computer	O
graphics	O
,	O
Catmull	O
–	O
Rom	O
splines	B-Algorithm
are	O
frequently	O
used	O
to	O
get	O
smooth	O
interpolated	O
motion	O
between	O
key	O
frames	O
.	O
</s>
<s>
For	O
example	O
,	O
most	O
camera	O
path	O
animations	O
generated	O
from	O
discrete	O
key-frames	O
are	O
handled	O
using	O
Catmull	O
–	O
Rom	O
splines	B-Algorithm
.	O
</s>
<s>
A	O
Kochanek	O
–	O
Bartels	O
spline	B-Algorithm
is	O
a	O
further	O
generalization	O
on	O
how	O
to	O
choose	O
the	O
tangents	O
given	O
the	O
data	O
points	O
,	O
and	O
,	O
with	O
three	O
parameters	O
possible	O
:	O
tension	O
,	O
bias	O
and	O
a	O
continuity	O
parameter	O
.	O
</s>
<s>
If	O
a	O
cubic	B-Algorithm
Hermite	I-Algorithm
spline	I-Algorithm
of	O
any	O
of	O
the	O
above	O
listed	O
types	O
is	O
used	O
for	O
interpolation	B-Algorithm
of	O
a	O
monotonic	O
data	O
set	O
,	O
the	O
interpolated	O
function	O
will	O
not	O
necessarily	O
be	O
monotonic	O
,	O
but	O
monotonicity	O
can	O
be	O
preserved	O
by	O
adjusting	O
the	O
tangents	O
.	O
</s>
<s>
This	O
writing	O
is	O
relevant	O
for	O
tricubic	O
interpolation	B-Algorithm
,	O
where	O
one	O
optimization	O
requires	O
computing	O
CINTu	O
sixteen	O
times	O
with	O
the	O
same	O
u	O
and	O
different	O
p	O
.	O
</s>
