<s>
The	O
Helmert	B-Algorithm
transformation	I-Algorithm
(	O
named	O
after	O
Friedrich	O
Robert	O
Helmert	O
,	O
18431917	O
)	O
is	O
a	O
geometric	B-Algorithm
transformation	I-Algorithm
method	O
within	O
a	O
three-dimensional	O
space	O
.	O
</s>
<s>
The	O
Helmert	B-Algorithm
transformation	I-Algorithm
is	O
also	O
called	O
a	O
seven-parameter	B-Algorithm
transformation	I-Algorithm
and	O
is	O
a	O
similarity	O
transformation	O
.	O
</s>
<s>
–	O
translation	B-Algorithm
vector	I-Algorithm
.	O
</s>
<s>
–	O
scale	B-Algorithm
factor	I-Algorithm
,	O
which	O
is	O
unitless	O
;	O
if	O
it	O
is	O
given	O
in	O
ppm	O
,	O
it	O
must	O
be	O
divided	O
by	O
1,000,000	O
and	O
added	O
to	O
1	O
.	O
</s>
<s>
–	O
rotation	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
rotation	B-Algorithm
matrix	I-Algorithm
is	O
an	O
orthogonal	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
A	O
special	O
case	O
is	O
the	O
two-dimensional	O
Helmert	B-Algorithm
transformation	I-Algorithm
.	O
</s>
<s>
Here	O
,	O
only	O
four	O
parameters	O
are	O
needed	O
(	O
two	O
translations	B-Algorithm
,	O
one	O
scaling	O
,	O
one	O
rotation	O
)	O
.	O
</s>
<s>
Sometimes	O
it	O
is	O
sufficient	O
to	O
use	O
the	O
five	O
parameter	O
transformation	O
,	O
composed	O
of	O
three	O
translations	B-Algorithm
,	O
only	O
one	O
rotation	O
about	O
the	O
Z-axis	O
,	O
and	O
one	O
change	O
of	O
scale	O
.	O
</s>
<s>
The	O
Helmert	B-Algorithm
transformation	I-Algorithm
only	O
uses	O
one	O
scale	B-Algorithm
factor	I-Algorithm
,	O
so	O
it	O
is	O
not	O
suitable	O
for	O
:	O
</s>
<s>
The	O
comparison	O
of	O
paper	O
deformations	O
while	O
scanning	B-Application
old	O
plans	O
and	O
maps	O
.	O
</s>
<s>
In	O
these	O
cases	O
,	O
a	O
more	O
general	O
affine	B-Algorithm
transformation	I-Algorithm
is	O
preferable	O
.	O
</s>
<s>
The	O
Helmert	B-Algorithm
transformation	I-Algorithm
is	O
used	O
,	O
among	O
other	O
things	O
,	O
in	O
geodesy	O
to	O
transform	O
the	O
coordinates	O
of	O
the	O
point	O
from	O
one	O
coordinate	O
system	O
into	O
another	O
.	O
</s>
<s>
scale	B-Algorithm
factor	I-Algorithm
(	O
with	O
a	O
value	O
near	O
1	O
)	O
and	O
the	O
addition	O
of	O
the	O
three	O
translations	B-Algorithm
,	O
,	O
,	O
.	O
</s>
<s>
To	O
bring	O
them	O
into	O
agreement	O
,	O
the	O
small	O
inconsistencies	O
(	O
usually	O
only	O
a	O
few	O
cm	O
)	O
are	O
adjusted	B-Algorithm
using	O
the	O
method	B-Algorithm
of	I-Algorithm
least	I-Algorithm
squares	I-Algorithm
–	O
that	O
is	O
,	O
eliminated	O
in	O
a	O
statistically	O
plausible	O
manner	O
.	O
</s>
<s>
The	O
translations	B-Algorithm
,	O
,	O
are	O
sometimes	O
described	O
as	O
,	O
,	O
,	O
or	O
,	O
,	O
.	O
</s>
<s>
This	O
is	O
not	O
precise	O
enough	O
for	O
surveying	O
,	O
and	O
the	O
Ordnance	O
Survey	O
supplements	O
these	O
results	O
by	O
using	O
a	O
lookup	O
table	O
of	O
further	O
translations	B-Algorithm
in	O
order	O
to	O
reach	O
accuracy	O
.	O
</s>
<s>
Since	O
a	O
total	O
of	O
seven	O
parameters	O
(	O
three	O
translations	B-Algorithm
,	O
one	O
scale	O
,	O
three	O
rotations	O
)	O
have	O
to	O
be	O
determined	O
,	O
at	O
least	O
two	O
points	O
and	O
one	O
coordinate	O
of	O
a	O
third	O
point	O
(	O
for	O
example	O
,	O
the	O
Z-coordinate	O
)	O
must	O
be	O
known	O
.	O
</s>
<s>
For	O
transformations	O
between	O
conformal	O
map	B-Application
projections	I-Application
near	O
an	O
arbitrary	O
point	O
,	O
the	O
Helmert	B-Algorithm
transformation	I-Algorithm
parameters	O
can	O
be	O
calculated	O
exactly	O
from	O
the	O
Jacobian	O
matrix	O
of	O
the	O
transformation	O
function	O
.	O
</s>
<s>
In	O
this	O
case	O
,	O
the	O
calculation	O
is	O
adjusted	B-Algorithm
with	O
the	O
Gaussian	O
least	B-Algorithm
squares	I-Algorithm
method	I-Algorithm
.	O
</s>
