<s>
Parallel	B-Application
coordinates	I-Application
are	O
a	O
common	O
way	O
of	O
visualizing	O
and	O
analyzing	O
high-dimensional	B-General_Concept
datasets	I-General_Concept
.	O
</s>
<s>
Parallel	B-Application
coordinates	I-Application
were	O
often	O
said	O
to	O
be	O
invented	O
by	O
Philbert	O
Maurice	O
d'Ocagne	O
in	O
1885	O
,	O
but	O
even	O
though	O
the	O
words	O
"	O
Coordonnées	O
parallèles	O
"	O
appear	O
in	O
the	O
book	O
title	O
this	O
work	O
has	O
nothing	O
to	O
do	O
with	O
the	O
visualization	O
techniques	O
of	O
the	O
same	O
name	O
;	O
the	O
book	O
only	O
describes	O
a	O
method	O
of	O
coordinate	O
transformation	O
.	O
</s>
<s>
But	O
even	O
before	O
1885	O
,	O
parallel	B-Application
coordinates	I-Application
were	O
used	O
,	O
for	O
example	O
in	O
Henry	O
Gannetts	O
"	O
General	O
Summary	O
,	O
Showing	O
the	O
Rank	O
of	O
States	O
,	O
by	O
Ratios	O
,	O
1880	O
"	O
,	O
or	O
afterwards	O
in	O
Henry	O
Gannetts	O
"	O
Rank	O
of	O
States	O
and	O
Territories	O
in	O
Population	O
at	O
Each	O
Census	O
,	O
1790-1890	O
"	O
in	O
1898	O
.	O
</s>
<s>
Some	O
important	O
applications	O
are	O
in	O
collision	O
avoidance	O
algorithms	O
for	O
air	B-Application
traffic	I-Application
control	I-Application
(	O
1987	O
—	O
3	O
USA	O
patents	O
)	O
,	O
data	B-Application
mining	I-Application
(	O
USA	O
patent	O
)	O
,	O
computer	B-Application
vision	I-Application
(	O
USA	O
patent	O
)	O
,	O
Optimization	O
,	O
process	O
control	O
,	O
more	O
recently	O
in	O
intrusion	O
detection	O
and	O
elsewhere	O
.	O
</s>
<s>
On	O
the	O
plane	O
with	O
an	O
xy	O
cartesian	O
coordinate	O
system	O
,	O
adding	O
more	O
dimensions	O
in	O
parallel	B-Application
coordinates	I-Application
(	O
often	O
abbreviated	O
||	O
-coords	O
or	O
PCP	O
)	O
involves	O
adding	O
more	O
axes	O
.	O
</s>
<s>
The	O
value	O
of	O
parallel	B-Application
coordinates	I-Application
is	O
that	O
certain	O
geometrical	O
properties	O
in	O
high	O
dimensions	O
transform	O
into	O
easily	O
seen	O
2D	O
patterns	O
.	O
</s>
<s>
For	O
example	O
,	O
a	O
set	O
of	O
points	O
on	O
a	O
line	O
in	O
n-space	O
transforms	O
to	O
a	O
set	O
of	O
polylines	O
in	O
parallel	B-Application
coordinates	I-Application
all	O
intersecting	O
at	O
n1	O
points	O
.	O
</s>
<s>
For	O
n	O
=	O
2	O
this	O
yields	O
a	O
point-line	O
duality	O
pointing	O
out	O
why	O
the	O
mathematical	O
foundations	O
of	O
parallel	B-Application
coordinates	I-Application
are	O
developed	O
in	O
the	O
projective	O
rather	O
than	O
euclidean	O
space	O
.	O
</s>
<s>
Hence	O
by	O
using	O
curves	O
in	O
parallel	B-Application
coordinates	I-Application
instead	O
of	O
lines	O
,	O
the	O
point	O
line	O
duality	O
is	O
lost	O
together	O
with	O
all	O
the	O
other	O
properties	O
of	O
projective	O
geometry	O
,	O
and	O
the	O
known	O
nice	O
higher-dimensional	O
patterns	O
corresponding	O
to	O
(	O
hyper	O
)	O
planes	O
,	O
curves	O
,	O
several	O
smooth	O
(	O
hyper	O
)	O
surfaces	O
,	O
proximities	O
,	O
convexity	O
and	O
recently	O
non-orientability	O
.	O
</s>
<s>
Hence	O
,	O
parallel	B-Application
coordinates	I-Application
is	O
not	O
a	O
point-to-point	O
mapping	O
but	O
rather	O
a	O
nD	O
subset	O
to	O
2D	O
subset	O
mapping	O
,	O
there	O
is	O
no	O
loss	O
of	O
information	O
.	O
</s>
<s>
The	O
rotation	O
of	O
the	O
axes	O
is	O
a	O
translation	O
in	O
the	O
parallel	B-Application
coordinates	I-Application
and	O
if	O
the	O
lines	O
intersected	O
outside	O
the	O
parallel	O
axes	O
it	O
can	O
be	O
translated	O
between	O
them	O
by	O
rotations	O
.	O
</s>
<s>
A	O
smooth	O
parallel	B-Application
coordinate	I-Application
plot	I-Application
is	O
achieved	O
with	O
splines	O
.	O
</s>
<s>
In	O
parallel	B-Application
coordinates	I-Application
,	O
each	O
axis	O
can	O
have	O
at	O
most	O
two	O
neighboring	O
axes	O
(	O
one	O
on	O
the	O
left	O
,	O
and	O
one	O
on	O
the	O
right	O
)	O
.	O
</s>
<s>
A	O
prototype	O
of	O
this	O
visualization	O
is	O
available	O
as	O
extension	O
to	O
the	O
data	B-Application
mining	I-Application
software	O
ELKI	B-Language
.	O
</s>
<s>
While	O
there	O
are	O
a	O
large	O
number	O
of	O
papers	O
about	O
parallel	B-Application
coordinates	I-Application
,	O
there	O
are	O
only	O
few	O
notable	O
software	O
publicly	O
available	O
to	O
convert	O
databases	O
into	O
parallel	B-Application
coordinates	I-Application
graphics	O
.	O
</s>
<s>
Notable	O
software	O
are	O
ELKI	B-Language
,	O
GGobi	B-Application
,	O
Mondrian	B-Application
,	O
Orange	B-Application
and	O
ROOT	O
.	O
</s>
<s>
Libraries	O
include	O
Protovis.js	O
,	O
D3.js	B-Language
provides	O
basic	O
examples	O
.	O
</s>
<s>
D3.Parcoords.js	O
(	O
a	O
D3-based	O
library	O
)	O
specifically	O
dedicated	O
to	O
parallel	B-Application
coordinates	I-Application
graphic	O
creation	O
has	O
also	O
been	O
published	O
.	O
</s>
<s>
The	O
Python	B-Language
data	O
structure	O
and	O
analysis	O
library	O
Pandas	B-Application
implements	O
parallel	B-Application
coordinates	I-Application
plotting	O
,	O
using	O
the	O
plotting	O
library	O
matplotlib	B-Language
.	O
</s>
