Graphics Reference
In-Depth Information
Parallel Coordinates: Quick Overview
14.5
Lines
14.5.1
An N-dimensional line ℓ can be described by the following N
−
linear equations:
ℓ
,
x
m
x
b
=
+
ℓ
,
x
=
m
x
+
b
ℓ
(
.
)
ℓ
i
−
,i
x
i
=
m
i
x
i
−
+
b
i
ℓ
N
−
,N
x
N
=
m
N
x
N
−
+
b
N
,
each with a pair of adjacently indexed variables. In the x
i
−
x
i
plane, the relation la-
beled ℓ
i
−
,i
, N
=
,...,N is a line; based on the line
point
-D correspondence,
this line can be represented by the point
b
i
ℓ
i
−
,i
=(
)
+(
i
−
)
,
)
)
(
.
)
(
−
m
i
(
−
m
i
is the distance between the y (or X
)and
Here the inter-axis distance is
, so i
−
X
i
−
axes. Actually, any N
−
independent equations like
ℓ
i, j
x
i
=
m
i, j
x
j
+
b
i, j
,
(
.
)
can equivalently specify the line ℓ, since Eq.
.
is the projection of ℓ onto the x
i
x
j
-Dplane and N
of these independent projections completely describe ℓ.hereis
a beautiful and very important relationship between three such points, as illustrated
in Fig.
.
(let).
−
Figure
.
.
(Let)hethreepointsℓ
i, j
, ℓ
j,k
, ℓ
i,k
are collinear for i
=
j
=
k.(Right) A line interval in
-D
