Geoscience Reference
In-Depth Information
7.3.2.2
Dual Interpretation
The following geometric interpretation is helpful when dealing with hyperplane
location problems: A non-vertical hyperplane H
a;b
(with a
D
D
1) may be inter-
preted as point .a
1
;:::;a
D1
;b/ in
D
. Vice versa, any point v
D
.v
1
;:::;v
D
/
may be interpreted as a hyperplane. Formally, we use the following transformation.
R
Definition 7.1
T
H
.v
1
;:::;v
D
/
WD
H
v
1
;:::;v
D1
;1;v
D
T
P
.H
a
1
;:::;a
D1
;1;b
/
WD
.a
1
;:::;a
D1
;b/
It can easily be verified that
d
ver
.H
a;b
;v/
D
d
ver
.T
H
.v/;T
P
.H
a;b
//
for non-vertical hyperplanes with a
D
D
1. In particular, we obtain
D
be a point. Then
Lemma 7.3
Let
H
be a non-vertical hyperplane and
v
2
R
v
2
H
()
T
P
.H/
2
T
H
.v/:
This means that H
a;b
passes through a point v if and only if T
H
.v/ passes through
.a
1
;:::;a
D1
;b/.
In the resulting
dual space
the goal is to locate a point which minimizes the sum
of distances to a set of given hyperplanes
f
T
H
.v/
W
v
2
V
g
. In the results of the next
sections it will become clear that this is a helpful interpretation.
Figure
7.2
shows an example of the dual interpretation in
2
. We consider five
points (depicted in the left part of the figure), namely v
1
D
.0;
2
/, v
2
D
.0;1/,
R
L
4
L
3
v
2
v
1
v
3
v
5
L
1
v
4
L
2
L
5
Fig. 7.2
Left
: Five existing points and a line in primal space.
Right
: The same situation in dual
space corresponds to five lines and one point
