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squared sine
1.5
1.0
5
4
3
0.5
2
p
1.0
0.8
1
0.6
0.4
0.0
0.2
0
0.0
zita
−
0.5
−
1.0
Figure 5.13
sin
2
α
as a function of
p
and
ζ
; its negative values determine the forbidden
volume for the skew distance representation. (
See insert for color representation of the
figure
.)
4
3
1.0
2
10
0.8
8
0.6
Z
1
6
0.4
4
zita
0.2
2
0
p
0
0.0
Z
−
1
Figure 5.14
Plot of sin 2
α
.(
See insert for color representation of the figure
.)
eq. (5.120), which corresponds to sin 2
.It is very flat when
p
exceeds 1, which
means that
p
does not greatly influence the value of
α
. Hence, in a first approx-
imation, for large
p
, eq. (5.6) represents the sum of skew distances.
α
Theorem 117 (OLS Distance as Reference)
If the distances are expressed
with respect to the OLS distances, the GeTLS error function (5.6) does not exactly
represent a sum of squared skew distances, except for
ζ
=
0
(OLS case).
Proof.
This proof is similar to the proof of Theorem 116 and uses the notation
of Figure
5.12. Using the sine theorem, it follows that
sin
(σ
−
π/
2
)
d
GeTLS
sin
(α
+
π/
2
)
d
OLS
=
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