18-1 General setup and distortion measures of pseudo-conic projections 397
Proof.
Λ
2
S
+ =
=
1
2
tr
C l G
−1
l
+
tr
C l G
−1
l
2 − 4det
C l G
−1
l
,
Λ
2
S
− =
=
1
2
tr
C l G
−1
l
−
tr
C l G
−1
l
2 − 4det
C l G
−1
l
,
(18.15)
det [C l ] =
=
1
4
(f
2 )
2 h
2 =
=
1
4
(f
2 )
2 cos
2 ∆ g
2 (∆) =
=
1
4
(f
2 )
2 cos
2 ∆ 4R
4 tan
2 ∆
(f 2 )
2 =
= R
4 sin
2 ∆ .
(18.16)
End of Proof.
Proof.
tr
C l G
−1
l
=
= c 11 G
−1
11 + c 22 G
−1
22 =
= f
2 h
2
R
2 sin
2 ∆
−1 +
f
2 h
2 Λ
2 + f
2
R
2
−1 =
=
1
R 2 sin
2 ∆
f
2 g
2 cos
2 ∆ +
1
R 2
f
2
g
cos ∆ − g sin ∆
2 Λ
2 + f
2
=
= 4R
2
f
2
(f 2 )
2 +
+
1
R 2
f
2
2R
2 cos ∆(1 + tan
2 ∆)
f 2
− 2R
2 sin ∆
(f 2 )
2
f
2
− 2R
2 sin ∆ tan ∆
f 2
2
Λ
2 + f
2
,
(18.17)
det [G l ] = R
4 sin
2 ∆ .
(18.18)
End of Proof.