1-11 Areal distortion 75
Box 1.39 (Areal distortion, representations of its multiplicative and additive deformation measures).
M
2
l → M
2
r :
(i)
p
det[G l ]dU ∧ dV =
p
det[G r ]du ∧ dv ,
p
det[G r ]du ∧ dv =
p
det[G l ]dU ∧ dV ;
p
det[G l ]dU ∧ dV =
p
det[G r − 2E r ]du ∧ dv ,
p
det[G r ]du ∧ dv =
p
det[G l − 2E l ]dU ∧ dV ;
p
det[G l ]dU ∧ dV =
1
det[F r ]
λ 1 λ 2 du ∧ dv ,
p
det[G r ]du ∧ dv =
1
det[F l ]
Λ 1 Λ 2 dU ∧ dV ;
(1.269)
˘ M
2
r , g µν
¯
=
˘ R
2 , δ µν
¯ ⇒
p
det[G l ]dU ∧ dV = λ 1 λ 2 du ∧ dv .
(ii)
Φ
2
l =
q
det[C l G
−1
l ] = Λ 1 Λ 2 ,
Φ
2
r =
q
det[C r G
−1
r ] = λ 1 λ 2 .
(1.270)
(iii)
S lr =
“ p
det[C l ] −
√
G l
”
dU ∧ dV ,
S rl =
“ p
det[C r ] −
√
G r
”
du ∧ dv ;
S lr = (Λ 1 Λ 2 − 1)
1
det[F l ]
dU ∧ dV ,
S rl = (λ 1 λ 2 − 1)
1
det[F r ]
du ∧ dv ;
(1.271)
˘ M
2
r , g µν
¯
=
˘ R
2 , δ µν
¯ ⇒ S rl = (λ 1 λ 2 − 1) du ∧ dv .
To give you again some breathing time, please enjoy Fig. 1.29, which presents the “quasicordiform”
Bonne-pseudo-conic projection.
Fig. 1.29. Bonne-pseudo-conic projection, with shorelines of a spherical Earth, equidistant mapping of the lineof-contact of a circular cone, “quasicordiform”. Tissot ellipses of distortion. (According to Rigobert Werner).
Box 1.39 (Areal distortion, representations of its multiplicative and additive deformation measures).
M
2
l → M
2
r :
(i)
p
det[G l ]dU ∧ dV =
p
det[G r ]du ∧ dv ,
p
det[G r ]du ∧ dv =
p
det[G l ]dU ∧ dV ;
p
det[G l ]dU ∧ dV =
p
det[G r − 2E r ]du ∧ dv ,
p
det[G r ]du ∧ dv =
p
det[G l − 2E l ]dU ∧ dV ;
p
det[G l ]dU ∧ dV =
1
det[F r ]
λ 1 λ 2 du ∧ dv ,
p
det[G r ]du ∧ dv =
1
det[F l ]
Λ 1 Λ 2 dU ∧ dV ;
(1.269)
˘ M
2
r , g µν
¯
=
˘ R
2 , δ µν
¯ ⇒
p
det[G l ]dU ∧ dV = λ 1 λ 2 du ∧ dv .
(ii)
Φ
2
l =
q
det[C l G
−1
l ] = Λ 1 Λ 2 ,
Φ
2
r =
q
det[C r G
−1
r ] = λ 1 λ 2 .
(1.270)
(iii)
S lr =
“ p
det[C l ] −
√
G l
”
dU ∧ dV ,
S rl =
“ p
det[C r ] −
√
G r
”
du ∧ dv ;
S lr = (Λ 1 Λ 2 − 1)
1
det[F l ]
dU ∧ dV ,
S rl = (λ 1 λ 2 − 1)
1
det[F r ]
du ∧ dv ;
(1.271)
˘ M
2
r , g µν
¯
=
˘ R
2 , δ µν
¯ ⇒ S rl = (λ 1 λ 2 − 1) du ∧ dv .
To give you again some breathing time, please enjoy Fig. 1.29, which presents the “quasicordiform”
Bonne-pseudo-conic projection.
Fig. 1.29. Bonne-pseudo-conic projection, with shorelines of a spherical Earth, equidistant mapping of the lineof-contact of a circular cone, “quasicordiform”. Tissot ellipses of distortion. (According to Rigobert Werner).