Geography Reference
In-Depth Information
18
“Sphere to Cone”: Pseudo-Conic Projections
Mapping the sphere to a cone: pseudo-conic projections. The Stab-Werner mapping and the
Bonne mapping. Tissot indicatrix.
First, let us develop the general setup of
pseudo-conic projections
from the
sphere to a cone
.
Second, let us present special pseudo-conic mappings like the
Stab-Werner mapping
and the
Bonne mapping
including illustrations.
18-1 General Setup and Distortion Measures of Pseudo-Conic
Projections
Conic projections, polyconic projections. Mapping equations, deformation tensor. Lemma of
Vieta and postulate of equal area mapping.
In general,
pseudo-conic projections
are based upon the setting (
18.1
) if we use spherical longitude
Λ
and spherical co-latitude
Δ
=
π/
2
Φ
and polar coordinates
α
and
r
. The next extension leaves
us with the polyconic projections of type (
18.2
). Our analysis is based upon Lemma
18.1
.
−
α
=
α
(
Λ, Δ
)=
Λ
cos
Δ, r
=
r
(
Δ
)=
f
(
Δ
)
,
(18.1)
α
(
Λ, Δ
)=
g
(
Δ
)
Λ
cos
Δ, r
=
r
(
Δ
)=
f
(
Δ
)
.
(18.2)
Lemma 18.1 (Equiareal mapping).
A general mapping of any surface to the plane is equiareal or area preserving if and only if
det[C
l
]
/
det[G
l
]=1
.
(18.3)
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