5-2 Special mapping equations 193
5-247 The Lagrange projection
The normal general perspective mapping of the sphere reduces to the Polar Stereographic Projection
(UPS) if we specialize H = 0 or D = R. A special variant already mentioned is achieved if we choose
the South Pole as the perspective center O
∗ (alternatively, the North Pole) and a projection plane
to coincide with the equatorial frame P
2
O . In Figs. 5.25 and 5.26, such a central perspective mapping
is illustrated. In Box 5.17, the characteristics of such a projection (namely, (i) the parameterized
mapping, (ii) the left principal stretches of the left Cauchy–Green eigenspace, (iii) the left maximal
angular shear, and (iv) the inverse parameterized mapping) are collected.
Historical
aside.
Such a central perspective mapping particularly is associated with the name of J. L. Lagrange
(1736–1813). Note that his works on map projections are published in A. Wangerin,
¨
Uber Kartenprojectionen, Abhandlungen von J. L. Lagrange and C. F. Gauss (Verlag
W. Engelmann, Leipzig 1894).
The basic results of the Lagrange projection of the sphere S
2
R to the equatorial plane are collected in
Lemma 5.6.
Lemma 5.6 (Special perspective mapping of the sphere: the Lagrange projection).
The Lagrange projection of the sphere S
2
R to the equatorial plane is parameterized by
x = R tan
∆
2
cos Λ = R tan
π
4
−
Φ
2
cos Λ ,
y = R tan
∆
2
sin Λ = R tan
π
4
−
Φ
2
sin Λ ,
(5.98)
subject to the left Cauchy–Green eigenspace
left CG eigenspace =
E Λ
1
2 cos 2 ∆
2
, E Φ
1
2 cos 2 ∆
2
.
(5.99)
The Lagrange projection is conformal.
End of Lemma.
Note that the northern hemisphere is conformally mapped from the southern projective center S = O
∗ ,
while the southern hemisphere is conformally mapped from the northern projective center N = O
∗ ,
namely generating northern and southern points within a circle of radius R. The union of these two
charts generates a minimal atlas of conformal type.
Question.
Question: “What makes the Lagrange projection particularly useful when compared with
the Universal Stereographic Projection (UPS)?” Answer: “It is the different factor of conformality Λ 1 = Λ 2 : the left principal stretches of the Lagrange projection are half of the left
principal stretches of the UPS: Λ 1 (Lagrange) = Λ 2 (Lagrange) =
1
2 Λ 1 (UPS) =
1
2 Λ 2 (UPS).”
5-247 The Lagrange projection
The normal general perspective mapping of the sphere reduces to the Polar Stereographic Projection
(UPS) if we specialize H = 0 or D = R. A special variant already mentioned is achieved if we choose
the South Pole as the perspective center O
∗ (alternatively, the North Pole) and a projection plane
to coincide with the equatorial frame P
2
O . In Figs. 5.25 and 5.26, such a central perspective mapping
is illustrated. In Box 5.17, the characteristics of such a projection (namely, (i) the parameterized
mapping, (ii) the left principal stretches of the left Cauchy–Green eigenspace, (iii) the left maximal
angular shear, and (iv) the inverse parameterized mapping) are collected.
Historical
aside.
Such a central perspective mapping particularly is associated with the name of J. L. Lagrange
(1736–1813). Note that his works on map projections are published in A. Wangerin,
¨
Uber Kartenprojectionen, Abhandlungen von J. L. Lagrange and C. F. Gauss (Verlag
W. Engelmann, Leipzig 1894).
The basic results of the Lagrange projection of the sphere S
2
R to the equatorial plane are collected in
Lemma 5.6.
Lemma 5.6 (Special perspective mapping of the sphere: the Lagrange projection).
The Lagrange projection of the sphere S
2
R to the equatorial plane is parameterized by
x = R tan
∆
2
cos Λ = R tan
π
4
−
Φ
2
cos Λ ,
y = R tan
∆
2
sin Λ = R tan
π
4
−
Φ
2
sin Λ ,
(5.98)
subject to the left Cauchy–Green eigenspace
left CG eigenspace =
E Λ
1
2 cos 2 ∆
2
, E Φ
1
2 cos 2 ∆
2
.
(5.99)
The Lagrange projection is conformal.
End of Lemma.
Note that the northern hemisphere is conformally mapped from the southern projective center S = O
∗ ,
while the southern hemisphere is conformally mapped from the northern projective center N = O
∗ ,
namely generating northern and southern points within a circle of radius R. The union of these two
charts generates a minimal atlas of conformal type.
Question.
Question: “What makes the Lagrange projection particularly useful when compared with
the Universal Stereographic Projection (UPS)?” Answer: “It is the different factor of conformality Λ 1 = Λ 2 : the left principal stretches of the Lagrange projection are half of the left
principal stretches of the UPS: Λ 1 (Lagrange) = Λ 2 (Lagrange) =
1
2 Λ 1 (UPS) =
1
2 Λ 2 (UPS).”