5-2 Special mapping equations 171
5-23 Equiareal mapping (Lambert projection)
Let us postulate an equiareal mapping by means of the canonical measure of areomorphism, i. e.
Λ 1 Λ 2 = 1. Such an equiareal mapping of the sphere to the tangential plane of the North Pole is
illustrated by means of Fig. 5.8 that follows after Box 5.6.
Question.
Question: “How can we construct the equiareal mapping equations?” Answer: “Following
the procedure of Box 5.6, we here depart from the general representation of Λ 1 and Λ 2 . The
postulate of an equiareal mapping leads us to the characteristic differential equation, which
we solve by separation of variables. We use the initial condition r(0) = f (0) = 2R sin ∆/2,
namely the polar coordinate r as a function of the colatitude ∆, also called polar distance.
The polar coordinate α = Λ is fixed by the postulate of an azimuthal projection. The
parameterized equiareal mapping is finally used to compute the left principal stretches,
namely Λ 1 = 1/ cos ∆/2, Λ 2 = cos ∆/2. They build up the left eigenvectors along the East
unit vector E Λ and the North unit vector E Φ (the South unit vector is E ∆ = −E Φ ).
These unit vectors are defined by E Λ := D Λ X/ D Λ X and E Φ := D Φ X/ D Φ X . In
addition, we have computed the left maximal angular shear as well as the parameterized
inverse mapping {Λ(x, y), Φ(x, y)}.”
The basic results of the equiareal azimuthal projection of the sphere to the tangential plane at the
North Pole are collected in Lemma 5.3.
Lemma 5.3 (Equiareal azimuthal projection of the sphere to the tangential plane at the North Pole).
The equiareal mapping of the sphere to the tangential plane at the North Pole is parameterized by
the two equations
x =
= 2R sin
∆
2
cos Λ =
= 2R sin
π
4
−
Φ
2
cos Λ ,
y =
= 2R sin
∆
2
sin Λ =
= 2R sin
π
4
−
Φ
2
sin Λ ,
(5.35)
subject to the left Cauchy–Green eigenspace
left CG eigenspace =
E Λ
1
cos
π
4 −
Φ
2
, E Φ cos
π
4
−
Φ
2
.
(5.36)
End of Lemma.
From the sketch that is shown in Fig. 5.8, we gain some geometric understanding of how to construct
the normal equiareal mapping by a “pair of dividers and a ruler”. The radial coordinate r = Np
coincides with the segment NP = 2R sin ∆/2, the peripheral point P within the vertical section
constitutes a rectangular triangle NSP subject to SN = 2R.
5-23 Equiareal mapping (Lambert projection)
Let us postulate an equiareal mapping by means of the canonical measure of areomorphism, i. e.
Λ 1 Λ 2 = 1. Such an equiareal mapping of the sphere to the tangential plane of the North Pole is
illustrated by means of Fig. 5.8 that follows after Box 5.6.
Question.
Question: “How can we construct the equiareal mapping equations?” Answer: “Following
the procedure of Box 5.6, we here depart from the general representation of Λ 1 and Λ 2 . The
postulate of an equiareal mapping leads us to the characteristic differential equation, which
we solve by separation of variables. We use the initial condition r(0) = f (0) = 2R sin ∆/2,
namely the polar coordinate r as a function of the colatitude ∆, also called polar distance.
The polar coordinate α = Λ is fixed by the postulate of an azimuthal projection. The
parameterized equiareal mapping is finally used to compute the left principal stretches,
namely Λ 1 = 1/ cos ∆/2, Λ 2 = cos ∆/2. They build up the left eigenvectors along the East
unit vector E Λ and the North unit vector E Φ (the South unit vector is E ∆ = −E Φ ).
These unit vectors are defined by E Λ := D Λ X/ D Λ X and E Φ := D Φ X/ D Φ X . In
addition, we have computed the left maximal angular shear as well as the parameterized
inverse mapping {Λ(x, y), Φ(x, y)}.”
The basic results of the equiareal azimuthal projection of the sphere to the tangential plane at the
North Pole are collected in Lemma 5.3.
Lemma 5.3 (Equiareal azimuthal projection of the sphere to the tangential plane at the North Pole).
The equiareal mapping of the sphere to the tangential plane at the North Pole is parameterized by
the two equations
x =
= 2R sin
∆
2
cos Λ =
= 2R sin
π
4
−
Φ
2
cos Λ ,
y =
= 2R sin
∆
2
sin Λ =
= 2R sin
π
4
−
Φ
2
sin Λ ,
(5.35)
subject to the left Cauchy–Green eigenspace
left CG eigenspace =
E Λ
1
cos
π
4 −
Φ
2
, E Φ cos
π
4
−
Φ
2
.
(5.36)
End of Lemma.
From the sketch that is shown in Fig. 5.8, we gain some geometric understanding of how to construct
the normal equiareal mapping by a “pair of dividers and a ruler”. The radial coordinate r = Np
coincides with the segment NP = 2R sin ∆/2, the peripheral point P within the vertical section
constitutes a rectangular triangle NSP subject to SN = 2R.