8-2 Special mapping equations 229
Box 8.4 (Two ellipsoidal coordinate systems parameterizing the oblate ellipsoid-of-revolution).
Oblate ellipsoid-of-revolution:
E A 1 ,A 1 ,A 2 :=
j
X ∈ R
3 X
2 + Y
2
A
2
1
+
Z
2
A
2
2
= 1, A 1 > A 2 ∈ R
+
ff
.
(8.28)
Ansatz 1 (surface normal coordinates):
Ansatz 2 (circle reduced coordinates):
X =
A 1 cos Φ cos Λ
p
1 − E 2 sin
2 Φ
,
X = A 1 cos Φ
∗ cos Λ ,
Y =
A 1 cos Φ sin Λ
p
1 − E 2 sin
2 Φ
,
Y = A 1 cos Φ
∗ sin Λ ,
Z =
A 1 (1 − E
2 ) sin Φ
p
1 − E 2 sin
2 Φ
,
Z = A 2 sin Φ
∗ ,
(8.29)
subject to
E
2 :=
A
2
1 − A
2
2
A
2
1
and
A 2
A 1
=
p
1 − E 2 .
(8.30)
Direct and inverse transformation of surface normal latitude Φ to circle reduced latitude Φ
∗ :
tan Φ =
1
1 − E 2
Z
√
X 2 + Y 2
versus tan Φ
∗ =
A 1
A 2
Z
√
X 2 + Y 2
=
1
√
1 − E 2
Z
√
X 2 + Y 2
,
tan Φ =
1
√
1 − E 2
tan Φ
∗
versus tan Φ
∗ =
p
1 − E 2 tan Φ ,
cos Φ =
√
1 − E 2
√
1 − E 2 cos 2 Φ ∗
cos Φ
∗
versus cos Φ
∗ =
1
p
1 − E 2 sin
2 Φ
cos Φ ,
sin Φ =
1
√
1 − E 2 cos 2 Φ ∗
sin Φ
∗
versus sin Φ
∗ =
√
1 − E 2
p
1 − E 2 sin
2 Φ
sin Φ .
(8.31)
In most practical cases, where we are aiming at an azimuthal projection of an equidistant type
of the ellipsoid-of-revolution representing the Earth, the planets, or other celestial bodies, a series
expansion of the meridian arc length has been a sufficient approximation. Accordingly, we are going
to outline the series expansion of the meridian arc length as a function of surface normal latitude Φ
or its complement, the polar distance ∆. In preparing such an series expansion, we have collected
auxiliary formulae in Corollary 8.1 to Corollary 8.7. First, we expand (1 + x)
y according to B. Taylor,
just representing the meridian arc length by x := −E
2 cos
2 ∆, y = −3/2, and |x| > 1. Second, we
represent (1 − E
2 cos
2 ∆)
3/2 in terms of powers {1, E
2 cos
2 ∆, E
4 cos
4 ∆, E
6 cos
6 ∆, . . .}. Third, we
transform the powers {cos
2 ∆, cos
4 ∆, cos
6 ∆, . . .} in terms of {1, cos 2∆, cos 4∆, cos 6∆, . . .}. Fourth,
an explicit version of the product sums is given in Corollary 8.4 to Corollary 8.6. Since the power series
are uniformly convergent, we can term-wise integrate in order to achieve the meridian arc length in
Corollary 8.7.
Box 8.4 (Two ellipsoidal coordinate systems parameterizing the oblate ellipsoid-of-revolution).
Oblate ellipsoid-of-revolution:
E A 1 ,A 1 ,A 2 :=
j
X ∈ R
3 X
2 + Y
2
A
2
1
+
Z
2
A
2
2
= 1, A 1 > A 2 ∈ R
+
ff
.
(8.28)
Ansatz 1 (surface normal coordinates):
Ansatz 2 (circle reduced coordinates):
X =
A 1 cos Φ cos Λ
p
1 − E 2 sin
2 Φ
,
X = A 1 cos Φ
∗ cos Λ ,
Y =
A 1 cos Φ sin Λ
p
1 − E 2 sin
2 Φ
,
Y = A 1 cos Φ
∗ sin Λ ,
Z =
A 1 (1 − E
2 ) sin Φ
p
1 − E 2 sin
2 Φ
,
Z = A 2 sin Φ
∗ ,
(8.29)
subject to
E
2 :=
A
2
1 − A
2
2
A
2
1
and
A 2
A 1
=
p
1 − E 2 .
(8.30)
Direct and inverse transformation of surface normal latitude Φ to circle reduced latitude Φ
∗ :
tan Φ =
1
1 − E 2
Z
√
X 2 + Y 2
versus tan Φ
∗ =
A 1
A 2
Z
√
X 2 + Y 2
=
1
√
1 − E 2
Z
√
X 2 + Y 2
,
tan Φ =
1
√
1 − E 2
tan Φ
∗
versus tan Φ
∗ =
p
1 − E 2 tan Φ ,
cos Φ =
√
1 − E 2
√
1 − E 2 cos 2 Φ ∗
cos Φ
∗
versus cos Φ
∗ =
1
p
1 − E 2 sin
2 Φ
cos Φ ,
sin Φ =
1
√
1 − E 2 cos 2 Φ ∗
sin Φ
∗
versus sin Φ
∗ =
√
1 − E 2
p
1 − E 2 sin
2 Φ
sin Φ .
(8.31)
In most practical cases, where we are aiming at an azimuthal projection of an equidistant type
of the ellipsoid-of-revolution representing the Earth, the planets, or other celestial bodies, a series
expansion of the meridian arc length has been a sufficient approximation. Accordingly, we are going
to outline the series expansion of the meridian arc length as a function of surface normal latitude Φ
or its complement, the polar distance ∆. In preparing such an series expansion, we have collected
auxiliary formulae in Corollary 8.1 to Corollary 8.7. First, we expand (1 + x)
y according to B. Taylor,
just representing the meridian arc length by x := −E
2 cos
2 ∆, y = −3/2, and |x| > 1. Second, we
represent (1 − E
2 cos
2 ∆)
3/2 in terms of powers {1, E
2 cos
2 ∆, E
4 cos
4 ∆, E
6 cos
6 ∆, . . .}. Third, we
transform the powers {cos
2 ∆, cos
4 ∆, cos
6 ∆, . . .} in terms of {1, cos 2∆, cos 4∆, cos 6∆, . . .}. Fourth,
an explicit version of the product sums is given in Corollary 8.4 to Corollary 8.6. Since the power series
are uniformly convergent, we can term-wise integrate in order to achieve the meridian arc length in
Corollary 8.7.