Ellipsoid-of-Revolution to Tangential Plane - Map Projections: Cartographic Information Systems - page 256

We depart from the representation of the meridian arc length as a function of the polar distance
Δ , the complement of the surface normal latitude Φ . Let us transform the integral kernel (which
is a function of Φ )to Φ ∗ (which is the circular reduced latitude). Such a polar coordinate is
generated by projecting a meridianal point P vertically onto a circle

A 1

of radius A 1 .Notethat
the geometrical situation is illustrated in Figs. 8.2 and 8.3 . Furthermore, note that the relations
Δ := π/ 2

S

Φ and Δ ∗ := π/ 2 Φ ∗ hold, and

− −

E 2 ) Δ
0

d Δ
(1 −E 2 cos 2 Δ ) 3 / 2

f ( Δ )= A 1 (1 −

⇔ (8.25)

1

f ( Δ ∗ )= A 1 Δ ∗

E 2 sin 2 Δ d Δ.

−

0

Φ ∗ and Φ ∗ →

The transformatio n formula e Φ → Φ , respectively, are summarized in Box 8.4 ,

originating from √ X 2 + Y 2 = A 1 cos Φ ∗ and Z = A 2 sin Φ ∗ , taking reference to the semi-major
axis A 1 and the semi-minor axis A 2 . Here, we refer to

sin Φ ∗

sin Φ =

√ 1

− E 2 cos 2 Φ ∗

or

cos Δ ∗

1

cos Δ = ,

E 2 sin 2 Δ ∗ −

and

E 2 ) A 1 (1 − A 1

E 2 sin 2 Δ ∗ ) 3 / 2 ,

√ 1

E 2 cos 2 Δ ) 3 / 2 =

E 2 (1

−

(1 −

−

and (8.26)

√ 1

− E 2

sin Δ ∗

1

sin Δ =

E 2 sin 2 Δ ∗

−

⇒

√ 1

− E 2

E 2 sin 2 Δ ∗ ) 3 / 2 cos Δ ∗ d Δ ∗ ,
⇒

cos Δ d Δ =

(1 −

√ 1 − E 2 √ 1 − E 2

d Δ = cos Δ ∗
cos Δ E 2 sin 2 Δ ∗ ) 3 / 2 d Δ ∗ =

d Δ ∗ ,

E 2 sin 2 Δ ∗

(1 − 1 −

in order to have derived

E 2 sin 2 Δ ∗ ) 3 / 2 √ 1

E 2 )

A 1 (1 − A 1 E 2

1 − E 2 sin 2 Δ ∗

−

d Δ ∗ ,

√ 1 − E 2 (1

E 2 cos 2 Δ ) 3 / 2 d Δ =

− (8.27)

(1 −

E 2 cos 2 Δ ) 3 / 2 d Δ = A 1 1 − E 2 sin 2 Δ ∗ d Δ ∗ .

A 1 (1 − E 2 )

(1 −