“Sphere to Tangential Plane”: Transverse Aspect - Map Projections: Cartographic Information Systems - page 240

Fig. 6.4 that meridians, except the central meridian, are complex curves unequally spaced at the
equator. Spacing decreases with increasing distance from the central meridian. Parallels, except
the equator which is a straight line, are as well complex curves. Distortions increase radially from
the point-of-contact which is mapped isometrically, i.e. free from any distortion.

Lemma 6.2 (Transverse equal area mapping of the sphere to a tangential plane at the meta-North
Pole Λ 0

[0 ◦ , 360 ◦ ] ,Φ 0 =0 ◦ ).

∈

x =2 R sin π

cos A, y =2 R sin π sin A

B
2

B
2

4 − 4 −

(6.13)

subject to
tan A = sin( Λ − Λ 0 )
−

, sin B =cos Φ cos( Λ − Λ 0 ) .

(6.14)

tan Φ

The left principal stretches and left Cauchy-Green eigenspace are specified through

2 , Λ 2 =cos π

,

1 B

2

cos 4 −

Λ 1 =

4 −

B

(6.15)

left CG eigenspace = E A

.

2 , E B cos π

1 B

2

cos 4 −

4 −

B

End of Lemma.

Fig. 6.4. Mapping the sphere to a tangential plane: transverse aspect, equal area mapping. Point-of-contact:
meta-North Pole at Λ 0 =20 ◦ ,Φ 0 =0 ◦