“Sphere to Cone”: Pseudo-Conic Projections - Map Projections: Cartographic Information Systems - page 449

D l :=
:= diag[ Λ min ,Λ max ]= 0 . 992 10 0

,

(18.36)

0 . 007 97

F l =
= 1 . 122 42 1 . 113 55

.

(18.37)

− 0 . 704 30 0 . 709 91

With the help of the left Jacobi matrix
J l = D Λ xD Φ x

= −

,

0 . 062 10 − 0 . 996 73

(18.38)

D Λ yD Φ y 0 . 629 42 − 0 . 082 38

Λ =6 ◦ 06 E ,Φ =50 ◦ 46 N

the transformation left-to-right is easily performed as
F r =J l F l D − l =J l F l D r = 0 . 770 59 0 . 637 33

,

(18.39)

− 0 . 637 33 0 . 770 59

and
F r C r F r =D r , F r G r F r =F r F r =I 2 . (18.40)

Indeed , F r is an orthonormal matrix.

End of Exercise.

Exercise 18.2 (Bonne projection).

Until recently, the pseudo-conic equal area Bonne projection (R. Bonne 1727-1795) was frequently
used for atlas maps of continents . The mapping equations are specified through polar coordinates
α and r or Cartesian coordinates x and y as follows. ( Λ and Φ describe spherical longitude and
spherical latitude. Φ 0 =60 ◦ N is the parallel circle which is mapped isometrically, i.e. without any
distortion.)

cos Φ

α = Λ

Φ +cot Φ 0 , r = R ( Φ 0

− Φ )+ R cot Φ 0 , (18.41)

Φ 0 −

Φ +cot Φ 0 )cos Λ

,

cos Φ

x = R ( Φ 0 −

Φ 0 − Φ +cot Φ 0

(18.42)

y = R ( Φ 0 − Φ +cot Φ 0 )sin Λ

.

cos Φ

Φ 0 − Φ +cot Φ 0

(i) Prove analytically that the Bonne projection is equal area and (ii) determine the numerical
values of the elements of the Tissot indicatrix (Tissot ellipse: minor distortions and major distor-
tions, and coordinates of the corresponding eigendirections in the map) for the point Alexandria,
Egypt ( Λ =29 ◦ 55 E ,Φ =31 ◦ 13 N).