Generalized Hammer Projection - Map Projections: Cartographic Information Systems - page 804

we recognize the homogeneous polynomial form of ( H.117 ) as soon as we substitute x := sin Φ ∗
by ( H.118 )and y by ( H.119 ). The inverse of the univariate homogeneous polynomial ( H.120 )
represented by ( H.121 ) is computed up to degree five. Forward and backward substitution amount
to ( H.129 ) reevaluated by means of the final solution to the inversion by ( H.131 ).

ar tanh x = x + x 3

3 + x 5 5 + x 7 7 +O( x 9 )(

| x | < 1) , (H.113)

(1 + x ) − 1 =1 x + x 2 x 3 +O( x 4 )(

− − − 1 <x< 1) , (H.114)

2 E artanh( E sin Φ ∗ )= 1

1

2 sin Φ ∗ + 1 6 E 2 sin 3 Φ ∗ + 1 10 E 4 sin 5 Φ ∗ +O( E 6 ) ,

(H.115)

sin Φ ∗

2(1 − E 2 sin 2 Φ ∗ ) = 1

2 sin Φ ∗ + 1 2 E 2 sin 3 Φ ∗ + 1

2 E 4 sin 5 Φ ∗ +O( E 6 ) ,

(H.116)

sin Φ ∗

1

2(1 − E 2 sin 2 Φ ∗ ) =sin Φ ∗ + 2 2 E ar tanh( E sin Φ ∗ )+ 3 E 2 sin 3 Φ ∗

+ 3

5 E 4 sin 5 Φ ∗ +O( E 6 ) .

(H.117)

Equation ( H.65 ) can now be written as univariate special homogeneous polynomial of degree n ,
namely

x := sin Φ ∗ , (H.118)

y := c 4 1

=

sin Φ

2 E ar tanh( E sin Φ )+ E 2 sin 2 Φ ) 2(1 −

= c 4 sin Φ + 2

5 E 4 sin 5 Φ +O( E 6 ) , 3 E 2 sin 3 Φ + 3

(H.119)

y ( x )= a 11 x + a 13 x 3 + a 15 x 5 + + a 1 n x n ··· ( n odd) , (H.120)

x ( y )= b 11 y + b 13 y 3 + b 15 y 5 + + b 1 n y n

··· ( n odd) , (H.121)

subject to
a 11 =1 . (H.122)

Following Grafarend ( 1996 ), we can immediately formulate the series expansion with respect to an
upper triangular matrix R A truncated up to degree five according to ( H.123 ) subject to ( H.124 )
as a forward substitution.
⎡

⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤

y
y 3
y 5

1 a 13 a 15
01 a 35
00 1

x
x 3
x 5

x
x 3
x 5

⎣ ⎦ = ⎣ ⎦ ⎣ ⎦ + r =R A ⎣ ⎦ + r,

(H.123)

a 13 = 2

3 E 2 , 15 = 3 5 E 4 , 35 =3 a 13 =2 E 2 .

(H.124)

In contrast, the backward substitution leads to ( H.125 )or( H.126 ) .
⎡

⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤

x
x 3
x 5

1 b 13 b 15
01 b 35
00 1

y
y 3
y 5

y
y 3
y 5

⎣ ⎦ = ⎣ ⎦ ⎣ ⎦ + s =R B ⎣ ⎦ + s,

(H.125)