5-1 General mapping equations 163
In Box 5.1, the ID card of the sphere S
2
R is summarized. According to the above considerations, F
(with elements a, b, c, d) is the Frobenius matrix, G (with elements e, f, g) is the Gauss matrix, H
(with elements l, m, n) is the Hesse matrix, J is the Jacobi matrix, and K is the curvature matrix,
leading to the mean curvature h and to the Gaussian curvature k, and
G = J
∗ J , H =
G 3
∂
2 X
∂U K ∂U L
=
G 3
X KL
, J =
∂X
J
∂U K
,
K = −H G
−1 , h = −
1
2
tr[K] , k = det[K] .
(5.2)
Box 5.1 (ID card of the sphere S
2
R ).
Spherical coordinates (1st chart: Λ, Φ):
{Λ, Φ, R} → {X, Y, Z} :
X (Λ, Φ, R) = E 1 R cos Φ cos Λ + E 2 R cos Φ sin Λ + E 3 R sin Φ ∈
˘ R
3 , I 3
¯
;
{X, Y, Z} → {Λ, Φ, R} :
Λ(X ) = arctan
Y
X
+ 180
◦
»
−
1
2
sgnY −
1
2
sgnY sgnX + 1
–
, Φ(X ) = arctan
Z
√
X 2 + Y 2
,
R =
p
X 2 + Y 2 + Z 2 .
(5.3)
Matrices F, G, H, J, K, and I (elements: a, b, c, d; e, f, g; l, m, n):
F =
» a b
c d
–
=
2
6
6
4
1
R cos Φ
0
0
1
R
3
7
7
5 =
2
6
6
4
1
√
G 11
0
0
1
√
G 22
3
7
7
5 ∈ R
2×2 ,
(5.4)
G =
» e f
f g
–
=
"
R
2 cos
2 Φ 0
0
R
2
#
∈ R
2×2 , H =
» l m
m n
–
=
"
−R cos
2 Φ 0
0
−R
#
∈ R
2×2 ,
(5.5)
J =
2
4
−R cos Φ sin Λ −R sin Φ cos Λ
+R cos Φ cos Λ −R sin Φ sin Λ
0
R cos Φ
3
5 ∈ R
3×2 , K =
2
6
6
4
1
R
0
0
1
R
3
7
7
5 ∈ R
2×2 ,
(5.6)
h = −
1
R
, k =
1
R 2 , I = I 2 =
»
1 0
0 1
–
∈ R
2×2 .
(5.7)
Christoffel symbols:
j
1
1 1
ff
=
j
1
2 2
ff
=
j
2
1 2
ff
=
j
2
2 2
ff
= 0 ,
j
1
1 2
ff
= − tan Φ ,
j
2
1 1
ff
= sin Φ cos Φ =
1
2
sin 2Φ . (5.8)
5-1 General mapping equations
Setting up general equations of the mapping “sphere to plane”: the azimuthal projection in the normal
aspect (polar aspect).
There are two basic postulates which govern the setup of general equations of mapping the sphere S
2
R
of radius R to a tangential plane T X 0 S
2
R , which is attached to a point X 0 ∈ T S
2
R . Let the tangential
plane be covered by polar coordinates {α, r}. Then these postulates read as follows.
In Box 5.1, the ID card of the sphere S
2
R is summarized. According to the above considerations, F
(with elements a, b, c, d) is the Frobenius matrix, G (with elements e, f, g) is the Gauss matrix, H
(with elements l, m, n) is the Hesse matrix, J is the Jacobi matrix, and K is the curvature matrix,
leading to the mean curvature h and to the Gaussian curvature k, and
G = J
∗ J , H =
G 3
∂
2 X
∂U K ∂U L
=
G 3
X KL
, J =
∂X
J
∂U K
,
K = −H G
−1 , h = −
1
2
tr[K] , k = det[K] .
(5.2)
Box 5.1 (ID card of the sphere S
2
R ).
Spherical coordinates (1st chart: Λ, Φ):
{Λ, Φ, R} → {X, Y, Z} :
X (Λ, Φ, R) = E 1 R cos Φ cos Λ + E 2 R cos Φ sin Λ + E 3 R sin Φ ∈
˘ R
3 , I 3
¯
;
{X, Y, Z} → {Λ, Φ, R} :
Λ(X ) = arctan
Y
X
+ 180
◦
»
−
1
2
sgnY −
1
2
sgnY sgnX + 1
–
, Φ(X ) = arctan
Z
√
X 2 + Y 2
,
R =
p
X 2 + Y 2 + Z 2 .
(5.3)
Matrices F, G, H, J, K, and I (elements: a, b, c, d; e, f, g; l, m, n):
F =
» a b
c d
–
=
2
6
6
4
1
R cos Φ
0
0
1
R
3
7
7
5 =
2
6
6
4
1
√
G 11
0
0
1
√
G 22
3
7
7
5 ∈ R
2×2 ,
(5.4)
G =
» e f
f g
–
=
"
R
2 cos
2 Φ 0
0
R
2
#
∈ R
2×2 , H =
» l m
m n
–
=
"
−R cos
2 Φ 0
0
−R
#
∈ R
2×2 ,
(5.5)
J =
2
4
−R cos Φ sin Λ −R sin Φ cos Λ
+R cos Φ cos Λ −R sin Φ sin Λ
0
R cos Φ
3
5 ∈ R
3×2 , K =
2
6
6
4
1
R
0
0
1
R
3
7
7
5 ∈ R
2×2 ,
(5.6)
h = −
1
R
, k =
1
R 2 , I = I 2 =
»
1 0
0 1
–
∈ R
2×2 .
(5.7)
Christoffel symbols:
j
1
1 1
ff
=
j
1
2 2
ff
=
j
2
1 2
ff
=
j
2
2 2
ff
= 0 ,
j
1
1 2
ff
= − tan Φ ,
j
2
1 1
ff
= sin Φ cos Φ =
1
2
sin 2Φ . (5.8)
5-1 General mapping equations
Setting up general equations of the mapping “sphere to plane”: the azimuthal projection in the normal
aspect (polar aspect).
There are two basic postulates which govern the setup of general equations of mapping the sphere S
2
R
of radius R to a tangential plane T X 0 S
2
R , which is attached to a point X 0 ∈ T S
2
R . Let the tangential
plane be covered by polar coordinates {α, r}. Then these postulates read as follows.