1-15 Exercise: the Armadillo double projection 93
Box 1.58 (Armadillo double projection. 1st: sphere to torus. 2nd: torus to oblique plane).
Left manifold S
2
r ,
Right manifold T
2
a,b ∼ S
1
a × S
1
b ,
left coordinates
right coordinates
(spherical longitude Λ,
(toroidal longitude λ,
(spherical latitude Φ):
(toroidal latitude φ):
S
2
r :=
T
2
a,b :=
:=
n
X ∈ R
3
˛
˛
˛X
2 + Y
2 + Z
2 − R
2 = 0,
:=
n
x ∈ R
3
˛
˛
˛
“ p
x 2 + y 2 − a
” 2
+ z
2 − b
2 = 0,
R ∈ R
+ , R > 0
o
,
a ∈ R
+ , b ∈ R
+ , b ≤ a
o
,
Φ
−1
l
: X (Λ, Φ) =
Φ
−1
r
: x(λ, φ) =
= E 1 R cos Φ cos Λ + E 2 R cos Φ sin Λ+
=e 1 (a + b cos φ) cos λ + e 2 (a + b cos φ) sin λ+
+E 3 R sin Φ .
+e 3 b sin φ .
(1.343)
1st mapping:
»
λ
φ
–
=
»
Λ/2
Φ
–
, a = b = R .
(1.344)
2nd mapping (oblique orthogonal projection).
Left manifold T
2
a,b :
Right manifold (oblique plane) P
2
O :
X = R(1 + cos Φ) cos Λ/2 ,
x = Y
,
Y = R(1 + cos Φ) sin Λ/2 ,
y = Z
;
Z = R sin Φ ;
(1.345)
2
4
X
Y
Z
3
5 = R 2 (β)
2
4
X
Y
Z
3
5
⇔
2
4
X
Y
Z
3
5 = R 2 (−β)
2
4
X
Y
Z
3
5 ,
R 2 (β) =
2
4
cos β 0 − sin β
0 1
0
sin β 0 sinβ
3
5
⇔
2
4
cos β 0 sin β
0
1 0
− sin β 0 cos β
3
5 = R
T
2 (β) = R
T
2 (−β) ;
(1.346)
x := Y = Y
, y := Z
= − sin βX + cos βZ ,
x = R(1 + cos Φ) sin Λ/2 , y = −R(1 + cos Φ) sin β cos Λ/2 + R cos β sin Φ .
(1.347)