“Ellipsoid-of-Revolution to Cylinder”: Oblique Aspect - Map Projections: Cartographic Information Systems - page 410

The rotation around the 3 axis, we have denoted by Ω , the “right ascension of the ascending
node”, while the rotation around the intermediate 1 axis by i , the “inclination”. R 1 ( i )and R 3 ( Ω )
are orthonormal matrices such that ( 16.21 )holds.

⎡ ⎤

cos Ω sin Ω 0
− sin Ω cos i +cos Ω cos i sin i
+sin Ω sin i − cos Ω sin i cos i

⎣ ⎦ ∈ R

3 × 3 .

R 1 ( i ) R 3 ( Ω )= (16.21)

Accordingly, ( 16.22 ) is a representation of the placement vector x in the orthonormal bases
{ e 1 , e 2 , e 3 , O} and { e 1 , e 2 , e 3 , O} , respectively. Note that ( 16.23 )and( 16.24 )hold.

3 3

e i x i , e i x i =

x = (16.22)

i =1 i =1

x 1 = x 1 cos Ω x 2 sin Ω cos i + x 3 sin Ω sin i,

x 2 = x 1 sin Ω + x 2 cos Ω cos i

−

x 3 cos Ω sin i, − (16.23)

x 3 = x 2 sin i + x 3 cos i,
x 1 =+ x 1 cos Ω + x 2 sin Ω =: x ,
x 2 = −x 1 sin Ω cos i + x 2 cos Ω cos i + x 3 sin i =: y ,

(16.24)

x 3 =+ x 1 sin Ω sin i − x 2 cos Ω sin i + x 3 cos i =: z .

A 1 ,A 2 2

O

Corollary 16.1 (Intersection of E and L ).

2
A 1 ,A 2

2
O

The intersection of the ellipsoid-of-revolution E and the central oblique plane L (two-

) is th e ellipse ( 16.25 ) of semi-major axis A 1 = A 1

dimensional linear manifold thr ough th e origin O

and semi-minor axis A 2 = A 1 √ 1 E 2 / √ 1 − − E 2 cos 2 i .

1
A 1 ,A 2

E :=

√ 1

:= x

E 2 cos 2 i ,A 1 >A 2 .

A 1 + y 2

x 2

− E 2

2

=1 , A 1 = A 1 ,A 2 = A 1

∈ R | √ 1 (16.25)

A 2

−

End of Corollary.

Proof.

[( x 1 ) 2 +( x 2 ) 2 ] A − 1 +( 16.23 )
⇒ (16.26)

[( x 1 ) 2 +( x 2 ) 2 ] A − 1 = A − 1 [ x 2 + y 2 cos 2 i + z 2 sin 2 i 2 y z sin i cos i ] , −

[( x 3 ) 2 ] A − 2 +( 16.23 )
⇒

(16.27)

[( x 3 ) 2 ] A − 2 = A − 2 [ y 2 sin 2 i + z 2 cos 2 i ] ,
if x =0 , then