Geography Reference
In-Depth Information
{
are called surface normal longitude and surface normal latitude , respectively. The orthonor-
mal basis vector with respect to the origin
l, b
}
O
spans a three-dimensional Euclidean space.
End of Definition.
Question: “How can we find with given parameterized
structure of the surface normal vector
ν
( l, b )thesetof
of the embedding of
the ellipsoid-of-revolution f ( x, y, z ):=( x 2 + y 2 ) /A 1 +
z 2 /A 2 ?”Answer: “Starting from the gradient grad of the
of the gradient function f ( x, y, z ). this set of functions
is immediately derived as shown by the calculations that
follow.”
The surface normal vector of an algebraic surface (“polynom representation”) has the representa-
tion ( J.26 ), where
functions
{
x ( l, b ) ,y ( l, b ) ,z ( l, b )
}
is identified as the l 2 norm of Euclidean length of the gradient
function f ( x, y, z ). In detail, we compute ( J.27 ).
grad f ( x, y, z )
grad f ( x, y, z )
grad f ( x, y, z ) 2 ,
ν
( x, y, z ); =
(J.26)
2 x/A 1
2 y/A 1
2 z/A 2
,
grad f ( x, y, z )=[ e 1 , e 2 , e 3 ]
(J.27)
= 2 A 2 ( x 2 + y 2 )+ A 1 z 2
A 1 A 2
grad f ( x, y, z )
.
We need the relative eccentricity of the ellipsoid-of-revolution ( J.28 ) for representing finally the
surface normal vector.
E 2 := A 1 A 2
A 1
1
E 2 =: A 1
or
A 2 .
(J.28)
1
2
A 1 ,A 2
Lemma J.2 (
E
surface normal vector).
Let f ( x, y, z ):=( x 2 + y 2 ) /A 1 + z 2 /A 2 1 be a polynomial representation of the ellipsoid-of-
revolution. Then ( J.29 )-( J.31 ) are Cartesian forms of the surface normal vector.
ν ( x, y, z ):= grad f ( x, y, z )
,
(J.29)
grad f ( xyz )
2
2 x/A 1
2 y/A 1
2 z/A 1
A 1 A 2
ν
( x, y, z ):=[ e 1 , e 2 , e 3 ]
A 2 ( x 2 + y 2 )+ a 1 Z 2 ,
(J.30)
x 2 + y 2 + z 2 / (1 −E 2 )
y
x
x 2 + y 2 + z 2 / (1 −E 2 )
z
ν ( x, y, z ):=[ e 1 , e 2 , e 3 ]
.
(J.31)
(1 −E 2 ) 2 ( x 2 + y 2 )+ z 2
End of Lemma.
 
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