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{
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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