Geoscience Reference
In-Depth Information
z
P
p
h
n
Q
Q
X
x
'
x, plane
N
Fig. 5.5. Ellipsoidal and rectangular coordinates
where
N
is the normal radius of curvature (2-149):
a
2
a
2
cos
2
ϕ
+
b
2
sin
2
ϕ
N
=
.
(5-21)
These equations are known from ellipsoidal geometry; it may also be verified
by direct substitution that a point with
xyz
-coordinates (5-20) satisfies the
equation of the ellipsoid (5-19) and so lies on the ellipsoid. The components
of the unit normal vector
n
are
n
=
cos
ϕ
cos
λ,
cos
ϕ
sin
λ,
sin
ϕ
,
(5-22)
because
ϕ
is the angle between the ellipsoidal normal and the
xy
-plane,
which is the equatorial plane (Fig. 5.5). Now let the coordinates of a point
P
outside the ellipsoid form the vector
X
=[
X, Y , Z
] ;
(5-23)
similarly we have, for the coordinates of the point
Q
on the ellipsoid,
x
=[
x, y, z
]
.
(5-24)
From Fig. 5.5, we read
X
=
x
+
h
n
,
(5-25)
