Geoscience Reference
In-Depth Information
Assume a trapezoidal map sheet at 1:1,000,000 scale, the difference in the south
and north bounding latitude is B 2
4 and the difference in the east and west
B 1
6 ; substituting into the above equation yields the area of
bounding longitude is l
this trapezoid:
15 A 0 sin2 cosB m
b 2
ˀ
B 0 sin6 cos3B m þ
C 0 sin10 cos5B m
D 0 sin14 cos7B m
P
E 0 sin18 cos9B m :
þ
90 , and multiply by 2, one obtains
the formula for total surface area of the ellipsoid:
In ( 5.47 ), if we put l
360 , B 1
0 , B 2
b 2 A 0
B 0
C 0
D 0
E 0
ʣ ᄐ
4
ˀ
þ
þ
þ
þ
þ
2
3
2
3 e 2
3
5 e 4
4
7 e 6
5
9 e 8
ð
5
:
48
Þ
4
5 :
b 2
4
ˀ
1
þ
þ
þ
þ
þ
The total area of GRS80 Ellipsoid is approximately 510,065,597 km 2 . It thus
follows that the total surface area of the Earth is about 5.1
10 8 km 2 .
5.4.4 Reciprocal Normal Sections
Let us first consider the normal sections formed by reciprocal observations from
two points on the surface of the ellipsoid. As shown in Fig. 5.18 , A and B are two
points on the surface of the ellipsoid. Their normals to the ellipsoid AK a and BK b
and the corresponding plumb lines are assumed to be coincident at the two points. If
A and B are the observation points, then the vertical plane will be the normal section
plane. The instrument is set up at A and then aimed at point B defining the normal
section by marking a point a; that is, AaB. The normal section AaB from point A to
B is formed by the intersection of a plane containing the normal at point A and that
passes through point B (the vertical plane AK a B) with the surface of the ellipsoid.
Similarly, the instrument set up at B and A is sighted defining the normal section
marking a point b; that is, BbA. The normal section BbA from point B to A is formed
by the intersection of the sighting plane BK b A with the surface of the ellipsoid. The
normal sections originating at points A and B, and terminating at B and A, respec-
tively, namely AaB and BbA, generally do not coincide. They are termed the
reciprocal normal sections between points A and B.
We can imagine that if the surface normals at two points A and B are in the same
plane, then the two vertical planes at one point and passing through the other point
in the reciprocal observations coincide. Only one normal section is formed. Other-
wise, the two vertical planes do not coincide and at any of the points there are two
Search WWH ::




Custom Search