Geoscience Reference
In-Depth Information
Fig. 6.16 Relationship
between s and D
P
¢
2
P
¢
1
From the definition of the scale factor, we have:
ds
ﾼ
mdS
:
Integrating produces:
ð
P
2
ð
S
s
ﾼ
mdS
ﾼ
mdS
:
ð
6
:
67
Þ
P
1
0
In practice, sometimes the integrals of some functions are hard to obtain, and in
such cases we can approximate a definite integral according to the desired compu-
tational accuracy. In (
6.67
), m changes with the position of the point. When the
projection area is not large, this change is slow. For instance, when y
300 km, and
the difference in latitude between points P
1
and P
2
reaches 1
, the difference in
scale between the two points is less than 4
ﾼ
10
7
. Therefore, by approximating
integrals, we can obtain satisfactory accuracy. Now, we shall find the solution of
(
6.67
) according to Simpson's rule for numerical integration.
Simpson's rule for approximate integration uses parabolic arcs to replace the area
bounded by curve y
f(x) over an interval [x
1
, x
2
] so as to find the definite integral.
This parabola passes through three points, i.e., y
1
ﾼ
ﾼ
f(x
1
), y
2
ﾼ
f(x
2
), y
0
ﾼ
f(x
0
),
x
2
2
, dividing the interval [x
1
, x
2
] into two subintervals. It is easy to
obtain the definite integral of the parabola on the interval [x
1
, x
2
], which is
x
2
x
1
þ
with x
0
ﾼ
ð
x
1
Þ
. Taking the definite integral of this parabola as the approx-
imation to the integral of f(x), we have:
ð
y
1
þ
4y
0
þ
y
2
Þ
6
ð
x
2
ð
x
2
x
1
Þ
fx
ðÞ
dx
ﾼ
ð
y
1
þ
4y
0
þ
y
2
Þ:
6
x
1
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