CALCULATION OF SMALL DISTANCES*
By C. F. RlcttT~a
ABSTRACT
TABLES are presented which facilitate the calculation of small distances between points
on the e a r t h ' s surface whose latitudes and longitudes are given, for mean latitudes from
30 ° to 40 °. The error does not exceed one-tenth of a kilometer in 500 kilometers, and m a y
be reduced by applying specified corrections.
IN WOrK relating to earthquakes local to an area under intensive seismological
and geological investigation, epicenters are usually placed to within a few
kilometers. Times of seismic waves are determined to the tenth of a second.
Distances from stations to epicenters assumed for trial are required to the
nearest kilometer; and it is desirable that the method used should be capable
of furnishing such distances to the tenth of a kilometer, in order to avoid
rounding-off errors during the calculations.
The last requirement usually causes trouble. To read distances of the order
of 300 kilometers from a map, with an accuracy of 0.1 km., means working on
an awkwardly large scale, and checking the measurements with such care that
they become very time-consuming. Computation from the coSrdinates of the
stations and epicenter is much more desirable, provided tables and other material are available for carrying through the process with reasonable speed.
The following discussion contains no new contribution, but presents wellknown material in the form which has been found most convenient at Pasadena, for use with a computing machine.
Distances up to about 5° (555 km.) are given with an error not exceeding 0.1
km. by a formula of the simple form
A2 = A x 2-~-Ay2;
Ax=AAX, A y = B A ~
(1)
Here A is the required distance in kilometers, AX and A~ are differences in
longitude and latitude between the two given points (epicenter and station),
and A and B are conversion factors from circular measure to kilometers. If
AX and A~ are in minutes of longitude and latitude, then sufficient accuracy is
provided by taking A and B as the lengths in kilometers of one minute of are
of the parallel and meridian, respectively, taken at the mean latitude between
the two points.
This method is equivalent to that known in navigation as "middle latitude
sailing." It is a close approximation to that given by Wiechert (1925). His
method, involving tables for finding factors equivalent to A and B, is reasonably rapid in practice, especially after the user is familiar with it. The definition
* Manuscript received for publication January 20, 1943.
[ 243 ]
244
BULLETIibT OF T H E S E I S M O L O G I C A L S O C I E T Y OF A M E R I C A
here given for the factors A and B leads to the preparation of simple tables for
m u c h more rapid computation.
The Smithsonian Geographical Tables (Woodward, 1894) contain d a t a from
which table 1 is excerpted and derived.
For the accuracy required, five significant figures are usually sufficient. I t is
necessary then to interpolate the kilometer equivalents given in table 1 to
a p p l y to the m e a n latitude between the two points considered. For the factor
B, applying to differences in latitude, this interpolation is simple, since the
TABLE 1
LENGTHSOF TERRESTRIALARCS, IN KILOMETERS(CLARKESPHEROID)
Mean
latitude
(~)
Degree
Arc of meridian, B
30°
31
32
33
34
35
36
37
38
39
40
110.8497
110.8669
110.8844
110.9023
110.9204
110.9388
110.9574
110.9763
110.9953
111.0145
111.0339
~inute
1.847495
1.847781
1.848073
1.848372
1.848673
1.848980
1.849290
1.849605
1.849922
1.850242
1.850562
Arc of parallel, A
Degree
96.4893
95.5073
94.4962
93.4563
92.3879
91.2913
90.1668
89.0148
87.8356
86.6296
85.3970
A/cos
Minute
Degree
Minute
1.608155
1.591788
1.574937
1.557605
1.539798
1.521522
1.502780
1.483580
1.463927
1.443827
1.423283
111.4162
111.4220
111.4279
111.4339
111.4399
111.4461
111.4523
111.4586
111.4650
111.4715
111.4779
1.856937
1.857033
1.857132
1.857231
1.857331
1.857435
1.857538
1.857643
1.857750
1.857858
1.857965
tabular differences in the second column of table 1 are v e r y regular. There
results the working table which is given as table 2.
The factor A, applying to differences in longitude, is less simple. The differences in the third and fourth columns of table 1 are not smooth enough for
easy interpolation. E a c h of these figures has accordingly been divided b y the
cosine of the corresponding latitude, which gives the quotients in the fifth and
sixth columns. The differences in these are now regular enough for simple
interpolation; this was performed for each 10' of latitude, the interpolated
values multiplied b y their corresponding cosines, and a final interpolation performed to give values of A for each minute of latitude. The final working table
is given as table 3.
Using tables 2 and 3, with the aid of a computing machine and a good table
of squares, routine calculation of distances is ,very rapid and accurate. With
the machine, it is more convenient to form the sums of squares directly instead
of referring to the table of squares, which is used only in finding A at the last
step. If no machine is available, the multiplications b y A and B are not par-
C A L C U L A T I O N OF S M A L L D I S T A N C E S
245
t i c u l a r l y o n e r o u s ; t h e s q u a r i n g s will t h e n be p e r f o r m e d w i t h t h e aid of t h e
t a b l e of squares. T h e following is a n example.
S t a t i o n , P a s a d e n a , 34 ° 08'.9 N, 118 ° 10'.3 W
T r i a l epicenter, 32 ° 00 t N, 119 ° 00' W
A t = 128~9
Ak = 49'.7
M e a n l a t i t u d e ¢ = 33 ° 04'. 45
F r o m t a b l e 2, B = 1.8484
F r o m t a b l e 3, A = 1.5562
H e n c e Ax = 77.3 km., Ay = 238.3 km.
A = ~/77.32 -t- 238.32 = 250.5 k m .
TABLE 2
COEFFICIENTS B
(Lengths in kilometers of 1' of the meridian)
Mean latitude •
29° 51'
3O 12
33
54
31 15
35
55
32 15
36
56
33 16
36
56
34 16
35
55
to
to
to
to
to
to
to
to
to
to
to
to
to
to
to
to
30° 11~
32
53
31 14
34
54
32 14
35
55
33 15
35
55
34 15
34
54
35 13
B
1.8475
76
77
78
79
80
81
82
83
84
85
86
87
88
89
1.8490
Mean latitude •
35° 14' to 35° 33~
34 to
52
35 53 to 36 11
36 12 ~o
30
31 to
49
50 to 37 08
37 09 to
27
28 to.
46
47 to 38 06
38 07 to
24
25 to
43
44 to 39 01
39 02 to
20
21 to
39
40 to
57
58 to 40 16
B
1.8491
92
93
94
95
96
97
98
1.8499
1.8500
01
02
03
04
05
1.8506
I t r e m a i n s to discuss the errors of the m e t h o d . These fall i n t o three classes:
(1) errors due to the c u r v a t u r e of the e a r t h considered as a sphere, (2) errors
due to t h e d e p a r t u r e of the Clarke spheroid (on which the geographical t a b l e s
used are based) from a sphere, a n d (3) errors due to d e p a r t u r e of t h e Clarke
spheroid from the t r u e figure of the earth, or f r o m p r e s u m a b l y b e t t e r approxim a t i o n s such as t h e I n t e r n a t i o n a l Ellipsoid.
Since errors of the first class v a n i s h w h e n the two p o i n t s are o n t h e same
m e r i d i a n (hk = 0), while those of the o t h e r classes are t h e n a t m a x i m u m , a
direct c o m p a r i s o n is of interest. T h e following figures are t y p i c a l :
L e n g t h of arc of t h e m e r i d i a n b e t w e e n l a t i t u d e s 31 ° 30' a n d 36 ° 30P:
1) m u l t i p l y i n g b y 300 the l e n g t h g i v e n for 1t of t h e m e r i d i a n a t 34 ° i n
t a b l e 2, 554.61 km.
TABLE 3
COEFFICIENTsAFoRGI~N MEANLATITUDE~ ( L e n g t h s i n k i l o m e t e r s o f l ' o f t h e p a r a l l e ! )
00'
01
02
03
04
05
06
07
08
09
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
30°
31°
32°
33°
34°
35°
36°
37°
38°
39°
1.6082
79
76
74
71
1.6068
65
63
60
57
1.6055
52
49
46
44
1.6041
38
36
33
30
1.6028
25
22
19
17
1.6014
11
08
06
03
1.6000
1.5998
95
92
89
1.5987
84
81
78
76
1.5973
70
68
65
62
1.5959
57
54
51
48
1.5946
43
40
37
34
1.5932
29
26
23
20
1.5918
1.5918
15
12
10
07
1.5904
1.5901
1.5898
96
93
1.5890
87
85
82
79
1.5876
73
71
68
65
1.5862
59
57
54
51
1.5848
45
43
40
37
1.5834
31
29
26
23
1.5820
17
15
11
09
1.5806
03
1.5800
1.5798
95
1.5792
89
86
83
81
1.5778
75
72
69
66
1.5764
61
58
55
52
1.5749
1.5749
47
44
41
38
1.5735
32
29
27
24
1.5721
18
15
12
09
1.5706
04
1.5,701
1,5698
95
1.5692
89
86
83
81
1.5678
75
72
69
66
1.5663
60
58
55
52
1.5649
46
43
40
37
1.5634
31
29
26
23
1.5620
17
14
11
08
1.5605
1.5602
1.5599
97
94
1.5591
88
85
82
79
1.5576
1.5576
73
70
67
64
1.5561
58
56
53
50
1.5547
44
41
38
35
1.5532
29
26
23
20
1.5517
14
11
08
05
1.5502
1.5499
96
93
91
1.5488
85
82
79
76
1.5473
70
67
64
61
1.5458
55
52
49
46
1.5443
40
37
34
31
1.5428
24
22
19
16
1.5413
10
07
04
1.5401
1.5398
1.5398
95
92
89
86
1.5383
80
77
74
71
1.5368
65
62
59
56
1.5353
50
47
44
41
1.5338
35
32
28
25
1.5322
19
16
13
10
1.5307
04
1.5301
1.5298
95
1.5292
89
86
83
80
1.5277
74
71
67
64
1.5261
58
55
52
49
1.5246
43
40
37
34
1.5231
28
24
21
18
1.5215
1.5215
12
09
06
03
1.5200
1.5197
94
90
87
1.5184
81
78
75
72
1.5169
66
63
60
56
1.5153
50
47
44
41
1.5138
35
31
28
25
1.5122
19
16
13
10
1.5106
03
1.5100
1.5097
94
1.5091
88
85
81
78
1.5075
72
69
66
63
1.5059
56
53
50
47
1.5044
40
37
34
31
1.5028
1.5028
25
21
18
15
1.5012
09
06
1.5002
1.4999
1.4996
93
90
87
83
1.4980
77
74
71
67
1.4964
61
58
55
52
1.4948
45
42
39
36
1.4932
29
26
23
20
1.4916
13
10
07
04
1.4900
1. 4897
94
91
87
1.4884
81
78
75
71
1.4868
65
62
58
55
1.4852
49
45
42
39
1.4836
1.4836
33
29
26
23
1.4820
16
13
10
07
1.4803
1.4800
1.4797
94
90
1.4787
84
81
78
74
1.4771
68
64
61
58
1.4754
51
48
45
41
1.4738
35
32
28
25
1.4722
18
15
12
09
1.4705
1.4702
1. 4699
95
92
1.4689
86
82
79
76
1.4672
69
66
62
59
1.4656
52
49
46
43
1.4639
1.4639
36
33
29
26
1.4623
19
16
13
09
1.4606
1.4603
1.4599
96
93
1.4589
86
83
79
76
1.4573
69
66
63
59
1.4556
53
49
46
43
1.4539
36
33
29
26
1.4523
19
16
13
09
1.4506
1. 4502
1. 4499
96
92
1.4489
86
82
79
75
1.4472
69
65
62
59
1.4455
52
48
45
42
1.4438
1.4438
35
32
28
25
1.4421
'18
15
11
08
1.4404
1.4401
1.4398
94
91
1.4387
84
81
77
74
1.4370
67
63
60
56
1.4353
50
46
43
40
1.4336
33
29
26
22
1.4319
16
12
09
05
1.4302
1.4298
95
92
88
1.4285
81
78
74
71
1.4267
64
60
57
54
1.4250
47
43
40
36
1.4233
CALCULATION OF SMALL DISTANCES
247
2) multiplying b y 5 the length given for 1° of the meridian at 34° in table 1,
554.6020 kin.
3) b y summing the lengths given at 32 °, 33 °, 34° , 35 °, and 36 ° in table 1,
554.6033 kin.
4) from tables based on the International Ellipsoid (Lambert and Swick,
1935), 554.6267 kin.
The third result is the correct length for the Clarke Spheroid. The first and
second computations differ from this b y the deviation of the spheroid from a
sphere, plus rounding-off errors. This deviation is seen to be actually less than
that between lengths computed On the two spheroids, and amounts to less than
0.01 kin., or less than ten meters.
The deviation between distances computed from formula (1) and distances
over the sphere is increased when the end points also differ in longitude, while
the effect of the ellipticity of the earth becomes less. Hence, in estimating
errors of the method it will be sufficient to determine the errors when applied
to points at the same latitude and longitude on a true sphere of nearly the
size of the earth--conveniently, a sphere of 40,000 kin. circumference. The
theoretical foundation is as follows.
Exact angular distances over the sphere are given b y
cos A = cos ftl cos ~2 cos AX + sin ~1 sin ~2
(2)
where ~i and ft~ are the latitudes of the two given points, the longitudes of
which differ b y AX. If we put ftl - ~ = A~ and ~1 + ~ = 2ft, and suppose
Aft, and AX to be small compared to one radian, then we find
cos A = 1
A~
2
cos2 ¢ -Ak
+ s
2
A¢ 4 + - -ACsAk
+cos s
s
s ¢ A,I~4 + .
24
• • (3)
B y applying the expansion of cos A and using the method of undetermined
coefficients it is easy to derive
As = A~2 + c°sS ~AX2 +
-- 4 +
A~s AXs -- sins ~12c°sS~
(4)
Writing A0~ -- A~ 2 + cos2 ~AXs, there results
A -- A0 - 1 sins ~AoAX2 _ 1 (1 + sin 2 ~) A~2AX-------~+
24
24
&0
• • •
(5)
A0 is the value of A given b y the method of this paper. The following terms give
the correction, including small quantities of the third order. If A~, AX, and A0
B U L L E T I IOF
~ THE SEISMOLOGICALSOCIETY
248
OF AMERICA
are n e a r 6 ° t h e y are of the order of o n e - t e n t h of a r a d i a n , a n d t h e correction
t e r m s g i v e n a b o v e are of the order of o n e - t h o u s a n d t h of a r a d i a n d i v i d e d b y
24, or a b o u t o n e - f o u r t h of a kilometer.
I n c o m p u t i n g tables showing these errors it has b e e n desirable to h a v e A~
a n d ' A k expressed i n i n t e g r a l n u m b e r s of degrees, a n d A, w i t h t h e corrections,
i n kilometers. I f A~ = m degrees a n d AX = n degrees, t h e n
n2
A ---- A0 -- 0.00140
[ (1 zr 2 sin 2 ~ ) m ~ ~- sin 2 ~ cos 2 ~ m 2]
(6)
~/m 2 + cos 2
for a sphere of 40,000 kin. circumference. T h e exact v a l u e of t h e n u m e r i c a l
coefficient i n t h e correction t e r m is v~/6998.4.
TABLE 4
ERRORS OF FORMULA (1), IN KILOMETERS
a. Mean latitude ~I, = 30°
nk
A~
0o
1
2
3
4
5
10
1°
.0003
.002
.004
.006
.008
.01
.02
2°
3°
4°
5°
•002
•006
.01
.02
.03
.04
.08
•008
.01
.03
.05
.07
.09
.19
.02
.03
.05
.08
.12
.15
.32
.04
.05
.08
.12
.17
.22
50
•
10 °
.31
.33
.39
50
.63
.79
1.80
•
b. Mean latitude (I, = 45°
AX
A~
0o
1
2
3
4
5
10
1°
.0005
•003
•005
.008
.01
.01
.03
2°
.0004
.01
.02
.03
.04
.06
.11
3°
4°
5°
.01
.02
.05
.07
.10
.12
25
.03
.05
.08
.12
.17
.21
.44
.06
.08
.12
.18
.25
.32
.69
•
10 °
.50
.53
.63
.79
.99
1.22
2.59
T a b l e s 4a a n d 4b h a v e b e e n c o m p u t e d from f o r m u l a (6). T a b l e 4a gives t h e
v a l u e of t h e t h i r d - o r d e r t e r m s for m e a n l a t i t u d e • = 30 °; t a b l e 4b, for • = 45 °.
I n t e r p o l a t i o n c a n be m a d e v e r y exact, if desired, b y n o t i n g t h a t w h e n b o t h A~
a n d Ak are m u l t i p l i e d b y the same c o n s t a n t k, t h e errors as t a b u l a t e d are
249
CALCULATION OF S ~ A L L DISTANCES
multiplied b y ]c3. These positive numbers are errors of formula (1); to be applied as corrections they m u s t be subtracted, in accordance with the negative
sign in (6).
T h e errors near 45 ° are rougly half again as large as those near 30 °, and
errors increase further at higher latitudes. The following are the errors in
meters for A~ = AX = 1°, at the indicated m e a n latitudes:
Error
30 °
1.80
35 °
2.05
40 °
2.32
45 °
2.59
50 °
2.87
55 °
3.14
60 °
3.39meters
T o c o m p a r e w i t h t a b l e 4, d i v i d e b y 1000.
While these corrections are computed for a sphere of 40,000 km. circumference,
so t h a t 1° of a great circle has the length 111.1111 km., t h e y will a p p l y with
their tabulated accuracy to the true earth and to the results from formula (1)
and tables 2 and 3.
The method of construction of tables 2 and 3 eliminates t h a t p a r t of the
effect of the ellipticity of the earth which expresses itself in the difference
between the ordinary, geodetic or geographical latitude (which is t h a t used in
the tables and on all ordinary maps), and the geocentric latitude. This effect
m u s t be considered in a n y a t t e m p t to m a k e direct use of equation (2), since
neglecting it will introduce comparatively large errors. As an example, suppose
formula (1) and the tables used to find a distance between two points at
latitudes 32 ° and 36 °, differing 4 ° in longitude. The m e a n latitude • is then 34 °.
Tables 2 and 3 give Ax -- 369.552 and Ay = 443.688, whence A ---- 577.432
kilometers. On the other hand, equation (2) with ~1 = 32 °, ~ = 36 °, and
A), = 4 °, gives cos A = 0.9958928, whence A = 5° 11' 40'.'9 = 5.19469 °. N o w
the length of 1° of a great circle at the m e a n latitude is t h a t given in table 1
for 1° of the meridian centering at 34 °, namely, 110.9204 km. Multiplying this
b y the last value for A expressed in degrees, we find A = 576.197. There is a
discrepancy of 1.235 km. between the two results.
The error originates in the use of geodetic latitudes in equation (2). The
computed value of cos A, and of A derived from it, should refer to points
at geocentric latitudes 32 ° and 36 ° . The corresponding geodetic latitudes are
32 ° 10' 30" and 36 ° 11' 07 ". The arc of the meridian between these latitudes is
found b y interpolation in table 1 to be 444.814 kin., so t h a t 1° of the great
circle is 111.203 km. Multiplying this b y 5.19469 gives A = 577.665 km., which
is probably the best result obtainable b y simple means. The values of Ax and
Ay used in formula (1) also need correction, since the geodetic mean latitude
should now be t a k e n as 34 ° 10'.8, while the difference in latitude is not 240'
b u t 240!61. Tables 2 and 3, then, give Ax = 368.760 kin., Ay = 444.816 kin.,
whence A ---- 577.793 km. A more exact value of Ax, derived from table 1, is
368.770 km., while more exactly Ay = 444.814 km. as found earlier. These
250
BULLETIN"OF TIIE SEIS1VIOLOGICALSOCIETY OF A1KERICA
give A = 577.798 kin. as the best result using formula (1). The correction to
this derived from formula (6) is - 0 . 0 1 2 7 kin., giving a corrected result of
A = 577.671 km. This differs from the result of (2) b y 0.006 kin., which is due
to slight rounding-off errors.
The quantities Ax and Ay are ~ d e p e n d e n t l y useful in least-square determinations of epicenter and origin time. Their quotient gives the tangent of the
angle which measures the azimuth of the station at the epicenter.
Mr. John M. Nordquist has assisted by verifying and extending some of
the numerical examples and formt/las, and also with the computation of tables
based on formula (6).
BALCHGRADUATESCHOOLOFTHE GEOLOGICALSCIENCES
CALIFORNIAINSTITUTEOF TECHNOLOGY,PASADENA,CALIFORNIA
CONTRIBUTIONNO. 351.
REFERENCES
LAMBERT,W. D., and SWICK, C. H. "Formulas and Tables for the Computation of Geodetic Positions on the International Ellipsoid," U. S. Coast and Geodetic Survey,
Spec. Publ. No. 200 (1935).
WIECItERT, E. (1925). "Entfernungsberechnungen yon Orten auf der Erde bei kleineren
Abstanden," Zeitschr.f. Geophysik, I, 176-191.
WOODWARD,R. S. (1894). Smithsonian Geographical Tables, Washington, Smithsonian Institution. (3d ed., 1918).