GEOGRAPHICAL COORDINATE COMPUTATIONS
Part I
General Considerations
Technical Report No. 2
ONR Task No. 389137
Contract Nonr 1224 (48)
Office of Naval Research
Geography Branch
W. R. Tobler
Department of Geography
University of Michigan
Ann Arbor, Michigan
December, 1964
This report has been made possible through support and sponsorship
by the United States Department of the Navy, Office of Naval
Research, under ONR Task Number 389137, Contract Nonr 1224(48).
Reproduction in whole or in part is permitted for any purpose
by the United States Government.
REPORT AVAILABILITY NOTICE
The following report has been issued by the University of Michigan under
Contract Nonr1224(48), ONR Task No. 389137, sponsored by the Geography
Branch of the Office of Naval Research. Copies are available from the
Defense Documentation center for Scientific and Technical Information.
GEOGRAPHICAL COORDINATE COMPUTATIONS
Part I
General Considerations
By W. R. Tobler
Technical Report Number 2
December, 1964
ABSTRACT
Part I provides a discussion of the usefulness of coordinate models
for studies of geographically distributed phenomena with comments on
specific coordinate systems and their relevance for the analysis and inventorying of geographical information. Appendices include equations for
conversion from the Public Land Survey system into latitude and longitude
and to rectangular map projection coordinates. Part II considers map
projections in greater detail, including estimates of the errors introduced
by the substitution of map projection coordinates for spherical coordinates.
Statistical computations of finite distortion are related to Tissot's Indicatrix as a general contribution to the analysis of map projections.
ACKNOWLEDGEMENTS
The preparation of this report has been facilitated by the
assistance of several individuals. The University of Michigan
Computation Center contributed in the numerical processing, and
the University's Office of Research Administration and the Geography Branch of the Office of Naval Research both provided
valuable administrative advice and support. Messrs E. Franckowiak,
D. Kolberg, F. Rens, and R. Yuill, graduate students in the Department of Geography, contributed in several ways and were largely responsible for the illustrations and computer programs. The project
has also benefited greatly from discussions with Professors
R. Berry, L. Briggs, D. Marble, and J. Nystuen.
INTRODUCT ION
In recent years there has been a rapid increase in the use of formal
mathematical and statistical methods for the analysis of terrestrial distributions. Such procedures have been found to be of considerable assistance in fields such as city and regional planning, demography, ecology,
geography, geology, and regional science. The present study is concerned
with only one of the several mathematical strategies which have been
utilized for such analyses; the "coordinate model". This term is taken
to include that class of studies which specifically refers to the location
of observational phenomena by some system of coordinates.
As an example, a technique associated with contemporary theories in
geology consists of estimating the departures of empirical geological
observations from a "regional trend". Here one has a collection of
numerical observations (zi) at specific terrestrial locations (xi, yi),
i  1,2,..., n. The procedure begins by estimating a specific portion
of the locational trend of the observations as a least squares equation
z = f(x, y). The trend is then subtracted from the observations to obtain
the residual, which is subsequently interpreted in terms of geological
theory. In this instance the recording of locations in some system of
coordinates is an essential prerequisite for the analysis. There are
many other such examples. The coordinate model is widely used, and is
enjoying increasing popularity since the advent of electronic computers,
which permit facile manipulation of the metricized locations. The researcher now also has available to him a rapidly increasing amount of
information recorded in terms of coordinates; the U.S. Bureau of the
Census, for example, now provides population statistics in terms of
latitude and longitude coordinates. It is difficult to overestimate the
usefulness of this manner of recording information since a large number
of the analytical methods designed for the analysis of distributions
assume the existence of a system of coordinates.
As a system of locational labels the specific coordinates employed
for the recording of observations are not of direct or inherent interest,
but rather only what they enable one to deduce regarding interrelations
among the phenomena observed. In this sense the particular coordinate
system utilized is irrelevant. On the other hand, computations may often
be simplified by the choice of a convenient coordinate notation. From a
scientific point of view descriptions of phenomena and their interrelations
are often simplified by appropriate formalizations involving a "natural"
coordinate system for that phenomena; the use of geomagnetic coordinates
in the study of terrestrial magnetism, for example. The present study,
however, considers only systems which appear to be of practical utility
for the large scale recording of terrestrial observations, with some
emphasis on systems available in the United States.
The actual surface of the earth can be referred to as the topographic
surface. This bumpy twodimensional surface is difficult to describe in
all its detail. Theoretically it is possible to introduce a system of
coordinates on this surface such that ground distances, etc., between all
2
points can be calculated. In practice this is not attempted. As an
approximation to the topographic surface the geodesist utilizes a surface
of constant gravitational potential, the geoid. This bumpy but rather
smooth surface is still too complicated for practical computations. A
further simplification is made by assuming the earth to have the shape
of an ellipsoid, generally an ellipsoid of revolution. Geodetic coordinates
are then defined for positions on this ellipsoid. An even simpler model
of the topographic surface is to consider the earth to be a perfect sphere.
One can continue thusly, finally arriving at the assumption that the
earth is a flat plane. Each of these assumed models of the earth has its
advantages and disadvantages since realism may lead to extreme cumbrousness.
In practical terms, the following (somewhat contradictory) criteria
seem appropriate:
a) The coordinates should permit accurate and economical formulae
for computation. The highest level of precision available today can be
achieved only through the use of geodetic formulae. These formulae are
fairly complicated. Computational simplicity can be obtained, with a
consequent reduction in accuracy, by employing spherical formulae.
Further computational simplicity can be achieved by the use of plane
coordinates based on an appropriate map projection, again with some
loss of accuracy.
Computational simplicity is important for two reasons. The cost
and time required for computation, particularly when large amounts of
information are to be processed, can be reduced by significant amounts
through the use of simplified formulae. In addition, the number of
persons and agencies who can effectively make use of the information is
significantly increased by the use of simplified formulae.
It is more difficult to discuss accuracy, upon which the level of
computational simplificty depends, since the degree of precision required
in particular studies varies considerably. There is no reason to employ
a method which results in accuracies greater than required or greater
than those with which the information was recorded. An objective of
this study has been to estimate the accuracy obtainable by employing
several alternate methods. This allows the individual researcher to
choose the simplest computational method which yields the requisite
level of accuracy.
b) A rapid and accurate method of determining the coordinates of a
position should be available. In general a system of coordinates which
requires a carefully executed geodetic survey can be considered highly
accurate, but relatively slow. A system which enables one to read
coordinates from an aerial photograph or map (either manually, mechanically,
or electronically) is more rapid but the accuracy is dependent on the
map scale. The convenience of this method is also dependent on the
availability of maps or photographs to which the coordinate system has
been affixed. Between the extremes of geodetic surveying and map scaling
are a number of intermediate systems, including automatic navigation
devices which permit virtually instantaneous, in situ position determination,
with fair accuracy. Emphasis in this study is on map scaling procedures.
3
c) The coordinates should be widely available and should be equally
convenient for use at a local, national, and international level. This
objective arises since most types of information collected at a national
level are used both nationally and locally. Census records provide a
good example. National use of local records also is increasing. The
accuracy requirements at these two levels generally differ, however.
At the local level an accuracy of fifty feet may be insufficient, whereas
at the national level an accuracy of five miles may suffice.
TERRESTRIAL COORDINATE SYSTEMS
There are many locational coordinate systems in use throughout the
world. Emphasis in the current discussion is on systems available in the
United States.
Geodetic Coordinates:
Geodetic latitude and longitude provide the traditional method of
identifying locations on the surface of the earth. The earth is assumed
to be an oblate spheroid and the geodetic coordinates are based on actual
measurement (triangulation) between sets of locations on the topographic
surface. These values are then adjusted to fit an ellipsoid representative
of the region in question. Different ellipsoids are employed for the
several continents of the world, with an International Ellipsoid in use
for worldwide computations. Geodetic coordinates, based on the Clarke
Ellipsoid of 1866 (1927 adjustment), are indicated on maps published by
the U.S. Geological Survey and by the U.S. Coast and Geodetic Survey.
Latitude and longitude scaled from such maps are geodetic coordinates,
but such scaling will not yield the same accuracy as when the positions
are established in the field by an expensive first order geodetic survey.
Computations employing geodetic coordinates usually take into account
the ellipsoidal shape assumed for the earth. The relevant formulae are
fairly complicated. For precise geodetic work it is necessary to carry
approximately fifteen significant digits. Experience with a digital
computer, however, indicates that, once programmed, the ellipsoidal
formulae do not require appreciably more effort than the simpler spherical
formulae. A floating point program with seven significant digits (as employed)
for this study) yielded values which differed less than 100 meters from more
precise values over a range of 6000 kilometers.
Assumptions required to apply geodetic computations to the surface of
the earth are that (a) the geodetic latitudes and longitudes are known
without error, (b) the ellipsoid chosen is representative of the region
in question, and (c) the points involved lie on the surface of the ellipsoid
(roughly, at sea level). On the other hand, this is the most accurate
system available. The actual proportional error in distance, based on
misclosures of the U.S. continental triangulation network, appears to be
on the order of
1 D1/3
20000
where D is the computed distance in miles on the Clarke Ellipsoid of
4
1866 (1927 adjustment). The differences between the several ellipsoids
in use throughout the world are small; on the order of three kilometers
per 6000 miles. Connections of this length on one ellipsoidal datum
are rare and the figure given does not take into account the fact that
the relation between the several datums in actual use are not yet known
in detail; in other words, distances between positions whose geodetic
coordinates are referred to different ellipsoids may be in error by a
larger figure.
Astronomic Coordinates:
Astronomic latitude and longitude are based on celestial observations
and may depart from geodetic coordinates by as much as two kilometers
at any point, due to departure of the geoid from the ellipsoid. Astronomical
observations are usually available only for isolated points, and will not
be considered in this report.
The U.S. Public Land Survey
The Public Land Survey system is based on a set of six mile squares
numbered as townships north and south of a base parallel, and as ranges
east and west of a base meridian. These six mile squares are then subdivided into 36 sections, each one mile square, and numbered in serpentine
fashion. Each section can be further subdivided into quartersections,
each one sixteenth of a mile in area. Several systems similar to the
Public Land Survey exist in various parts of the United States; these
are not considered here.
Strictly speaking, the Public Land Survey is an areal identification
scheme and not a metrical coordinate system, though it is often regarded
as such. As a partitioning of areas the system does not differ from
county or census units, except that the elemental areas are roughly of
equal size and are labeled in a more convenient fashion. The system is
not complete, in the sense that it is defined only for certain portions
of the western United States. In these areas large amounts of information
have been (and continue to be) collected and recorded in terms of Sections,
Townships, and Ranges. These collections of information provide a valuable
source of raw data for research workers. The Public Land Survey, however,
was not designed for the analytical manipulations usually required in
research work. For example, statistical analyses of spatial distributions
may require calculation of the average location (and its variance, and
so on) of phenomena. For such computations the distances between observed
locations may be needed. The distance between the SWk, Sec. 25, T5S, R7W,
Willamette Meridian, and the NEI, Sec. 2, T6N, R8E, Black Hills Meridian,
is not immediately apparent, nor is there any simple formula which can be
employed to obtain this difference. Observations recorded in the Public
Land Survey System, however, can be convered to a coordinate system having
the requisite metrical properties. This is because the Public Land Survey
has many of the topological ordering properties of a coordinate system.
The most direct and convenient conversion is to latitude and longitude.
This can be effected in several ways. The system of Townships and Ranges
is shown on U.S. Geological Survey topographic maps and approximate
5
coordinates could be scaled from there maps. A more convenient
procedure is to attempt a direct calculation. The equations for such
a conversion are given in the Appendix, along with an estimate of their
validity. The errors are fairly small so that they might be of little
consequence when working with observations from the entire United
States. The urban researcher working within one city, on the other
hand, might find these errors intolerable.
The GEOREF and Marsden Squares Systems
The GEOREF System is used by the U.S. Air Force to identify
locations. It is a modification of latitude and longitude in which
letters are substituted for the numerical values. Every combination of
letters is taken to represent a quadrilateral bounded by latitude
and longitude. In this sense the system is a partitioning of area
rather than a true coordinate system. The same results can be achieved
by using latitude and longitude with a convention regarding the quadrant
in which the quadrilateral of area lies. The system has certain advantages
in applications which require errorfree rapid verbal communications (e.g.
radio). The system is shown on maps published by the U.S. Air Force.
The Marsden Squares system employed by the National Oceanographic
Data Center is similar to the GEOREF System in that a numbering of
latitude and longitude quadrilaterals is substituted for the geodetic
coordinates. There are many other such systems available, including
the World Aeronautical Chart designations and the International Millionth
Map of the World system. The advantage of these systems is largely one
of bookkeeping. Such systems are not further considered in this report.
THE SPHERICAL ASSUMPTION
The various computations are simplified if it is assumed that the
earth is a sphere. The results of such computations do not differ by
large amounts from the corresponding ellipsoidal values  the polar
flattening of the earth, after all, is quite small. On the basis of a
number of computations it appears that a reasonable and convenient rule
of thumb is that the flattening of the earth can be taken as an approximate
upper bound on the percentage error of measures calculated on a spherical
as compared to an ellipsoidal assumption. This is about one part in 300.
An even safer estimate is that the error will be less than one percent.
For some purposes this is intolerably large, but for the majority of
requirements it is far more accurate than are the data or theories now
available. Detailed numerical differences between an ellipsoid and
sphere for distances and angles also have recently been published.
Computation of the differences for a random sample of 200 pairs of points
within the continental United States resulted in the following values;
6
Distance Differences (miles) Angular differences (degrees)
Average: 0.046 Average: 0.006
Standard Deviation: 1.896 Standard Deviation: 0.083
Minimum: 3.788 Minimum: 0.150
Maximum: 4.871 Maximum: 0.159
A second sample might yield somewhat different results, but the sample is
probably representative for the country as a whole. As expected, the
differences depend on both the distance and on the direction of the point
pairs. A comparison of surface areas is given in the accompanying table.
In performing these computations it has been assumed that the earth
is a sphere whose radius is equal to the equatorial radius of the Clarke
Ellipsoid of 1866, and that the geodetic latitude and longitude can,
without modification of their numerical values, be considered to be
spherical coordinates. These assumptions have the advantage of extreme
simplicity. A slight improvement in accuracy can be obtained if they
are not retained. For example, the spherical radius employed might be
the average radius of terrestrial ellipsoid in the vicinity of the area
of interest, rather than the equatorial radius. It can be proven that
the average radius at any latitude is the geometric mean of the radii of
curvature along the meridian and normal to the meridian, This average
radius is given in the accompanying tables. Conversion of geodetic
latitude to spherical latitude can also be accomplished in a large number
of ways. Four of the simpler methods are illustrated in the figure.
Mathematical treatments can be found in works on geodesy and map projections.
THE PLANE ASSUMPTION
It often is convenient to employ plane coordinates for the inventorying
of analysis of terrestrially distributed phenomena. In particular, many
of the numerous statistical and analytical methods which have been devised for the analysis of two dimensional distributions assume the existence of a system of Cartesian coordinates. As a very simple example,
suppose that an objective is to compute the average location and the
locational variance of a set of discrete phenomena on the surface of a
sphere. One can proceed in several ways:
a) Record the observations in latitude and longitude and then perform
the calculations using the spherical formulae for average and
variance.
b) Plot the distribution on a map, assign arbitrary rectangular
coordinates to the map, record the observations in these
coordinates, and then perform the calculations using the plane
formulae for the average and variance.
COMPARISON OF AREAS FOR A ONE DEGREE ZONE
OF LONGITUDE WITHIN THE UNITED STATES
(Values in square miles, rounded to the nearest square mile)
Latitude Ellipsoidal Area Spherical Area*
26N to 27N 4265 4282
27N to 28N 4228 4244
28N to 29N 4189 4205
29N to 30N 4150 4164
30N to 31N 4109 4123
31N to 32N 4067 4080
32N to 33N 4024 4035
33N to 34N 3979 3990
34N to 35N 3934 3943
35N to 36N 3887 3895
36N to 37N 3839 3846
37N to 38N 3789 3796
38N to 39N 3739 3744
39N to 40N 3687 3692
40N to 41N 3634 3638
41N to 42N 3581 3583
42N to 43N 3526 3528
43N to 44N 3469 3471
44N to 45N 3412 3413
45N to 46N 3354 3354
46N to 47N 3295 3294
47N to 48N 3234 3232
*Radius equal to equatorial radius of Clarke ellipsoid of 1866.
CLARKE ELLIPSOID OF 1866
RADII IN MILES RADII IN KILOMETERS
LATITUDE RADIUS RADIUS MEAN RADIUS RADIUS RADIUS MEAN RADIUS
OF THE NORMAL RADIUS OF THE OF THE NORMAL RADIUS OF THE
MERIDIAN TO THE PARALLEL MERIDIAN TO THE PARALLEL
MERIDIAN MERIDIAN.0 3936.4000 3963.2258 3949.7901 3963.2258 6335.0344 6378.2064 6356.5837 6378.2064
2.5 3936.4761 3963.2513 3949.8410 3959.4791 6335.1569 6378.2474 6356.6656 6372.1767
5.0 3936.7036 3963.3276 3949.9932 3948.2459 6335.5231 6378.3703 6356.9106 6354.0986
7.5 3937.0810 3963.4543 3950.2456 3929.5463 6336.1304 6378.5741 6357.3168 6324.0045
10.0 3937.6055 3963.6303 3950.5964 3903.4138 6336.9745 6378.8574 6357.8814 6281.9481
12.5 3938.2731 3963.8542 3951.0429 3869.8950 6338.0489 6379.2178 6358.6000 6228.0048
15.0 3939.0788 3964.1245 3951.5818 3829.0502 6339.3456 6379.6528 6359.4672 6162.2714
17.5 3940.0167 3964.4391 3952.2090 3780.9528 6340.8549 6380.1591 6360.4766 6084.8658
20.0 3941.0799 3964.7957 3952.9200 3725.6892 6342.5659 6380.7330 6361.6208 5995.9276
22.5 3942.2603 3965.1915 3953.7092 3663.3592 6344.4656 6381.3699 6362.8909 5895.6169
25.0 3943.5491 3965.6236 3954.5709 3594.0755 6346.5398 6382.0652 6364.2777 5784.1154
27.5 3944.9368 3966.0886 3955.4985 3517.9636 6348.7730 6382.8137 6365.7706 5661.6249
30.0 3946.4128 3966.5832 3956.4852 3435.1618 6351.1485 6383.6097 6367.3584 5528.3680
32.5 3947.9662 3967.1036 3957.5233 3345.8212 6353.6484 6384.4472 6369.0292 5384.5881
35.0 3949.5852 3967.6458 3958.6052 3250.1051 6356.2540 6385.3198 6370.7703 5230.5477
37.5 3951.2578 3968.2058 3959.7227 3148.1893 6358.9457 6386.2209 6372.5687 5066.5297
40.0 3952.9710 3968.7792 3960.8672 3040.2613 6361.7029 6387.1439 6374.4106 4892.8360
42.5 3954.7122 3969.3619 3962.0303 2926.5205 6364.5051 6388.0815 6376.2824 4709.7877
45.0 3956.4680 3969.9492 3963.2029 2807.1780 6367.3308 6389.0268 6378.1696 4517.7242
47.5 3958.2252 3970.5369 3964.3763 2682.4558 637C.1588 6389.9725 6380.0579 4317.0029
50.0 3959.9702 3971.1203 3965.5413 2552.5869 6372.9670 6390.9114 6381.9329 4107.9987
52.5 3961.6898 3971.6950 3966.6893 2417.8147 6375.7345 6391.8364 6383.7803 3891.1034
55.0 3963.3709 3972.2567 3967.8113 2278.3928 6378.4399 6392.7403 6385.5861 3666.7252
57.5 3965.0004 3972.8010 3968.8988 2134.5844 6381.0624 6393.6163 6387.3362 3435.2876
60.0 3966.5660 3973.3239 3969.9435 1986.6619 6383.5820 6394.4577 6389.0175 3197.2289
62.5 3968.0557 3973.8212 3970.9373 1834.9064 6385.9794 6395.2581 6390.6170 2953.0015
65.0 3969.4577 3974.2891 3971.8727 1679.6072 6388.2357 6396.0112 6392.1223 2703.0712
67.5 3970.7614 3974.7242 3972.7423 1521.0611 6390.3339 6396.7114 6393.5218 2447.9155
70.0 3971.9566 3975.1230 3973.5395 1359.5722 6392.2574 6397.3531 6394.8047 2188.0237
72.5 3973.0342 3975.4824 3974.2581 1195.4507 6393.9916 6397.9316 6395.9612 1923.8952
75.0 3973.9856 3975.7997 3974.8925 1029.0127 6395.5226 6398.4422 6396.9822 1656.0388
77.5 3974.8036 3976.0724 3975.4380 860.5796 6396.8391 6398.8812 6397.8600 1384.9715
80.0 3975.4817 3976.2986 3975.8901 690.4770 6397.9304 6399.2451 6398.5876 1111.2173
82.5 3976.0147 3976.4763 3976.2454 519.0344 6398.7882 6399.5310 6399.1595 835.3065
85.0 3976.3984 3976.6042 3976.5013 346.5839 6399.4057 6399.7369 6399.5712 557.7739
87.5 3976.6297 3976.6813 3976.6555 173.4605 6399.7780 6399.8610 6399.8195 279.1581
90.0 3976.7071 3976.7071 3976.7070.0001 6399.9025 6399.9025 6399.9024.0001
INTERNATIONAL ELLIPSOID
RADII IN MILES RADII IN KILOMETERS
LATITUDE RADIUS RADIUS MEAN RADIUS RADIUS RADIUS MEAN RADIUS
OF THE NORMAL RADIUS OF THE OF THE NORMAL RADIUS OF THE
MERICIAN TO THE PARALLEL MERIDIAN TO THE PARALLEL
MERIDIAN MERIDIAN.0 3936.6635 3963.3075 3949.9630 3963.3075 6335.4584 6378.3380 6356.8619 6378.3380
2.5 3936.7391 3963.3329 3950.0136 3959.5606 6335.5801 6378.3787 6356.9434 6372.3079
5.0 3936.9650 3963.4087 3950.1647 3948.3267 6335.9437 6378.5008 6357.1866 6354.2286
7.5 3937.3400 3963.5345 3950.4155 3929.6259 6336.5471 6378.7032 6357.5902 6324.1325
10.0 3937.8609 3963.7093 3950.7639 3903.4916 6337.3854 6378.9846 6358.1509 6282.0734
12.5 3938.5240 3963.9317 3951.2074 3869.9707 6338.4526 6379.3425 6358.8647 6228.1266
15.0 3939.3242 3964.2002 3951.7426 3829.1233 6339.7405 6379.7746 6359.7260 6162.3890
17.5 3940.2558 3964.5127 3952.3656 3781.0229 6341.2397 6380.2775 6360.7286 6084.9787
20.0 3941.3118 3964.8668 3953.0717 3725.7560 6342.9391 6380.8474 6361.8650 5996.0350
22.5 3942.4842 3965.2599 3953.8557 3663.4225 6344.8260 6381.4800 6363.1266 5895.7188
25.0 3943.7644 3965.6891 3954.7115 3594.1348 6346.8862 6382.1707 6364.5040 5784.2109
27.5 3945.1426 3966.1510 3955.6329 3518.0189 6349.1043 6382.9141 6365.9868 5661.7139
30.0 3946.6087 3966.6422 3956.6127 3435.2129 6351.4638 6383.7046 6367.5637 5528.4503
32.5 3948.1517 3967.1591 3957.6439 3345.8680 6353.9470 6384.5364 6369.2233 5384.6633
35.0 3949.7598 3967.6976 3958.7185 3250.1476 6356.5349 6385.4031 6370.9526 5230.6160
37.5 3951.4209 3968.2538 3959.8284 3148.2274 6359.2083 6386.2982 6372.7388 5066.5909
40.0 3953.1227 3968.8234 3960.9652 3040.2950 6361.9470 6387.2148 6374.5684 4892.8904
42.5 3954.8520 3969.4020 3962.1203 2926.5501 6364.7301 6388.1461 6376.4273 4709.8353
45.0 3956.5960 3969.9854 3963.2849 2807.2036 6367.5367 6389.0849 6378.3016 4517.7653
47.5 3958.3411 3970.5689 3964.4503 2682.4775 6370.3453 6390.0241 6380.1771 4317.0377
50.0 3960.0743 3971.1484 3965.6075 2552.6050 6373.1346 6390.9567 6382.0394 4108.0278
52.5 3961.7822 3971.7192 3966.7476 2417.8295 6375.8832 6391.8753 6383.8742 3891.1271
55.0 3963.4518 3972.2771 3967.8619 2278.4045 6378.5701 6392.7731 6385.6676 3666.7440
57.5 3965.0704 3972.8177 3968.9421 2134.5934 6381.1750 6393.6431 6387.4059 3435.3020
60.0 3966.6253 3973.3370 3969.9797 1986.6685 6383.6775 6394.4788 6389.0758 3197.2394
62.5 3968.1047 3973.8309 3970.9668 1834.9109 6386.0583 6395.2737 6390.6644 2953.0088
65.0 3969.4972 3974.2957 3971.8957 1679.6100 6388.2993 6396.0217 6392.1592 2703.0756
67.5 3970.7920 3974.7277 3972.7594 1521.0625 6390.3831 6396.7170 6393.5493 2447.9177
70.0 3971.9791 3975.1238 3973.5511 1359.5725 6392.2936 6397.3544 6394.8234 2188.0242
72.5 3973.0492 3975.4807 3974.2647 1195.4501 6394.0157 6397.9289 6395.9719 1923.8944
75.0 3973.9942 3975.7959 3974.8949 1029.0117 6395.5365 6398.4361 6396.9861 1656.0372
77.5 3974.8066 3976.0668 3975.4366 860.5784 6396.8439 6398.8721 6397.8578 1384.9695
80.0 3975.4800 3976.2913 3975.8856 690.4758 6397.9277 6399.2334 6398.5804 1111.2153
82.5 3976.0093 3976.4678 3976.2385 519.0333 6398.7796 6399.5175 6399.1484 835.3047
85.0 3976.3904 3976.5948 3976.4926 346.5831 6399.3928 6399.7219 6399.5573 557.7726
87.5 3976.6203 3976.6715 3976.6458 173.4601 6399.7629 6399.8452 6399.8040 279.1575
90.0 3976.6971 3976.6971 3976.6971.0001 6399.8864 6399.8864 6399.8864.0001
Four simple methods for the
CONVERSION OF GEODETIC LATITUDE TO SPHERICAL LATITUDE
_______/ /9 _ _ _ ______/ s / 9
7
c) Record the observations in latitude and longitude, apply a
transformation to obtain rectangular coordinates, and then
perform the claculations using the plane formulae for the
average and variance.
Procedure (a) has the disadvantage of being more complicated. A
sufficiently small portion of the earth's surface can be considered a
plane and the additional complication introduced by the use of spherical
versions of the statistical formulae may not be warrented. Somewhat
similar problems have been investigated in the field of land surveying
and are reported in most works on geodesy. Procedures (b) and (c),
above, are mathematically equivalent since maps are made by transforming
latitude and longitude to plane coordinates via a map projection. Hence
a study of the numerical differences between computations on a plane and
on the earth becomes a study of map projection distortions.
The official map producing agencies of the various countries of the
world have recognized the advantages of rectangular coordinates for local
purposes and save the map user the trouble of assigning his own system
of rectangular coordinates. They do this by publishing maps which have
the official plane coordinates printed directly on the maps. Two map
projection systems of this type are available in the United States, and
comparable systems exist in most other countries of the world. The use
of these systems is not restricted to calculations; they might also be
used to record and index information in terms of the plane coordinates,
perhaps scaled from topographic maps. The systems now available have
several features in common. The coordinates are usually given as rectangular coordinates, often chosen so that all values are positive. More
importantly, the errors in computing as though the earth were a plane
disk can be evaluated. This implies that the region within which one
can perform plane computations with a specified level of precision can
be defined on an a priori basis. If the allowable error is small, the
region must be small or several map projection systems (called zones)
must be used within the region. In the latter event conversion between
zones may be required. This conversion may be directly from zone to
zone or may involve reconversion to geodetic coordinates as an intermediate
step. There are certain advantages in using a conformal map projection
for such a system since the scale errors are then independant of direction
and a scale factor can be applied to improve the accuracy of short lengths
The two systems employed in the United States are:
1) The State Plane Coordinate System: This system comprises
approximately 120 zones covering the entire United States, with the
orientation toward individual states. The accuracy within each zone is
one part in 10,000. The larger states therefore require several zones.
The zones overlap,with boundaries between zones lying along minor civil
divisions (usually counties). The Lambert Conformal Conical projection
and the Transverse Mercator projection are employed (with only one exception),
8
depending on the shape of the individual states. This system is admirably
suited to the needs of the local land surveyor and has been officially
adopted by many local governmental units. In many states it has legal
status, is used for land ownership, and appears on large scale maps. Conversion tables are available and simple to use for any particular zone.
Conversion between zones, and especially between states, is somewhat more
inconvenient. The location of the zones occasionally is awkward. In
Washington state, for example, the two merging metropolitan areas of
Seattle and Tacoma each lie in a separate zone. The system of State Plane
Coordinates appears on all recent U.S. geological Survey topographic maps.
2) The Transverse Mercator System:
Known as the Universal Transverse Mercator grid system (UTM)
this system is employed by the U.S. Army. The UTM grid extends
to eighty degrees north and south latitude, beyond which a
Polar Stereographic grid is employed. The UTM grid extends around
the world in sixty northsouth zones, each covering six degrees
of longitude with an overlap of one half degree. The accuracy
within each zone is one part in 2500. Since different areas of
the world are based on distinct ellipsoidal datums, separate
tables are required for various parts of the world. Procedures
are available for converting directly from onezone to adjacent
zones. The zonal nature of the system is occasionally inconvenient.
An alphanumeric partitioning of areas is available in the system.
The UTM grid appears on all Army Map Service topographic maps, on
some foreign maps, and on all recent U.S. Geological Survey
topographic maps.
Both of the foregoing systems have several advantages. They can
be employed for virtually all computations without serious error. Further,
any information recorded in either of these systems can be related to
geodetic coordinates and hence to information collected anywhere else in
the world. Also, these coordinates are already shown on published maps,
and most photogrammetric firms are sufficiently familiar with these
systems to add them to aerial photographic or maps compiled by photogrammetric methods. The disadvantages of these systems stem largely from their
advantages. The very refinement required to provide coordinates of high
accuracy restrict these systems to relatively small portions of the earth's
surface and the transformation equations, either between zones, or to
and from geodetic coordinates, are relatively complicated. These difficulties
can be circumvented in several ways.
When a map projection system is to be used soley for computational
purposes, and not necessarily to be indicated on published maps, the
choice of a particular projection depends on the type of computation
contemplated. The systems cited above are so refined that they yield a
stated level of accuracy for virtually all computations. This is
a restriction which narrows th e of suitable projections and results
in projection which require fairly involved computations. For a given
9
problem there may be a specific projection which is computationally
much simpler but which yields results which are of equal accuracy. For
example, a problem which requires interpolation between two points
on a sphere might be attacked by using the gnomonic projection (see
Appendix) since all great circles are straight lines on this projection;
linear interpolation in gnomonic coordinates will yield a point lying
on the arc connecting the two given points. Similarly, problems involving
circles on a sphere may be attacked using the stereographic projection.
In other situations computational simplicity and speed may be more
important than a few tens of meters of accuracy. Kao, for example, has
recently shown that the geometric (perspective) projections are especially
well suited for calculation by digital computer, particularly when large
amounts of locational information are required within fractions of
a second (i.e., in real time problems). Clearly the choice depends on
the nature of the problems and the volume of the information to be processed.
Computer calclation of distance and direction on a sphere (or ellipsoid)
may, in many instances, be easier than attempting to convert to plane coordinates. On the other hand a more complicated problem, as for example,
occurs in weather prediction, may advantageously be solved by the use of
an appropriate map projection. In this instance the problem is to construct contourtype maps of the entire northern hemisphere from information received from locations scattered within this region. Rather than
attempting to solve the contour interpolation problem on a sphere, the
Weather Bureau employs stereographic map projection coordinates with
a local correction for the projection distortion and solves the problem
in plane coordinates.
If one has information recorded in latitude and longitude simple
conversions to map projection coordiantes are available. For example,
one can pretend that these are already the ordinate and abscissa
of a plane coordinate system. The resulting projection is known as the
square projection. Computations performed in this manner will differ
from the true values by amounts which depend on the size of the region
and on the latitude. Another simple, but slightly better, conversion is
to multiply all the abscissas (longitudes) by a constant equal to the
cosine of the average latitude of the region in question (the square
projection with a standard parallel; also known as the rectangular projection). Such a procedure might for example, by employed in urban
analysis, depending on the size of the area. Another alternative would
be to transform to rectangular coordinates by converting all values
into distances north and east (that is, measured along a parallel)
from some arbitrary point within the region. This yields the sinusoidal projection. The equations for all of the above projections are
extremely simple. Somewhat more refined, but also more complicated, solutions take into cosideration the shape of the area of concern. Albers'
equal area conical projection with standard parallels at 29~ 30'N and 45~
30'N, and Lambert's conformal conical projection with standard parallels
at 33~N and 45~N, for example, are two systems which might be suitable
SQ UARE PROJECTION
1 Co
    v    ^ ^     ^ ^ ^ ^ ^       F\ ^ ^?^7! r,.
RECTANGU LAR PROJECTION
[:X2CI~ my'vIiII irv
Standard parallels at 60N and 60S
10
for the continental United States. The distance error in computing
with these latter systems is not likely to exceed fifty miles.
The use of latitude and longitude, while advantage* from the point
of view of long run national needs, entails some local difficulties. In
the process of recording it may be necessary to interpolate between curved
lines, and the system of minutes and seconds is awkward (Decimal
degrees are more convenient). The complete number of digits required
to specify a given location in its world context is excessively large
for local use, and the northsouth and eastwest designation is often
superflouous (a mathematical convenience is obtained if south latitudes
and west longitudes are considered negative). Finally, and perhaps
most important, it is often difficult to determine the latitude and
longitude of a particular spot.
The direct recording and storage of geographical information in
terms of rectangular coordinates circumvents some of these difficulties,
but introduces others. The majority of the electro  mechanical data
reduction devices (specifically, coordinate readers) which are now on
the market utilize rectangular coordinates. These instruments reduce
the teduim of coordinate reading, even when the desired result is
latitude and longitude coordinates. In this case the inverse map projection equations are required. Curiously, these are not widely available in the literature on the subject of map projections (with a few
exceptions) since the previous technology prohibited their extensive
use.
If the objective of the study does not include subsequent conversion
to latitude and longitude, a convenient procedure is to draw arbitrary
rectangular (or polar) coordinates on whatever maps or aerial photographs
are available. One advantage is that this can be done by persons with
no training and with virtually no intellectual effort or financial
expenditure. When the map used is accurate and at a "sufficiently large"
scale these arbitrary coordinates may be employed as are the map projection
coordinates discussed above. If the information collected has no permanent
value, this procedure is perfectly satisfactory.
A disadvantage is that the errors introduced are not known. The limits
within which a certain level of accuracy obtains is uncertain an one never
knows whether the system can be extended to include a neighboring
territory. A second major disadvantage is that it may not be possible
to use information collected for ore study in a second study which either
(a) encompasses a larger area than the original study, or (b) which is
subsequent in time to the original study, especially if the original map
has been lost, or (c) which requires a higher level of accuracy than the
11
the original study. One can imagine the difficulty of analyzing the
greater metropolitan area of Kansas City if Kansas City, Missouri and
Kansas City, Kansas, used two different and unrelated grid systems.
Or if each bureau of a city government employed a distinct system of
coordinates. The actual occurance of situations of this very nature
in the field of civil engineering is what gave impetus to the establishment of the system of State Plane Coordinates by the U.S. Coast and
Geodetic Survey in the 1930's.
Conversion between arbitrary map coordinates can be effected with
relative ease if the relation between the two systems is known, or if
both systems are related to latitude and longitude by known inverse
equations. If the relation between systems is not known it is theoretically possible to estimate the relation if the coordinates of a sufficient number of points are accurately known in both systems (see
Appendix). Such conversions may occasionally be required but are
expensive.
A final distinction should be made between coordinates and areas.
Coordinates describe points, not areas, and one must distinguish between an areal recording unit such as a census tract and between the
coordinate system used to pinpoint some centroid taken to represent
that areal unit. Areal information recording units are extremely
numerous and differ widely in size and shape. As a consequence it
is often necessary to convert from one areal unit (e.g. census tract)
to other areal units (school district, political precint, and so on).
These areal conversions differ somewhat from the coordinate conversions
discussed in this report. In general, specification of the areal boundaries must be included in the mathematical conversion statements.
There are then again several procedures, of varying accuracy and
complexity, which may be employed for the conversions.
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Cartography, Coast and Geodetic Survey Special Publication
No. 67 (Washington, Government Printing Office, 1921), 132 pp.
R. Bachi, "Standard Distance Measures and Related Methods for Spatial
Analysis", Papers, Regional Science Assn., X(1962), pp. 83132.
G. V. Bagratuni, "On the Accuracy of Distances and Azimuths obtained from
the solution of the Inverse Geodetic Problem," AERDLT1081, 1961.
H. P. Bailey, "Two Grid Systems that Divide the Entire Surface
of the Earth into Quadrilaterals of Equal Area", Transactions,
American Geophysical Union, XXXVII (1956), pp. 628635.
B. Berry, "Sampling, Coding, and Storing Flood Plain Data", Agriculture
Handbook No. 237, U.S. Department of Agriculture, Washington, 1962,
27 pp.
B. Berry, et. al., "Geographic Ordering of Information: New
Opportunities", The Professional Geographer, 16, 4 (July 1964) pp.
3640.
W. Bowie and 0. S. Adams, Grid System for Progressive Maps in
the United States, U.S. Coast and Geodetic Survey Special
Publication #59 (Washington, Government Printing Office, 1919),
227 pp.
R. M. Brooks, Coordinate Transformation Formulas, Pacific Missile
Range Technical Note. 3280220, 1962.
R. A. Bryson, "Fourier Analysis of Spatial Series," in Quantitative
Geography, W. L. Garrison, ed., Forthcoming.
Bureau of Land Management, Manual of Instructions for the Survey
of the Public Lands of the United States (Washington, Government
Printing Office, 1947), 613 pp.
J. D. Carroll, Jr., Chicago Area Transportation Study, Final
Report, Vol. I: Survey Findings (Chicago, CATS, 1959) 126 pp.
D. Clark, Plane and Geodetic Surveying, Vol. II, 4th ed.,
London, Constable and Co., 1951.
Coast and Geodetic Survey, PlaneCoordinate Systems, Serial 562
(Washington, Government Printing Office, 1948) 5 pp.
F. H. Collins, Coordinate Transformation, Technical Report
NAVTRADEVCEN 19077315, 1963.
R. L. Creighton, J. D. Carroll, Jr., and G. S. Finney, "Data
Processing For City Planning", Journal (American Institute of
Planners), XXV, 2 (1959), pp. 96103.
C. H. Deetz, and 0. S. Adams, Elements of Map Projection, Coast
and Geodetic Survey Special Publication No. 68, 5th ed.
(Washington, Government Printing Office, 1945), 60 pp.
S. C. Dodd and F. R. Pitts, "Proposals to Develop Statistical
Laws of Human Geography", Proceedings, IGU Regional Conference
in Japan (Tokyo, Kasai, 1959), pp. 302309.
F. Fiala, Mathematische Kartographie, (Berlin, Verlag Technik,
1957), 316 pp.
G. A. Ginzburg, "A Practical Method of Determining Distortion
On Maps", Geodezist, 10 (1935), pp. 4957.
D. I. Good, "Mathematical Conversion of Section, Township, and
Rxnge Notation to Cartesian Coordinates", Bulletin 170, part 3,
State Geological Survey of Kansas, 1964, 30 pp.
N. D. Haasbrock, Investigation of the Accuracy of Plotting and
Scalingoff, Netherlands Geodetic Commission, Delft, 1955.,i
T. HJgerstrand, "Statistika Primaruppgifter, Flygkartering
Och'Data Processing'  Maskiner: Ett Kombineringsprojekt",
Meddelanden Fran Lunds Geografiska Institution, Nr. 344 (Lund,
University of Lund, 1955), pp. 233255.
E. T. Homewood, "The Computation of Geodetic Areas...", Empire
Survey Review, XIII, 101, pp. 309321.
A. J. Hoskinson and J. A. Duerksen, Manual of Geodetic Astronomy,
Coast and Geodetic Survey Special Publication No. 237 (Washington;
Government Printing Office, 1947), 219 pp.
G. L. Hosmer, Geodesy (New York, Wiley, 1946).
B. R. Ingalls, Washington's Extended Use of State Plane Coordinates
(Olympia, Bureau of Surveys and Maps, 1957), llpp.
R. C. Kao, "Geometric Projections of the Sphere and the Spheroid",
The Canadian Geographer, v. 3. (Autumn 1961), pp. 1221.
R. C. Kao, Geometric Projections and Radar Data, (Santa Monica,
System Development Corp., 1959), 47 pp.
R. C. Kao, "The Use of Computers in the Processing and Analysis
of Geographic Information", The Geographical Review, 53(1963).
pp. 530547.
W. C. Krumbein, "Trend Surface Analysis of Contourtype Maps with
Irregular ControlPoint Spacing," Journal of Geophysical Research,
Vol. 64, 7 (July 1959), pp. 823834.
W. D. Lambert, Effect of Variations in the Assumed Figure of the
Earth on the Mapping of a Large Area, Coast and Geodetic Survey
Special Publication No. 100, Serial No. 258 (Washington,
Government Printing Office, 1924), 35pp.
W. D. Lambert, "The Distance between two Widely Separated Points
on the Surface of the Earth", Journal, Washington Academy of Sciences,
XXXII,5, (1942), pp. 125130.
E. A. Lewis, "Parametric Formulas for Geodesic Curves and Distances
on a Slightly Oblate Earth", Air Force Cambridge Research Laboratories,
April 1963, 37 pp.
A. Libault, Les Measures sur les Cartes et leur Incertitude,
Paris, 1961.
K. A. MacLachlan, "The Coordinate Method of 0 and D Analysis",
Highway Research Board Proceedings, 29th Annual Meeting (Washington
National Research Council, 1949) pp. 349367.
F. J. Marschner, "Structural Properties of Medium and small scale
maps", Annals, Association of American Geographer, XXXIV, 1, pp.
146.
F. J. Marschner, Boundaries and Records....(Washington, Farm
Economics Research Division, Department of Agriculture, 1960),
73 pp.
H. C. Mitchell and Lansing G. Simmons, The Plane Coordinate Systems,
Coast and Geodetic Survey Special Publication No. 235, (Washington,
Government Printing Office, 1945) 62 pp.
F. Moser, "A Computer Oriented System in Stratigraphic Analysis",
Ann Arbor, Institute of Technology, 1963.
D. Neft, "Statistical Analysis for Areal Distributions", Ph.D. Thesis,
Columbia University, 1962, 286 pp.
S. Nordbeck, Location of Areal Data For Computer Processing,
Lund Studies in Geography, Series C, 2, 1962, 41 pp.
J. O'Keefe "The New Military Grid of the Department of the Army",
Surveying and Mapping VIII, 4 (1948), pp. 214216.
J. A. O'Keefe, "The Universal Transverse Mercator Grid and
Projection", The Professional Geographer, N. S., IV, 5 (1952),
pp. 1924.
W. D. Pattison, Beginnings of the American Rectangular Land
Survey System, 17841800, Research paper no. 50 (Chicago; Department
of Geography, University of Chicago, 1957), 248 pp.
F. R. Pitts, "Committee on the Utilization of Stored Data Systems",
The Professional Geographer, 16, 4 (July, 1964) pp. 4144.
Radio Technical Comission for Aeronautics, "Coordinate System Aspects
of Position Identification", Journal of the Institute of Navigation,
Vol. 8, #1, Spring 1961, pp. 4858.
W. F. Reynolds, Relation Between Plane Rectangular Coordinates
and Geographic Positions, Coast & Geodetic Survey Special Publication
#71, (Washington, G. P. P., 1936), 90 pp.
E. Schmid, Transformation of Rectangular Space Coordinates, U.S.
Coast and Geodetic Survey Technical Bulletin No. 15 (Washington,
Government Printing Office, 1961), 13 pp.
A. I. Shevanova, "On the Accuracy of SmallScale Maps," Geodesy and
Cartography, (OTS, JPRS, L1389D.). 1957, pp. 3644.
L. G. Simmons, "How Accurate is FirstOrder Traingulation?",
Coast and Geodetic Survey Journal, 3 (April, 1950), pp. 5356.
B. W. Sitterly and J. A. Pierce, "Simple Computation of Distances
over the Earth", Journal, Institute of Navigation, 1, 4 (Dec. 1946),
pp. 6267.
E. M. Sodano, "General NonIterative Solution of the Inverse and
Direct Geodetic Problems" paper presented at 1963 IGU meeting.
Berkeley, California.
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(S. B. 83), "Washington Coordinate System" (Olympia, State Printer,
1945) 2 pp.
P. Thompson, Numerical Weather Analysis and Prediction, MacMillan,
New York, 1961, 170 pp.
W. R. Tobler, "A Comparison of Spherical and Ellipsoidal Measures",
The Professional Geographer, XVI, 4 (1964), pp. 912.
W. R. Tobler, "A Polynomial Representation of Michigan Population,"
1963 Papers, Michigan Academy of Science, Arts, and Letters, XLIX
(1964), pp. 445452.
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Air Force Regulation No. 965, (Washington, Department of the
Air Force, 1956), 7 pp.
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Lines Under 500 Miles, ACIC Technical Report No. 59 (St. Louis,
Aeronautical Chart and Information Center, 1960), 77 pp.
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Lines Over 500 Miles, ACIC Technical Report No. 80 (St. Louis,
Aeronautical Chart and Information Center, 1959), 83 pp.
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Government Printing Office, 1951), 324 pp.
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Coast and Geodetic Survey Special Publication No. 238(Washington,
Government Printing Office, 1947), 246 pp.
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Manual fo Photographic Interpretation (Washington; American Society
of Photogrammetry, 1960) pp. 667716.
APPENDIX I
CONVERSION FROM THE PUBLIC LAND SURVEY
SYSTEM TO LATITUDE AND LONGITUDE
The simplest conversion begins with a procedure which assumes that
the Public Land Survey conforms to the exact specifications upon which
it is based. The system, as is well known, does not conform to these
specifications, for a number of reasons including measurement errors
unavoidable in any empirical work and a certain laxity of supervision
during the establishment of the system. For conversion into latitude
and longitude the following notation is convenient:
i is an index to indicate the initial point of the survey. It is
necessary to distinguish at least 37 initial points in the Western
United States.
0 i is the latitude of the ith base parallel.
>hi is the longitude of the ith base meridian, with west longitudes
negative.
a is the equatorial radius of the ellipsoid taken to represent the
earth. For the Clarke Ellipsoid of 1866, a=3963.2257 miles.
e is the eccentricity of the ellipsoid taken to represent the earth.
For the Clarke Ellipsoid of 1866 e = 0.0822718542.
Mi is the radius of the meridian at the ith initial point. Mi is given
by
a(l  e2)
Mi:  (1 e2 sin2 0p)3/2
T is the township number of the location in question, with north
townships taken as positive and south townships taken as negative.
R is the range number of the location in question, with east ranges
taken as positive and west ranges taken as negative.
Sn is the northing of the section in question, with the sign convention
as above.
Se is the section easting, with the sign convention as above.
Qn is the quarter section northing, with signs as above.
Qe is the quarter section easting.
is the latitude (to be found) of the location in question.
N is the radius of curvature perpendicular to the meridian at
latitude 0:
N _a
(1  e2 sin2 0)1/2
is the longitude (to be found) of the location in question.
The necessary equations are then:
6 T 3 + Sn + Qn
0: 0i 
Mi
and
6 R 3 + Se + 0Q
e e
N C cos
The formulae are established by observing that the center of the
township in question should be six miles times the number of the township
north (south) of the base parallel, minus three miles to obtain the
center of the township. The section northing and easting give the
distance of the center of the section from the center of the township,
and the quartersection northing and easting give the distance of the
center of the quartersection from the center of the section. For the
SW 1/4, Sec. 25, T 5 N, R 17 E,, one should have for example, that the
center of the township is 27 miles (5 x 6  3) north of the initial point.
Sn is 1.5 miles and Q is 0.25 miles. The total distance in the
northsouth direction from the initial point should therefore be + 25.25
miles. This distance must then be converted to the appropriate number of
degrees and added to the latitude of the initial point. A further refinement, though hardly necessary, would be to iterate on the latitude obtained
in the first step in order to adjust the meridional radius employed in the
computation. Determination of the longitude is similar but slightly
more difficult since the distances are measured along a parallel (a
loxodrome, not a great circle or geodesic), whose radius varies with the
latitude.
In programming the outlined procedure for a digital computer it is
simplest to employ radians instead of degrees and to store a table of
Sn, Se, Qn, Qn,, 0i, and Ai. The computer can perform the assignment
to the correct initial point by letter for letter examination of the name
of the principal meridian. A convention is necessary to distinguish
the two different initial points employed for the Fourth Principal Meridian.
The method detailed assigns latitude and longitude (to about the nearest
1/4 mile) on the assumption that the Public Land Survey designations are
where they should be. Of course they are not exactly there: the legal
strategy is to assign to the actual locations a status of incontestable
correctness, irrespective of any errors which may have been introduced
during the survey. To adjust the calculated values to conform to their
legal positions requires detailed historical and empirical corrections,
and can be quite tedious. For many research purposes, however, such a
refinement may not be necessary. To obtain an orderofmagnitude estimate
of the discrepancies, the actual latitude and longitude (as recorded on
large scale topographic maps) of a scattered set of locations have been
compared with the computed values. For a selection of 74 points within
the State of Michigan the errors are as follows:
Distribution of Errors: (N = 74)
Mean: 2.849 miles
Standard deviation: 1.827 miles
Maximum: 9.339 miles
60o of the errors are less than 2 miles
93% of the errors are less than 5.5 miles
The directional errors appear evenly distributed in all directions.
A random selection of points (N = 25) from other states indicates that
the errors are quite comparable and of the same order of magnitude. A
sample computation is as follows:
observed location: Si 1/4, Sec. 28, T 2 S, R 6 E, Michigan Meridian
calculated Lat/Lon: 42~ 16'07' N, 83~0 4' 08T1 U
observed Lat/Lon: 42~ 17'10' N, 63~ 44'4911"
difference: 1'03' 41'i
difference in miles: 2.53
direction of difference: 154.34~ (E of N)
The method given above does not include an adjustment for the convergence of the meridians. Since the edges of the ranges run due north
and south, the ranges become narrower as the meridians converge. To adjust
for this, standard parallels are established every twentyfour miles north
and south of the base parallel. The ranges are again made six miles wide at
these standard parallels. The system thus is self correcting every twentyfour miles. The order of magnitude of the difference in width of ranges,
separated by twentyfour miles in a northsouth direction, can be established
as follows: The radius of the parallel at 45~N latitude is 2807.178 miles.
At a distance of 24 miles north of 45~N it is approximately 2789.834 miles.
The eastwest width of the northern edge of the range 24 miles north of
the 45th parallel is therefore not six miles but 0.037104 miles (195.9 feet)
less than six miles. On this basis the error at R 50 E, an extreme value,
would be 1.86 miles. Another slight error is introduced by the topographic
elevation, since the radii employed apply to a mean sea level ellipsoid.
Empirical corrections for Michigan would need to include the fact that the
standard parallels are 60 (not 24) miles apart (in accord with the surveying
instructions in force at the time), and that R 1 E is consistently too
narrow from T 1 N to T 20 N. An adjustment for these, and other, systematic departures could be incorporated into the computer program. Conversion of the Section, Township, and Range information to latitude and
longitude can be followed by conversion to map projection coordinates
for map plotting or computational purposes. Direct conversion to
Cartesian map coordinates also is possible but is less convenient for
the entire Western United States. This is more appropriate for operations
restricted to a limited area, e.g., one individual state.
1
APPENDIX II
MAP PROJECTION EQUATIONS
The following list gives the mathematical rules for the most commoo
map projections of a sphere. The following notation is standard.
Latitude of a point whose projection coordinates are desired.
\ Longitude of a point whose projection coordinates are desired.
) Abscissa of a plane cartesian coordinate system.
Ordinate of a plane cartesian coordinate system.
C Radial distance of a plane polar coordinate system.
0 Angular direction of a plane polar coordinate system.' Latitude of the center of the map; either the point of "tangency",
or a single standard parallel.
lp Southerly standard parallel for projections having two standard
parallels.
(/ Northerly standard parallel for projections having two standard
L parallels.
0 Longitude of the center of the map; either the point of "tangency",
or the central meridian.
C The constant of the cone for conic projections.
Y The radial distance from the origin to the image of the southerly
standard parallel in plane polar coordinates.
All equations are given for a sphere of unit radius (R = 1) and all
values are assumed to be in radians. Conversion to scale can be achieved
by multiplying all distances by the appropriate scale factor. North
latitudes and east longitudes are taken to be positive, i.e.
i\ ~ A
The equations are given in their most commonly applied form. The
conical projections, for example, are not given in their oblique cases.
2
(1) Albers' equal area conic projection with two standard parallels:
s9= C (X^)
This puts the origin of the coordinates somewhat beyond the north
pole, which is rather inconvenient. The origin can be shifted to the
intersection of the southern standard parallel with the central meridian
by using
)= r 5sn
y V I  CoSS
(2) Azimuthal equidistant projection:
r= arc. os I ~n' <'" +$ tosv x qoS h: 1 A  D) J
e cs\= acC S . S \ I.f  j
The origin of the coordinates is at to ) o
(3) Bonne's Equal Area projection.
^ (p Op t, a(^ y> 2 )
) )

3
To place the origin at the intersection of the standard parallel and
the central meridian use:
)(z ~c^''^)
\/= r S& 5
y = r,;  lr <Lo 8
(4) Cassini Projection
Ovc   V Ic 4\I CP t"' ~dn Co^ (XXo).
x~~~~~
(5) Gnomonic Projection:
51iv n 5i (4\ + Cos4( Cos % a, sOC<> X,)
Stn 4 C>S 4^  Cl x ^ ^ ^ ^s ^  \^
1 "~h ^~ Sin dx ) Los cf CoS i0 Co 0 A o'
(6) Lambert's azimuthal equal area projection:
6 =rc o' + (261 Q.Co9o9
I Sini C\ C
oe= ci c ear (roec )
(7) Lambert's cylindrical equal area projection:
x^ \ >^
^ = <.~^
4
Or, with a standard parallel:
X 2 (  N) Cos TO
(8) Lambert's conformal conic with two standard parallels:
In Cos {,  Lh CO,,S (z.
L, ( + ) L h tA >\ ( L  )
^Co n (s )
= lC ( 2)
X = r Y  C3
y = r,  v <os e
(9) Mercator's conformal cylindrical projection:
X X X X
Y: L(. if ( )
(10) Miller's Cylindrical projection:
X   Xo
NNe
5
(11) Mollweide's equal area elliptical projection:
Define chi by 1 ~/ + 3 Sin 4 1 S sI Lf ) H,
X^.^ (A>.)(^S y
X J7 s;Y(12) Orthographic projection:
X= San ^ Cjs ip,  tl< ^^ ^ eos ( \A \.
X1 Co$ s f Q (Q(,)
(13) Polyconic projection (American polyconic):
r — d^~ ^
y) r +' P C
Which puts the origin at the equator.
(14) Sinusoidal equal area projection:
y (>.>^) C ^
(15) Square projection:
or, with a standard parallel (also known as the rectangular projection):
X — (xy,) os cP
y CP
6
(16) Stereographic projection:
4^ SlvCos 4 sl < (Poiy l, $f J os Cos (>.X.)
y I s^^ S si`"o + (yt t cp Cos.p. C.s(A A,)
(17) Transverse Mercator projection.
X 7  14 Cos Sih b^ co )(
i o< p ge c (._ <,)
 ~r^
APPENDIX III
LEAST SQUARES CONVERSION FROM ONE SYSTEM OF
RECTANGULAR COORDINATES TO ANOTHER
Given two sets of coordinates on the same map, with a minimum
of five points identified in both systems of coordinates, it is possible
to convert the coordinates of one set to the other by a twodimensional
version of a leastsquares "line'l The procedure is most easily effected
using complex numbers.
Let x,y be one set of coordinates and u,v be the other set, and
let W: x + iy and Z* u — iv, where i2 1, be the complex numbers
representing the ith point. The objective is then to find the complex
constants A = a,+ ia2 and B = bI+ ib2 in the equation W 1 A I BZ such
that the squared residual
2I \&.jvw^i
J='1
is a minimum. The normal equations are readily obtained by differentiation. The equation can be rewritten as a pair of transformation equations
by separating the real and imaginary parts, viz:
A A
Re (W) = x = a + b u + b2v
A A
Im (W) y a2 4 b2 u  blV
/A A
where x and y are the estimates of the x,y coordinates as obtained from
the known u,v coordinates. The standard error, etc., of the estimate
can be obtained in a manner analogous to that employed for ordinary
least squares procedures.
A similar, but considerably more complicated, procedure must be
employed if the two sets of coordinates do not come from the same map,
or if the relation to latitude and longitude is to be estimated, or
if an attempt is made to determine the map projection of an arbitrary
map.
APPENDIX IV
R CLARKE ELLIPSOID OF 1866
R DISTANCE AND DIRECTION / SODANO METHOD
R W. R. TOBLER / UNIVERSITY OF MICHIGAN / GEOGRAPHY
$COMPILE MAD, PUNCH OBJECT
EXTERNAL FUNCTION (LT1.LG1*LT29LG2#DISsDIRD)
R ENTRY IN RADIANS
R RETURNS DISTANCE IN KILOMETERS
R RETURNS DIRECTION IN DECIMAL DEGREES
R ACIC TR 809 PAGES 4147.
R NECESSARY CONSTANTS
VECTOR VALUES PI=314159265E8
VECTOR VALUES TPI=628318531E8
VECTOR VALUES ARAD=63782064E4
VECTOR VALUES BRAD=63565838E4
VECTOR VALUES BOVRA=9966099247E10
VECTOR VALUES VK1=2356218428E7
VECTOR VALUES VK2=6956258069E5
VECTOR VALUES VK3=4986428206E9
VECTOR VALUES VK4=4010886986E10
VECTOR VALUES VK5=7994556507E10
VECTOR VALUES VK6=3986428206E9
VECTOR VALUES E1=17036962E10
VECTOR VALUES E2=21769E10
VECTOR VALUES E3=29026E10
VECTOR VALUES E4=3628E10
VECTOR VALUES RAD=174532925E10
R BEGIN COMPUTATION
ENTRY TO CLARKE.
INDEX=1.
TANB1=BOVRA*(SIN (LT1)/COS (LT1 ))
TANB2=BOVRA*(SIN (LT2)/COS(LT2) )
COSB11l./SQRT. (1.+ TANB1*TANB1 )
COSB2=1./SORT.(1.+(TANB2*TANB2))
SINB =TANB 1COSB1
SINB2=TANB2*COSB2
C1=SINB1*SINB2
D1=COSB1*COSB2
DIFLON=LG2LG1
WHENEVER DIFLON.L.O.* INDEX=1
DIFLON=.ABS DIFLON
CDIF=COS.( DIFLON)
CDIS=C1+D1*CDIF
SDIS=SQRT. (1CDIS*CDIS)
CA=D1*SIN (DIFLON)/SDIS
CB=CA*CA
CC=CDIS (1.CB)/VK3
CD=VK4C 1
CE=VK5*C1
CF=VK6*CC
CG1=2.*ATAN.(SQRT.((1.CDIS)/(1.+CDIS)))
CG=CG1/SDI S
CX=CA*((CG1*(VK1+CB)+SDIS*(CC+CD)+CG*(CE+CF))/VK2)
DELTAL=CX+D IFLON
SDELTL=SIN. (DELTAL)
CDELTL=COS. (DELTAL)
DEN=TANB2*COSB1S INB1*CDELTL
DI RATN1 ( SDELTL DEN)
WHENEVER DIR.G.PIDIR=DIRTPI
DIRD=DIR/RAD
DIRD=DIRD*INDEX
CPHO=C1+D1*CDELTL
SPHO=SQRT. ( 1.CPHO*CPHO)
CBO=D1*SDELTL/SPHO
APHO=2.*ATAN (SQRT. ((1.CPHO) / (*+CPHO)))
SB02=1.CBO*CBO
C2DEL= (2 *Cl/SB02)CPHO
C4DEL= (2 *C2DEL*C2DEL) 1
SB04=SB02*SB02
S2PHO=SIN.(2.*APHO)
AO=1.+E1*SB02E2*SB04
BO=E1*SB02E3*SB04
CO=E4*SB04
DIS=BRAD( AO*APHO+BO*SPHO*C2DELCOS*2PHO*C4DEL)
FUNCTION RETURN
END OF FUNCTION
APPENDIX V
R CONVERSION OF PUBLIC LAND SURVEY INFORMATION
R INTO LATITUDE AND LONGITUDE
R SUBROUTINES NEEDED ARE DEGRAD, RADEG, SPHEREt AVERAD
R W. R. TOBLER /UNIVERSITY OF MICHIGAN / GEOGRAPHY
$COMPILE MADIPRINT OBJECT.PUNCH OBJECT.EXECUTE TRC
INTEGER MER.C1lC2,C3,C4,C5
INTEGER COMPAR NTWP RNGQl1Q2,SPRINCNl1
INTEGER R.ST
D'N SECE(37) SECN(37),PMERID(36) BLINE(36) RMER(36)
V'S DLT(1)=43.0943.0,35.0,31.0,360,961.0,64*0,34*0,40*0,40.0O
1 420.C33.0,40.0B34.0,34.0,31.0,42.0,37*0,35.0,34*0,45.0,40.0,
2 34.0,38 0,60*0,400,30.0,30.0,30.0,38O0,40*0,39*0,30eO,450O,
3 43.0
V'S MLT(1)=59*0,22.0,1.0952.0,30o.049.0,51.0,38.0,59*.O,00,
1 30.0922.0,25.0,59.0,29.0,0.0,25.0,52.0,44.0,15.0,47*.046.0,
2 7.0,28.0,7.0,0.05959.0,59.0,26.0,28.0,25.0,6.0.59.0,31.00.0
V'S SLT(1)=44.0,21*0,53.032.0,5.0,21.0*50.0,45.0922.0,50.0,
1 27.0.38.0,2.0,27,0,32.0,31.0,28.0,54.0,56.0,35.0,13*0,11.0,
2 20.0,14.0,36.0,7.0,56.0,51.0,3.0,27.0,59.0,23.0,560,11.0,
3 41.0
V'S DLG(1)=104.0,116.0,89.0,90.0,103.0,145.0,147*O091.0,84*.0
1 90.0,90.0,112.0,124.0,86.0,97.0,92.0,84.0,121.0,108.0,106.0,
2 111.0,111.0,116.0,86.0 149.0,97.0,91.088.0,84.0,89.0,109.0,
3 108.0,91.0,122.0108.0
V'S MLG(1)=3.0,23.00,14.014.0,0.0,18.0,38*.03.048.0,27*.0
1 25.0,18.0,7.0,34.0,14.024.0o21.0,54.0,31.0,53.0939.0,53.0,
2 55.0,27.0,21.0,22.09.0,1.0,16.0,8.0,56.031.09.0,944.048.0
V'S SLG(1)=16.0,35.047.0,41.0,7.0,13.026.0,7.0011.0,11.0,
1 37.0,19.0.10.0,l6.0,49.055.0,53.0,47*.059.0,12.033.0,27.0
2 17.0.21.0,24.0,8.0,36.020.038.0,540,6.0,59.0,36*.034.0,
3 49.0
SECE(0)=0.0
V'S SECE(1)=2.5i1.5,0.5,0.5i1.5,..5i2.5i1.5.0.5.0.5,
11.5,2.5.2.5,1.5,0.5,0.51.52.5,2.51.5.0.50.5.15,~
22.5,25,1.5,05,0.5,0.5,1.52.5,2.5i1.5,0.5.0.51.5.2.5
SECN(O)=0.0
V'S SECN(1)=25,25,2.255.22.5,2525.52.51.5.1.5,1.5,15,1.5.1.5
10.5,0.5.0.50.55 05,0 i50.50.5i,0.5*0.5505.5 0 5. 1.5,
21.5,1.5,1.51.5,1.5,2.5,2.5,2.52.5,2.5,2.5
R1=63782064E04
MILE=0.62136994
ESQR=6768658E09
RAD=174532925E10
T'H INITAL, FOR I=1,1lI.G.35
DLG(I)=DLG(I)
EXECUTE DEGRAD.(DLT(I),MLT(I),SLT(I),BLINE(I))
EXECUTE DEGRAD.(DLG(I),MLG(I),SLG(I)IPMERID(I))
SMLT=SIN.(BLINE(I))
DUM=SQRT(. 1ESQR*SMLT*SMLT)
DUMCUB=DUM P.3
DUMMY=(l ESQR)*Rl
RMER(I)=DUMMY*MILE/DUMCUB
INITAL CONTINUE
R'T CONS.COMPAR
V'S CONS=$S3,I1*$
N=0
N1=0
P'T SKIP
V'S SKIP=$1H1*$
START W'R COMPAR.GE.
R'T LATLON Q1t Q2.S.T*TWP R.RNGC1 tC2,C3*C4C5,
1DLAT MLATLASLAT,DLON MLONSLON
V'SLATLON=$2C1S0I 12.S3 I2.C1iS3,I2 C1.S2.5C1lS16,F3.00
12F2.0,S1,F4.02F2.0*$
EXECUTE DEGRAD.(DLATMLATSLATRLAT)
EXECUTE DEGRAD.(DLON.MLONSLON.RLON)
O'E
R'T INDATQl,Q2,STTWP,RRNG9C1l C2,C3.C4,C5
V'SINDAT =$2C1 SS0 I2,S3 I 2C1 S3. I2C1 S2 5C1 *$
E'L
N=N+1
N1=N1+1
W'R TWP.E.$N$
A=T*6.03.0
O'R TWP.E.$S$
A=( T*6 0 30)''E
T'0 ERR
E'L
W'R RNG.E.$E$
B=R*6*03.0
O'R RNG.E.$W$
B=R*6 0+3 0
O'E
T'0 ERR
E'L
W'R Q1.E.$N$
QN=0.25
O'R Q1.E*$S$
QN=0O25
0'R Q1.E$ $
QN=00' E
T'0 ERR
E'L
WIR Q2.E.$E$
QE=025
0'R Q2.E.$W$
QE=0.25
O'R Q2.E.$ $
QE=0.O
O'E
T'0 ERR
E'L
W'R C1.E.$B$
W'R C2.E.$0$
MER=2
O'E
MER=1
E'L
O'R C1.E.$C$
W'R C2.E.$I$
MER=5
O'R C2.E.$0$
MER=6
O'E
W'R C3.E.$I$
MER=3
O' E
MER4
E'L
E'L
O'R C1.E.$F$
MER=7
O'R ClES$5$
MER=8
O'R C1.E.$1$
MER=9
O'R C1.E.$4$
W'R (C4. E$A$) OR. (C5E.$A$)
MER=10
O' E
MER=11
E'L
O'R C1.E*$G$
MER=12
O'R C1.E.$H$
W R C3E. $M$
MER=13
O0'E
MER=14
E'L
O'R C1.E.$I$
MER=15
0'R C1.E.$L$
MER=16
O'R C1 E.$M$
W'R C2E.$I$
MER=17
O'E
MER=18
E'L
O'R C1.E.$N$
W'R C2.E.$A$
MER=19
O'E
MER=20
E'L
O'R C1.E.$P$
MER=21
O'R C1.E.$S$
W'R C3.E.$L$
MER=22
O'R C3.E.$N$
MER=23
O'R C3.E.$W$
MER=25
O' R ( C3.E.$H$) OR.(C4.E.$H$) OR (C5.E.$H$)
MER=27
0'E
MER=28
E'L
O'R C1.E.$2$
MER=24
O'R C1.E.$6$
MER=26
O'R C1.E.$T$
MER=29
O'R C1.E.$U$
W'R C2.E.$I$
MER=31
O' E
MER=32
E'L
O'R C1.E.$W$
W'R C3.E.$N$
MER=35
O'E
MER=34
W'R C2.E.$A$.MER=33
E'L
O'R C1.E.$3$
MER=30' E
T'O ERR
E'L
A=A+SECN(S) +QN
A=A/RMER (MER)
LATIT=BLINE(MER)+A
CLAT=COS.(LATIT)
SMLT=SIN.(LATIT)
DUM=SQRT.(1.ESQR*SMLT*SMLT)
RPAR=R1*MI LE*CLAT/DUM
B=(B+SECE(S)+QE)/RPAR
LONGIT=PMERID(MER)+B
EXECUTE RADEG.(LATITLTD9LTMLTS)
EXECUTE RADEG.(LONGIT.LGDLGMLGS)
P'T ONE,NI
V'S ONE=$1H ////S1.I4*$
P'TRIQ1,Q2,S,T,TWPRRNG,ClC2,C3,C4,C5,
lLTD LTMLTSiLGDLGMLGS
V'SRI=$1H,2C1,10H 1/4, SEC,I2,3H, TI2,C1,3H, RI2,C1.
12H,,S2,5C1,10H. MERIDIAN //S1,2(F5.0F3.0.F3OSS5),
219HCALCULATED LAT/LONG *$
W'R COMPAR.GE.1
DIFLON=LONGI TRLON
DIFLAT=LAT I TRLAT
EXECUTE RADEG.(DIFLATDLTDDLTMDLTS)
EXECUTE RADEG.(DIFLONDLGDLGDDLGMDLGS)
EXECUTE SPHERE.(RLATRLONLATIT,LONGITRHOALPHA)
EXECUTE AVERAD.(RLATLATIT,Rl,ESQRMRAD )
RHO=RHO*MRAD
ALPHA=ALPHA/RAD
P'TR2 DLATMLAT,SLAT,DLOONMLONSLONDLTDDLTMDLTS DLGD
1DLGM DLGSRHOALPHA
V'S R2=$1H,2(F5.0F3.0,F3.0,S5),17HOBSERVED LAT/LONG /S1,2(F
15.0,F3.0,F3.0oS5),10HDIFFERENCE /Sl,Fll.4,S6,F9.4,S6,
222HMILES AND DIRECTION *$
PUNCH FORMAT OUT,N1,RHOALPHA
V'SOUT=$I5,S2,F11.4,S2,F9.4*$
E'L
TRANSFER TO START
ERR PRINT FORMAT ONE,N
PRINT COMMENTS THIS OBSERVATION IS INCORRECTLY RECORDED$
N1=N11
TRANSFER TO START
E'M
$COMP I LEMAD PUNCHOBJECT AVERAD
EXTERNAL FUNCTION (LLTTULTR1,ESQR.R3)
R MEAN RADIUS ON ELLIPSOID
R LATITUDES IN RADIANS
ENTRY TO AVERAD.
SMLT=SIN.( (LLT+ULT)/2.)
DUM=SQRT. (1.ESQR*SMLT*SMLT)
DUMC U B DUM*DUM*DUM
DUMMY=(1.ESQR )*R
RMER=DUMMY/DUMCUB
RPAR=R1/DUM
R3=SQRT. (RMER*RPAR)
FUNCTION RETURN
END OF FUNCTION
$COMP I LEMAD PUNCHOBJECT DEGRAD
EXTERNAL FUNCTION (DEGMINSEC*RAD)
R SUBROUTINE TO CONVERT DEGREES TO RADIANS
ENTRY TO DEGRAD.
VECTOR VALUES RADIAN=174532925EO1
SIGN=RADIAN
WHENEVER DEG.L.O.. SIGN=RADIAN
RAD=SIGN*(.ABS (DEG)+(MIN/60.)+(SEC/3600) )
FUNCTION RETURN
END OF FUNCTION
$COMP I LEMADPUNCHOBJECT RADEG
EXTERNAL FUNCTION (RADDEGMIN*SEC)
R CONVERTS RADIANS TO DEGREES,MINUTES, AND DECIMAL SECONDS
INTEGER I
ENTRY TO RADEG.
VECTOR VALUES CONS=206264806E3
SEC=ABS.( RAD)*CONS
I=SEC/3600.
REMAIN=SEC (*3600.)
DEG=I*1
I=REMAIN/60.
MIN=I*1.
SEC=REMAIN( *60 )
WHENEVER RAD.L.*O*DEG=DEG
FUNCTION RETURN
END OF FUNCTION
$COMPILE MAD,PRINT OBJECT,PUNCH OBJECT SPHERE
R COMPUTES OBLIQUE SPHERICAL COORDINATES
EXTERNAL FUNCTION(NLTNLGLAT LONRHO2,GA)
VECTORVALUESPI=314159265E8
VECTORVALUESTP I =628318531E8
VECTORVALUESPIOVR2= 157079633E8
VECTORVALUESEPS=O.0000001
VECTORVALUESRAD= 174532925E10
ENTRY TO SPHERE.
WHENEVERNLTE. (90.*RAD)
GA=LONNLG
RH02zPIOVR2LAT
OTHERWISE
WHENEVER(LT.NE.NLT).OR.(LG.NE.NLG)
PI=314159265E8
TPI=2.*PI
PIOVR2P I/2.
EPS=0O0000001
CNLT=COS ( NLT)
SNLT=SIN ( NLT)
END OF CONDITIONAL
WHENEVER LON.NE.LON1
DIF=LONNLG
CDIF=COS.(DIF)
SDIF=SIN.( DIF)
END OF CONDITIONAL
CLT=COS (LAT)
SLT=SIN. (LAT)
Q=SLT*SNLT+CLT*CNLT*CDIF
WHENEVER Q.GE.1.
RH02=O0
ORWHENEVER Q.LE.1.
RH02=PI
OTHERWISE
RH02=ARCCO S (Q)
END OF CONDITIONAL
NUM=CLT*SDI F
DEN=CNLT*SLTSNLT*CLT*CDIF
WHENEVER.ABS.DEN.L.EPS
WHENEVER.ABS.NUM*.LEPS
GA=O
OTHERWISE
GA=PIOVR2
WHENEVER NUM.L.O. GA=GA
END OF CONDITIONAL
ORWHENEVER.ABS.NUM.L.EPS
GA=O.
WHENEVER DEN.L.O.,GA=PI
OTHERWISE
GA=ATN1.(NUMDEN)
WHENEVER GA.G.PI,GAGATPI
END OF CONDITIONAL
LON 1 =LON
LT=NLT
LG=NLG
END OF CONDITIONAL
FUNCTION RETURN
END OF FUNCTION
NW 1/4 SEC 04 T05S R07W MICHIGAN +420405 0850800 UNION CITY MI
DISTRIBUTION LIST
(One copy unless otherwise noted)
Chief of Naval Research 2 Commanding Officer
Office of Naval Research Army Map Service
Washington 25, D.C. 6500 Brooks Lane
Attn: Geography Branch Washington 25, D.C.
Defense Documentation Center 20 Dr. Reid A. Bryson
Cameron Station Department of Meteorology
Alexandria, Virginia 22314 University of Wisconsin
Madison 6, Wisconsin
Director 6
Naval Research Laboratory Mr. Robert Leland
Washington 25, D.C. Cornell Aeronautical Laboratory
Attn: Tech. Info. Officer P. 0. Box 235
Buffalo 21, New York
Director 2
Central Intelligence Agency Dr. Richard J. Russell
Washington 25, D.C. Coastal Studies Institute
Attn: Map Division Louisiana State University
Baton Rouge 3, Louisiana
Commanding Officer
Office of Naval Research Dr. Charles B. Hitchcock
Branch Office American Geographical Society
230 N. Michigan Avenue Broadway at 156th Street
Chicago, Illinois 60601 New York 32, New York
Commanding Officer Dr. Edward B. Espenshade
Office of Naval Research Department of Geography
Navy No. 100 Northwestern University
Fleet Post Office Evanston, Illinois
New York, New York
Dr. Brian J. L. Berry
The Oceanographer Department of Geography
U. S. Navy Oceanographic Office University of Chicago
Washington 25, D.C. Chicago 37, Illinois
Commanding Officer Dr. William L. Garrison
U. S. Naval Photo Interpretation Department of Geography
Centre Northwestern University
4301 Suitland Road Evanston, Illinois
Washington 25, D.C.
Dr. William C. Krumbein
Geography Division Department of Geology
Bureau of the Census Northwestern University
Washington 25, D.C. Evanston, Illinois
DISTRIBUTION LIST (Concluded)
Dr. Ruth M. Davis Professor John D. Nystuen
Office of Director of Defense Department of Geography
Research and Engineering The University of Michigan
Department of Defense Ann Arbor, Michigan
Washington 25, D.C.
Professor M. F. Dacey
Dr. Leslie Curry Department of Regional Science
Department of Geography University of Pennsylvania
Arizona State College Philadelphia 4, Pennsylvania
Tempe, Arizona
Professor Edwin Thomas
Dr. M. Gordon Wolman Department of Geography
Department of Geography Arizona State College
Johns Hopkins University Tempe, Arizona
Baltimore 18, Maryland
Professor Forrest R. Pitts
Dr. Richard C. Kao Department of Geography
Economics Department University of Oregon
The RAND Corporation Eugene, Oregon
1700 Main Street
Santa Monica, California Professor Edwin Taaffe
Department of Geography
U. S. Navy Oceanographic Office The Ohio State University
Washington 25, D.C. Columbus 10, Ohio
Attn: Code 5005
Dr. Lewis T. Reinwald
Professor J. Ross Mackay 10002 Cedar Lane
Department of Geography Kensington, Maryland
University of British Columbia
Vancouver, British Columbia, Canada Dr. Duane F. Marble
Department of Geography
Professor William Bunge Northwestern University
Department of Geography Evanston, Illinois
Wayne State University
Detroit, Michigan Dr. John C. Sherman
Department of Geography
Dr. Allen V. Hershey, Head University of Washington
Mathematical Physics Branch Seattle 5, Washington
Computation and Analysis Laboratory
U. S. Naval Weapons Laboratory
Dahlgren, Virginia
Unclassified
Security Classification
DOCUMENT CONTROL DATA  R&D
(Security classeiication of title, body of abstract and indexing annotation must be entered when the overall report ie classified)
1. ORIGINATING ACTIVITY (Corporate author) 2a. REPORT SECURITY C LASSIFICATION
The University of Michigan Unclassified
Ann Arbor, Michigan 2b GROUP
3 REPORT TITLE
GEOGRAPHICAL COORDINATE COMPUTATIONS
PART I: GENERAL CONSIDERATIONS
4. DESCRIPTIVE NOTES (Type of report and inclusive datee)
Technical Report No. 2
5. AUTHOR(S) (Laot name, first name, initial)
Tobler, W. R.
6. REPORT DATE 7a. TOTAL NO. OF PAGES 76. NO. OF REFS
December 1964 45 65
8a. CONTRACT OR GRANT NO. 9a. ORIGINATOR'S REPORT NUMBER(S)
Nonr 1224(48) 058242T
b. PROJECT NO.
c. Task No. 589157 9b. OTHER RE PORT NO(S) (Any other number that may be assined
this report)
d.
10. A VA IL ABILITY/LIMITATION NOTICES
Qualified requesters may obtain copies of this report from DDC.
II. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY
Office of Naval Research
Geography Branch
__ _ _ _ _ _ _  ~~~~~~~Washington, D. C.
13. ABSTRACT
Part I provides a discussion of the usefulness of coordinate models
for studies of geographically distributed phenomena with comments on
specific coordinate systems and their relevance for the analysis and inventorying of geographical information. Appendices include equations for
conversion from the Public Land Survey system into latitude and longitude
and to rectangular map projection coordinates. Part II considers map
projections in greater detail, including estimates of the errors introduced
by the substitution of map projection coordinates for spherical coordinates.
Statistical computations of finite distortion are related to Tissot's Indicatrix as a general contribution to the analysis of map projections.
DD, JAN 64 1473 Unclassified
Security Classification
Unclassified
Security Classification
14. KLINK A LINK B LINK C
KEY WORDS .ROLE WT ROLE WT ROLE WT
Geography
Coordinate conversion
Map projections
Spatial analysis
Information processing
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Unclassified
Security Classification