PRE-GEOMETRY DISCUSSION OF CALCULATING DISTANCES ON THE EARTH'S SURFACE

(The professional-level discussion follows this brief summary.)

The Earth is round, but big, so we can consider it flat for short

distances. But, even though the circumference of the Earth is about

25,000 miles (40,000 kilometers), flat-Earth formulas for calculating

the distance between two points start showing noticeable errors when

the distance is more than about 12 miles (20 kilometers). Of course,

how much error is "noticeable" depends on how you are going to use the

result.

If lots of distances are going to be computed, the amount of

calculation has to be considered. I studied this issue because I was

working on a computer program to support U.S. Army training exercises

- our program typically calculated hundreds of thousands of distances

every hour.

Cartesian coordinates express distances in two different directions,

such as north-south for one direction and east-west for the other.

The straight line distance between two points can then be thought of

as the long side of a right triangle with one of the short sides being

the north-south distance between the points and the other being the

east-west distance. (A right triangle is one that has a square

corner.) The usual formula for computing the length of the long side

of a right triangle is the Pythagorean Theorem. Using this formula

from geometry requires knowing about square roots.

Near the North Pole and near the South Pole, the lines that go

north-south (called the meridians) approach each other - in fact, they

meet at the pole. The lines that go east-west are circles around the

pole. Treating differences in locations along these directions as if

they were the sides of a right triangle leads to errors in the

resultant distance. Very close to the pole, the answer could be VERY

wrong - but a different flat-Earth approximation, obtained from plane

trigonometry, can be used for short distances: the Polar Coordinate

Flat-Earth Formula. Using this formula - and the others mentioned

below - requires knowing something about trigonometry.

Latitude and longitude are spherical coordinates, based on recognition

that the Earth is round. Their definition does not require that the

Earth be exactly spherical, but approximating the Earth as a sphere is

satisfactory for most needs. Converting from spherical coordinates to

flat (Cartesian) coordinates is what most map projections are all

about; a lot of calculation is sometimes needed. The easiest is to use

north-south for one direction and east-west for the other - but that

has problems near the poles, as I described above.

When you study spherical trigonometry, you learn a bunch of formulas.

One of them is called the Law of Cosines for Spherical Trigonometry.

It is a perfectly fine formula when it is used for the right purposes.

Solving for short distances on the surface of the Earth is not one of

those purposes. The problem is as follows: Suppose you have a right

triangle with a very small angle. The ratio between the looooooong

short side and the long side is very close to 1.0. The formula

computes that ratio first, then requires the computer to find the angle

that has that ratio. In principle, the computer can do so - after all,

the formula is mathematically correct - but ordinary computers

approximate all numbers to some number of what are called "significant

digits". With 7 or 8 significant digits, the computer cannot

distinguish between the ratios for angles smaller than about a minute

of arc (a minute is 1/60 of a degree). Since the angle being computed

has its apex at the center of the Earth, a minute of arc corresponds to

about a mile on the surface.

Since the formula is mathematically correct, it can be manipulated

into other forms. The Haversine Formula is one result of such

manipulations. It has a similar problem, but it is "poorly

conditioned" when the two points are all the way around the Earth

from each other, rather than when they are close to each other. The

discussion below gives a second version of the Haversine formula that

is easier to program on some computers.

It also discusses the fact that the Earth is not quite a sphere, and

gives several references for further reading. There is discussion of

where to look if even the ellipsoid approximation is too coarse. It

also has a pointer to one source of navigation formulas on the

Internet.

******* PROFESSIONAL-LEVEL DISCUSSION *******

******* CALCULATING DISTANCES ON THE SURFACE OF THE EARTH *******

What is the best way to calculate the distance between 2 points?

From: Bob Chamberlain <rgc@solstice.jpl.nasa.gov>

The distances considered here are along the surface of the Earth,

and deliberately ignore the effect of differences in elevation.

Distances on the surface of the terrain, whether geodesic, on roads,

or cross-country, depend on relief (including elevation differences),

the status of engineering projects, and perhaps even route selection.

Hence, computation is idiosyncratic and not well suited to simple

approximations.

If the distance is less than about 20 km (12 mi) and the locations of

the two points in Cartesian coordinates are X1,Y1 and X2,Y2 then the

Pythagorean Theorem

d = sqrt((X2 - X1)^2 + (Y2 - Y1)^2)

will require the least amount of computation and will be in error by

less than 30 meters (100 ft) for latitudes less than 70 degrees

less than 20 meters ( 66 ft) for latitudes less than 50 degrees

less than 9 meters ( 30 ft) for latitudes less than 30 degrees

(These error statements reflect both the convergence of the meridians

and the curvature of the parallels. The error is non-linear with

distance; shorter distances will have better percentage errors.)

The flat-Earth distance d will be expressed in the same units as the

coordinates.

If the locations are not already in Cartesian coordinates, the

computational cost of converting from spherical coordinates and then

using the flat-Earth model may exceed that of using the more accurate

spherical model.

Otherwise, presuming a spherical Earth with radius R (see below), and

the locations of the two points in spherical coordinates (longitude

and latitude) are lon1,lat1 and lon2,lat2 then the

Haversine Formula (from R.W. Sinnott, "Virtues of the Haversine",

Sky and Telescope, vol. 68, no. 2, 1984, p. 159):

dlon = lon2 - lon1

dlat = lat2 - lat1

a = (sin(dlat/2))^2 + cos(lat1) * cos(lat2) * (sin(dlon/2))^2

c = 2 * arcsin(min(1,sqrt(a)))

d = R * c

will give mathematically and computationally exact results. The

intermediate result c is the great circle distance in radians. The

great circle distance d will be in the same units as R.

When the two points are antipodal (on opposite sides of the Earth),

the Haversine Formula is ill-conditioned (see the discussion below

the Law of Cosines for Spherical Trigonometry), but the error, perhaps

as large as 2 km (1 mi), is in the context of a distance near

20,000 km (12,000 mi). Further, there is a possibility that roundoff

errors might cause the value of sqrt(a) to exceed 1.0, which would

cause the inverse sine to crash without the bulletproofing provided by

the min() function.

Most computers require the arguments of trigonometric functions to be

expressed in radians. To convert lon1,lat1 and lon2,lat2 from degrees,

minutes, and seconds to radians, first convert them to decimal

degrees. To convert decimal degrees to radians, multiply the number

of degrees by pi/180 = 0.017453293 radians/degree.

Inverse trigonometric functions return results expressed in radians.

To express c in decimal degrees, multiply the number of radians by

180/pi = 57.295780 degrees/radian. (But be sure to multiply the number

of RADIANS by R to get d.)

The Haversine Formula can be expressed in terms of a two-argument

inverse tangent function, atan2(y,x), instead of an inverse sine

as follows (no bulletproofing is needed for an inverse tangent):

dlon = lon2 - lon1

dlat = lat2 - lat1

a = (sin(dlat/2))^2 + cos(lat1) * cos(lat2) * (sin(dlon/2))^2

c = 2 * atan2( sqrt(a), sqrt(1-a) )

d = R * c

The problem of determining the great circle distance on a sphere has

been around for hundreds of years, as have both the Law of Cosines

solution (given below but not recommended) and the Haversine Formula.

Sinnott gets the credit here because he was quoted by Snyder (cited

below). Perhaps someone will provide the truly seminal reference so

the proper attribution can be given?

The Pythagorean flat-Earth approximation assumes that meridians are

parallel, that the parallels of latitude are negligibly different from

great circles, and that great circles are negligibly different from

straight lines. Close to the poles, the parallels of latitude are not

only shorter than great circles, but indispensably curved. Taking this

into account leads to the use of polar coordinates and the planar law

of cosines for computing short distances near the poles: The

Polar Coordinate Flat-Earth Formula

a = pi/2 - lat1

b = pi/2 - lat2

c = sqrt( a^2 + b^2 - 2 * a * b * cos(lon2 - lon1) )

d = R * c

is computationally only a little more expensive than the Pythagorean

Theorem and will give smaller maximum errors for higher latitudes and

greater distances. The maximum errors, which depend upon azimuth in

addition to separation distance, are equal at 80 degrees latitude when

the separation is 33 km (20 mi), 82 degrees at 18 km (11 mi),

84 degrees at 9 km (5.4 mi). But even at 88 degrees the polar error

can be as large as 20 meters (66 ft) when the distance between the

points is 20 km (12 mi).

The latitudes lat1 and lat2 must be expressed in radians (see above);

pi/2 = 1.5707963. Again, the intermediate result c is the distance in

radians and the distance d is in the same units as R.

An UNRELIABLE way to calculate distance on a spherical Earth is the

Law of Cosines for Spherical Trigonometry

** NOT RECOMMENDED **

a = sin(lat1) * sin(lat2)

b = cos(lat1) * cos(lat2) * cos(lon2 - lon1)

c = arccos(a + b)

d = R * c

Although this formula is mathematically exact, it is unreliable for

small distances because the inverse cosine is ill-conditioned. Sinnott

(in the article cited above) offers the following table to illustrate

the point:

cos (5 degrees) = 0.996194698

cos (1 degree) = 0.999847695

cos (1 minute) = 0.9999999577

cos (1 second) = 0.9999999999882

cos (0.05 sec) = 0.999999999999971

A computer carrying seven significant figures cannot distinguish the

cosines of any distances smaller than about one minute of arc.

The function min(1,(a + b)) could replace (a + b) as the argument for

the inverse cosine to guard against possible roundoff errors, but

doing so would be to "polish a cannonball".

5.1a: What value should I use for the radius of the Earth, R?

The historical definition of a "nautical mile" is "one minute of arc

of a great circle of the earth". Since the earth is not a perfect

sphere, that definition is ambiguous. However, the internationally

accepted (SI) value for the length of a nautical mile is (exactly, by

definition) 1.852 km or exactly 1.852/1.609344 international miles

(that is, approximately 1.15078 miles - either "international" or

"U.S. statute"). Thus, the implied "official" circumference is 360

degrees times 60 minutes/degree times 1.852 km/minute = 40003.2 km.

The implied radius is the circumference divided by 2 pi:

R = 6367 km = 3956 mi

5.1b: When is it NOT okay to assume the Earth is a sphere?

A quick (?) test is: Compare the results produced by using the two

extreme values of the radius of curvature for the Earth:

minimum radius of curvature: 6336 km (3937 mi)

maximum radius of curvature: 6399 km (3976 mi)

in your application. If the results are different enough to cause you

to change your action (or your recommendation, or your interpretation

of the implication of the results, etc.), then assuming the Earth is

spherical is NOT okay.

The shape of the Earth is well approximated by an oblate spheroid. The

radius of curvature varies with direction and latitude. According to

formulas given on pages 24 and 25 of the book by Snyder,

"Map Projections - A Working Manual", by John P. Snyder,

U.S. Geological Survey Professional Paper 1395,

United States Government Printing Office, Washington DC, 1987,

the radius of curvature of an ellipsoidal Earth in the plane of the

meridian is given by

R' = a * (1 - e^2) / (1 - e^2 * (sin(lat))^2)^(3/2)

where a is the equatorial radius,

b is the polar radius, and

e is the eccentricity of the ellipsoid = sqrt(1 - b^2/a^2)

and the radius of curvature in a plane perpendicular to the meridian

and perpendicular to a plane tangent to the surface is given by

N = a/sqrt(1-e^2*(sin(lat)^2))

A Swedish book by Ilmar Ussisoo, _Kartprojektioner_ [map projections]

(published by the National Land Survey, Sweden, Professional papers

1977/6) suggests use of the geometric mean of these two radii of

curvature for all azimuths, as it produces errors of order of

magnitude 0.1% for distances within 500 km (300 mi) at 60 degrees

latitude. The formula for that average is no more complicated than

either of its components. That is, r = sqrt(R' * N) becomes

r = a * sqrt(1 - e^2) / (1 - e^2 * (sin(lat))^2)

Using these formulas with

a = 6378 km (3963 mi) Equatorial radius (surface to center distance)

b = 6357 km (3950 mi) Polar radius (surface to center distance)

e = 0.081082 Eccentricity

gives the following table of values for the

Radii of Curvature:

latitude.......... r...................R'................. N..........

00 degrees . 6357 km (3950 mi) . 6336 km (3937 mi) . 6378 km (3963 mi)

15 degrees . 6360 km (3952 mi) . 6340 km (3940 mi) . 6379 km (3964 mi)

30 degrees . 6367 km (3957 mi) . 6352 km (3947 mi) . 6383 km (3966 mi)

45 degrees . 6378 km (3963 mi) . 6367 km (3957 mi) . 6389 km (3970 mi)

60 degrees . 6388 km (3970 mi) . 6383 km (3966 mi) . 6394 km (3973 mi)

75 degrees . 6396 km (3974 mi) . 6395 km (3974 mi) . 6398 km (3975 mi)

90 degrees . 6399 km (3976 mi) . 6399 km (3976 mi) . 6399 km (3976 mi)

Note that the radius of curvature for an ellipsoid is not the same as

the distance from the surface of the ellipsoid to the center. In fact,

the radius of curvature increases as the radius decreases. Also, be

aware that a variety of ellipsoids with slightly different parameters

have been fit to the Earth; the preferred ellipsoid may depend on the

region in which you are most interested.

Cliff Tiedemann, <clifft@uic.edu>, at the University of Illinois at

Chicago pointed out that spherical earth computations will provide

underestimates of real world distances measured in the direction of

the equator (and especially for trans-equatorial links) and

overestimates for those measured in the direction of the poles (and

especially for trans-polar ones).

For most purposes, it is quite satisfactory to treat the Earth as a

sphere. If not, an ellipsoid can provide a better approximation.

Some standard textbooks that may be helpful follow (reviews are by

Steve Robertson, <stever@mindlink.bc.ca>, of Tangent Survey Systems in

Canada):

Bomford, Guy 1980 _Geodesy_ Clarendon Press, Oxford

ISBN 0-19-851946-X

Review: For geodetic computations, this is pretty well the standard in

English. It's a cookbook and offers no development, however.

Vanicek, Petr, and Krakiwsky, Edward 1986 _Geodesy, the Concepts_

North-Holland, Amsterdam

ISBN 0-444-87775-4

Review: This offers a great, but quite involved, discussion of the

concepts behind geometrical (and all other) geodesy.

Torge, Wolfgang 1980 _Geodesy_ de Gruyter, Berlin (translated to

English by C. Jekeli)

ISBN 3-11-007232-7

Review: This concentrates mostly on gravimetric geodesy, but has some

geometrical stuff, well explained without too much mathematics.

Software for solving distance and azimuth problems on the ellipsoid

can be obtained (as of 10/10/96) by anonymous ftp from several

sources, two of which are listed below:

The URL of the National Geodetic Survey (of the National Oceanic and

Atmospheric Administration in the US Department of Commerce) is:

ftp://www.ngs.noaa.gov/pub/pcsoft/for_inv.3d/

Review (by Ronald C. McConnell, <rcmcc@cc.bellcore.com>, of Bellcore):

They have Fortran source and PC executable versions of both the normal

"inverse" great circle calculations (two lat/long pairs to distance

and bearing), and the less used "forward" calculation (one lat/long

pair plus bearing and distance to the second lat/long pair). They have

both 2-dimensional and 3-dimensional versions of each. The inverse

program works to within a few seconds or a few minutes, depending on

the Fortran compiler, of the antipodal points. The forward program

seems immune to any and all problem locations and pairs of locations.

You can choose among a couple of dozen ellipsoids.

See the read.me file for explanations. The NGS software directory may

contain other listing of interest. Its URL is:

http://www.ngs.noaa.gov/PC_PROD/pc_prod.shtml/

(Case is relevant in many URLs - e.g.: this one.)

The NGS has a public BBS at 301-713-4181 and provides FTP access at:

ftp://ftp.ngs.noaa.gov/pub/pcsoft

Another anonymous ftp source for ellipsoid software is the US

Geological Survey (of the US Department of the Interior), at:

http://kai.er.usgs.gov/ftp/PROJ.4/proj.html

Again, see the README file for explanations. The URLs for the USGS

directory and home page are:

http://kai.er.usgs.gov/ftp/index.html

http://kai.er.usgs.gov/homepage.html

5.1c: When is it NOT okay to assume the Earth is an ellipsoid?

The shape the Earth would assume if it were all measured at mean sea

level is called the geoid. The geoid varies no more than about a

hundred meters above or below a well-fitting ellipsoid, a variation

far less than the ellipsoid varies from the sphere. Terrain relief is

reported relative to the geoid. (Paraphrased from p. 11 of the book by

Snyder cited above.)

Distances on the surface of the geoid are not particularly meaningful.

However, there are applications, such as long-term prediction of

orbits of Earth satellites, that require better approximations than

are provided by an ellipsoid. Astrodynamics texts, such as

Kaula, William M. 1966 _Theory of Satellite Geodesy_ Blaisdell

Publishing Co., Waltham, MA (This book may be out of print.)

Battin, Richard H. 1964 _Astronautical Guidance_ McGraw-Hill,

New York (There may be later editions.)

may be consulted for further information.

5.1d: Where can I find formulas for great circle navigation problems,

like what course to follow and where will I be after following a known

azimuth for a given distance?

See Ed Williams' navigation formulas at

http://www.best.com/~williams/avform.htm

which can be accessed by clicking on 'FAQ Lists' at

http://vancouver-webpages.com/peter/#faq

5.1e: How can I calculate distance on an ellipsoid.

On May 18, 1999, Steven Michael Robbins <stever@jeff.cs.mcgill.ca>

posted a discussion of this issue. He said:

About two weeks ago, I posted the following question:

> I need to compute the distance between two points on the surface of

> an ellipsoid. I'm looking for the distance *on* the surface. In

> short, I'm after the length of the shortest geodesic curve joining

> the points.

I was naively expecting a nice solution, akin to the "great circles"

on a sphere. I was soon disabused of this notion.

A big thanks to all who replied. Here is a summary of the responses.

I found two approaches to this problem.

One approach involves computing a discretized approximation to a

geodesic. This works on any ellipsoid. The first two software

pointers follow this approach. As far as I can tell, all the other

references follow the second approach.

The second approach is designed for the Earth, exploiting two special

features: (1) it is an ellipsoid of revolution, and (2) it is nearly

spherical. These properties allow one to construct a solution as a

power series in the small "flattening" parameter. One truncates the

series depending upon the accuracy required.

I was looking for a solution that works for any ellipsoid. Most of

what follows is geodesy-centric, so I have only explored a handful of

the suggestions listed below. Also --- as one person noted --- a web

search with key words of "ellipsoid & distance & arc" will generate a

list of hundreds of sites. (All dealing with the Earth, I might add).

What follows is heavily cut'n'pasted from the responses or from web

pages. Don't take these as advice or endorsement from me -- these are

not my words below!

Available Software on the net:

http://www.magic-software.com/gr_surf.htm

The code here computes a polygonal approximation to a geodesic and

then refines it until the geodesic curvature vanishes at each vertex

on the path.

http://www.netlib.org/ode/geodesic/

Similar to the above, except that it actually works on any manifold --

you supply a routine to compute the metric at any point, etc.

ftp://kai.er.usgs.gov/pub/PROJ.4/

Well-proven code, forward and inverse problem.

ftp://ftp.ngs.noaa.gov/pub/pcsoft/for_inv.3d

>From the (U.S.) National Geodetic Survey:

INVERSE/FORWARD3D (Version 1.0)

Comprises four programs - Inverse (Version 2.0) which computes the

geodetic azimuth and distance between two points, given their

geographic positions; Forward (Version 2.0) which computes the

geographic position of a point, given the geodetic azimuth and

distance from a point with known geographic position; and the

three-dimensional versions of these programs . INVERS3D (Version 1.0)

and FORWRD3D (Version 1.0), which include the height component.

ftp://mapping.usgs.gov/pub/software/current_software/w5501

This program will solve either the geodetic position or inverse

problems in any quadrant of the Earth using the parameters of one of

five commonly-used ellipsoids.

http://www.globalserve.net/~nac/products.html

A product called NACNav (60USD/license) which can calculate the

shortest distance between two points anywhere on the earth surface and

the angle between the direction and the magnetic north (determined by

local magnetic declination).

Suggested References

If you need very high accuracy, I suggest reading B.R. Bowring, "Total

Inverse Solution for the Geodesic and Great Elliptic", Survey Review,

33, 261 (July 1996).

If you are satisfied with the relative error of the square of the

Earth's flattening, then see J. Meeus, Astronomical Algorithms, 1991.

Two people recommended: Bomford, G., 1952, Geodesy: London, England,

Oxford University Press.

Clark, D., 1963, Plane and Geodetic Surveying, II: London, England,

Constable & Co., Ltd.

Hosmer, G.L., 1930, Geodesy: New York, New York, John Wiley & Sons, Inc.

Lambert, W.D. and Swick, C.H.,1935, Formulas and Tables for the

Computation of Geodetic Positions on the International Ellipsoid:

U.S. Coast and Geodetic Survey Special Publication No. 200.

U.S.C.& G.S., Formulas and Tables for the Computation of Geodetic

Positions, U.S. Coast and Geodetic Survey Special Publication No. 8.

"Direct and Inverse Solutions of Geodesics on the Ellipsoid with

Application of Nested Equations", T. Vincenty, Survey Review 22, April

1975.

Thomas, Paul D., 1965, Geodesic arc-length on the reference ellipsoid

to second-order terms in the flattening, Journal of Geophysical

Research, vol.70(14), 3331-3340.

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