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<!DOCTYPE html>
<html>
<head>
<title>Geodesic calculations for an ellipsoid done right</title>
<meta name="description"
content="Geodesic calculations for an ellipsoid done right" />
<meta name="author" content="Charles F. F. Karney">
<meta name="keywords"
content="geodesics,
geodesic distance,
geographic distance,
shortest path,
direct geodesic problem,
inverse geodesic problem,
distance and azimuth,
distance and heading,
range and bearing,
geographic area,
geodesic polygon,
spheroidal triangle,
latitude and longitude,
online calculator,
WGS84 ellipsoid,
GeographicLib" />
<script type="text/javascript"
src="http://geographiclib.sf.net/scripts/geographiclib.js">
</script>
<script type="text/javascript">
<!--
var geod = GeographicLib.Geodesic.WGS84;
var dms = GeographicLib.DMS;
function formatpoint(lat, lon, azi, dmsformat, prec) {
prec += 5;
if (dmsformat) {
var trail = prec < 2 ? dms.DEGREE :
(prec < 4 ? dms.MINUTE : dms.SECOND);
prec = prec < 2 ? prec : (prec < 4 ? prec - 2 : prec - 4);
return (dms.Encode(lat, trail, prec, dms.LATITUDE) + " " +
dms.Encode(lon, trail, prec, dms.LONGITUDE) + " " +
dms.Encode(azi, trail, prec, dms.AZIMUTH));
} else {
return (lat.toFixed(prec) + " " +
lon.toFixed(prec) + " " +
azi .toFixed(prec));
}
}
function GeodesicInverse(input, dmsformat, prec) {
var result = {};
try {
// Input is a blank-delimited line: lat1 lon1 lat2 lon2
var t = new String(input);
t = t.replace(/^\s+/,"").replace(/\s+$/,"").split(/[\s,]+/,6);
if (t.length != 4)
throw new Error("Need 4 input items");
var p1 = GeographicLib.DMS.DecodeLatLon(t[0], t[1]);
var p2 = GeographicLib.DMS.DecodeLatLon(t[2], t[3]);
t = geod.Inverse(p1.lat, p1.lon, p2.lat, p2.lon);
result.status = "OK";
result.p1 = formatpoint(t.lat1, t.lon1, t.azi1, dmsformat, prec);
result.p2 = formatpoint(t.lat2, t.lon2, t.azi2, dmsformat, prec);
result.s12 = t.s12.toFixed(prec);
}
catch (e) {
result.status = "ERROR: " + e.message;
result.p1 = "";
result.p2 = "";
result.s12 = "";
}
return result;
}
function GeodesicDirect(input, dmsformat, prec) {
var result = {};
try {
// Input is a blank-delimited line: lat1 lon1 azi1 s12
var t = new String(input);
t = t.replace(/^\s+/,"").replace(/\s+$/,"").split(/[\s,]+/,6);
if (t.length != 4)
throw new Error("Need 4 input items");
var p1 = GeographicLib.DMS.DecodeLatLon(t[0], t[1]);
var azi1 = GeographicLib.DMS.DecodeAzimuth(t[2]);
var s12 = parseFloat(t[3]);
t = geod.Direct(p1.lat, p1.lon, azi1, s12);
result.status = "OK";
result.p1 = formatpoint(t.lat1, t.lon1, t.azi1, dmsformat, prec);
result.p2 = formatpoint(t.lat2, t.lon2, t.azi2, dmsformat, prec);
result.s12 = t.s12.toFixed(prec);
}
catch (e) {
result.status = "ERROR: " + e.message;
result.p1 = "";
result.p2 = "";
result.s12 = "";
}
return result;
}
function GeodesicInversePath(input, dmsformat, prec) {
var result = {};
try {
// Input is a blank-delimited line: lat1 lon1 lat2 lon2 ds12 maxnum
var t = new String(input);
t = t.replace(/^\s+/,"").replace(/\s+$/,"").split(/[\s,]+/,8);
if (t.length != 6)
throw new Error("Need 6 input items");
var p1 = GeographicLib.DMS.DecodeLatLon(t[0], t[1]);
var p2 = GeographicLib.DMS.DecodeLatLon(t[2], t[3]);
var ds12 = parseFloat(t[4]);
var maxnum = parseInt(t[5]);
var t = new String(input);
t = t.split(/[\s,]+/,8);
if (t[0] == "") t.shift();
t = geod.InversePath(p1.lat, p1.lon, p2.lat, p2.lon, ds12, maxnum);
result.status = "OK";
result.points = ""
for (var i = 0; i < t.length; ++i)
result.points +=
formatpoint(t[i].lat, t[i].lon, t[i].azi, dmsformat, prec) + "\n";
}
catch (e) {
result.status = "ERROR: " + e.message;
result.points = "";
}
return result;
}
function GeodesicArea(input, polyline) {
var result = {};
try {
// Input is a newline-delimited points;
// each point is blank-delimited: lat lon
var t = new String(input);
t = t.split(/[\r\n]/);
if (t[0] == "") t.shift();
if (t[t.length-1] == "") t.pop();
for (var i = 0; i < t.length; ++i) {
var pos = t[i].replace(/^\s+/,"").replace(/\s+$/,"").split(/[\s,]+/,4);
if (pos.length != 2)
throw new Error("Need 2 items on each line");
t[i] = dms.DecodeLatLon(pos[0], pos[1]);
}
t = geod.Area(t, polyline);
result.status = "OK";
result.area = t.number + " " +
t.perimeter.toFixed(8);
if (!polyline)
result.area += " " + t.area.toFixed(2);
}
catch (e) {
result.status = "ERROR: " + e.message;
result.area = "";
}
return result;
}
//-->
</script>
</head>
<body>
<div>
<h2>Geodesic calculations for an ellipsoid done right</h2>
</div>
<p>
This page illustrates the geodesic routines available in
<a href="http://geographiclib.sourceforge.net">GeographicLib</a>.
The C++ code has been converted to JavaScript so the calculations
are carried out on the client. The algorithms are considerably
more accurate than Vincenty's method, and offer more functionality
(an inverse method which never fails to converge, differential
properties of the geodesic, and the area under a geodesic). The
algorithms are derived in
<blockquote>
Charles F. F. Karney,<br>
<a href="http://dx.doi.org/10.1007/s00190-012-0578-z">
<i>Algorithms for geodesics</i></a>,<br>
J. Geodesy <b>87</b>(1), 43&ndash;55 (Jan. 2013);<br>
DOI:
<a href="http://dx.doi.org/10.1007/s00190-012-0578-z">
10.1007/s00190-012-0578-z</a>
(<a href="http://dx.doi.org/10.1007/s00190-012-0578-z">pdf</a>);
addenda: <a href="http://geographiclib.sf.net/geod-addenda.html">
geod-addenda.html</a>.
</blockquote>
This page just provides a basic interface. Enter latitudes,
longitudes, and azimuths as degrees and distances as meters using
spaces or commas as separators. (Angles may be entered as decimal
degrees or as degrees, minutes, and seconds, e.g. -20.51125,
20&deg;30&prime;40.5&Prime;S, S20d30'40.5&quot;, or
-20:30:40.5.) The results are accurate to about
15&nbsp;nanometers (or 0.1&nbsp;m<sup>2</sup> per vertex for
areas). A slicker page where the geodesics are incorporated into
Google Maps is given <a href="geod-google.html">here</a>.
<p>
Jump to:
<ul>
<li><a href="#inverse">Inverse problem</a></li>
<li><a href="#direct">Direct problem</a></li>
<li><a href="#path">Geodesic path</a></li>
<li><a href="#area">Polygon area</a></li>
<li><a href="geod-google.html">Geodesic lines, circles, and
envelopes in Google Maps</a></li>
</ul>
</p>
<hr>
<form name="inverse" >
<h3><a class="anchor" id="inverse">Inverse problem</h3>
<p>
Find the shortest path between two points on the earth. The
path is characterized by its length <i>s12</i> and its azimuth
at the two ends <i>azi1</i> and <i>azi2</i>. The sample
calculation finds the shortest path between Wellington, New
Zealand, and Salamanca, Spain. (For this example, the
<a href="http://www.ngs.noaa.gov/">NGS</a>
<a href="http://www.ngs.noaa.gov/cgi-bin/Inv_Fwd/inverse2.prl">
inverse geodesic calculator</a> returns a result which is 1.2 km
too long with an azimuth which is off by 3 degrees.) To perform
the calculation, press the &ldquo;COMPUTE&rdquo; button.
</p>
<p>Enter <i>&ldquo;lat1 lon1 lat2 lon2&rdquo;</i>:</p>
<p>input:
<input name="input" size=72 value="-41.32 174.81 40.96 -5.50" />
</p>
<p>
Output format:&nbsp;&nbsp;<label for="ig">
<input type="radio" value="g" name="format" id="ig" checked>
decimal degrees
</label>&nbsp;
<label for="id">
<input type="radio" value="d" name="format" id="id">
degrees minutes seconds
</label><br>
Output precision:&nbsp;&nbsp;<select name="prec" size=1>
<option value='0'> 1m 0.00001d 0.1"</option>
<option value='1'> 100mm 0.01"</option>
<option value='2'> 10mm 0.001"</option>
<option value='3' selected> 1mm 0.0001"</option>
<option value='4'> 100um 0.00001"</option>
<option value='5'> 10um 0.000001"</option>
<option value='6'> 1um 0.0000001"</option>
<option value='7'> 100nm 0.00000001"</option>
<option value='8'> 10nm 0.000000001"</option>
<option value='9'> 1nm 0.0000000001"</option>
</select>
</p>
<p>
<input type="button" value="COMPUTE"
onclick="var t = GeodesicInverse(document.inverse.input.value,
document.inverse.format[1].checked,
document.inverse.prec.selectedIndex);
document.inverse.status.value = t.status;
document.inverse.p1.value = t.p1;
document.inverse.p2.value = t.p2;
document.inverse.s12.value = t.s12;" />
</p>
<p>
status:
<input name="status" size=50 readonly />
</p>
<p>
lat1 lon1 <font color='blue'>azi1</font>:
<input name="p1" size=75 readonly />
</p>
<p>
lat2 lon2 <font color='blue'>azi2</font>:
<input name="p2" size=75 readonly />
</p>
<p>
<font color='blue'>s12</font>:
<input name="s12" size=25 readonly />
</p>
</form>
<hr>
<form name="direct">
<h3><a class="anchor" id="direct">Direct problem</h3>
<p>
Find the destination traveling a given distance along a geodesic
with a given azimuth at the starting point. The destination is
characterized by its position <i>lat2, lon2</i> and its azimuth
at the destination <i>azi2</i>. The sample calculation shows
the result of travelling 10000 km NE from JFK airport. To perform
the calculation, press the &ldquo;COMPUTE&rdquo; button.
</p>
<p>Enter <i>&ldquo;lat1 lon1 azi1 s12&rdquo;</i>:</p>
<p>input:
<input name="input" size=72 value="40.6 -73.8 45 10000e3" />
</p>
<p>
Output format:&nbsp;&nbsp;<label for="dg">
<input type="radio" value="g" name="format" id="dg" checked>
decimal degrees
</label>&nbsp;
<label for="dd">
<input type="radio" value="d" name="format" id="dd">
degrees minutes seconds
</label><br>
Output precision:&nbsp;&nbsp;<select name="prec" size=1>
<option value='0'> 1m 0.00001d 0.1"</option>
<option value='1'> 100mm 0.01"</option>
<option value='2'> 10mm 0.001"</option>
<option value='3' selected> 1mm 0.0001"</option>
<option value='4'> 100um 0.00001"</option>
<option value='5'> 10um 0.000001"</option>
<option value='6'> 1um 0.0000001"</option>
<option value='7'> 100nm 0.00000001"</option>
<option value='8'> 10nm 0.000000001"</option>
<option value='9'> 1nm 0.0000000001"</option>
</select>
</p>
<p>
<input type="button" value="COMPUTE"
onclick="var t = GeodesicDirect(document.direct.input.value,
document.direct.format[1].checked,
document.direct.prec.selectedIndex);
document.direct.status.value = t.status;
document.direct.p1.value = t.p1;
document.direct.p2.value = t.p2;
document.direct.s12.value = t.s12;" />
</p>
<p>
status:
<input name="status" size=50 readonly />
</p>
<p>
lat1 lon1 azi1:
<input name="p1" size=75 readonly />
</p>
<p>
<font color='blue'>lat2 lon2 azi2</font>:
<input name="p2" size=75 readonly />
</p>
<p>
s12:
<input name="s12" size=25 readonly />
</p>
</form>
<hr>
<form name="path">
<h3><a class="anchor" id="path">Geodesic path</h3>
<p>
Find intermediate points along a geodesic. In addition to
specifying the endpoints, give <i>ds12</i>, the maximum distance
between the intermediate points and <i>maxk</i>, the maximum
number of intervals the geodesic is broken into. The output
gives a sequence of positions <i>lat, lon</i> together with the
corresponding azimuths <i>azi</i>. The sample shows the path
from JFK to Singapore's Changi Airport at about 1000 km
intervals. (In this example, the path taken by Google Earth
deviates from the shortest path by about 2.9 km.) To perform
the calculation, press the &ldquo;COMPUTE&rdquo; button.
</p>
<p>Enter <i>&ldquo;lat1 lon1 lat2 lon2 ds12 maxk&rdquo;</i>:</p>
<p>input:
<input name="input" size=72 value="40.6 -73.8 1.4 104 1000e3 20" />
</p>
<p>
Output format:&nbsp;&nbsp;<label for="pg">
<input type="radio" value="g" name="format" id="pg" checked>
decimal degrees
</label>&nbsp;
<label for="pd">
<input type="radio" value="d" name="format" id="pd">
degrees minutes seconds
</label><br>
Output precision:&nbsp;&nbsp;<select name="prec" size=1>
<option value='0' selected> 1m 0.00001d 0.1"</option>
<option value='1'> 100mm 0.01"</option>
<option value='2'> 10mm 0.001"</option>
<option value='3'> 1mm 0.0001"</option>
<option value='4'> 100um 0.00001"</option>
<option value='5'> 10um 0.000001"</option>
<option value='6'> 1um 0.0000001"</option>
<option value='7'> 100nm 0.00000001"</option>
<option value='8'> 10nm 0.000000001"</option>
<option value='9'> 1nm 0.0000000001"</option>
</select>
</p>
<p>
<input type="button" value="COMPUTE"
onclick="var t = GeodesicInversePath(document.path.input.value,
document.path.format[1].checked,
document.path.prec.selectedIndex);
document.path.status.value = t.status;
document.path.points.value = t.points;" />
</p>
<p>
status:
<input name="status" size=50 readonly />
</p>
<p>
points (lat lon azi):<br>
<textarea name="points" cols=70 rows=21 readonly></textarea>
</p>
</form>
<hr>
<form name="area">
<h3><a class="anchor" id="area">Polygon area</h3>
<p>
Find the perimeter and area of a polygon whose sides are
geodesics. The polygon must be simple (i.e., must not intersect
itself). (There's no need to ensure that the polygon is
closed.) Counter-clockwise traversal of the polygon results in
a positive area. The polygon can encircle one or both poles.
The sample gives the approximate perimeter (in m) and area (in
m<sup>2</sup>) of Antarctica. (For this example, Google Earth
Pro returns an area which is 30 times too large! However this
is a little unfair, since Google Earth has no concept of
polygons which encircle a pole.) If the <i>polyline</i> option
is selected then just the length of the line joining the points
is is returned. To perform the calculation, press the
&ldquo;COMPUTE&rdquo; button.
</p>
<p>Enter points, one per line, as <i>&ldquo;lat lon&rdquo;</i>:</p>
<p>points (lat lon):<br>
<textarea name="input" cols=36 rows=13>-63.1 -58
-72.9 -74
-71.9 -102
-74.9 -102
-74.3 -131
-77.5 -163
-77.4 163
-71.7 172
-65.9 140
-65.7 113
-66.6 88
-66.9 59
-69.8 25
-70.0 -4
-71.0 -14
-77.3 -33
-77.9 -46
-74.7 -61
</textarea>
</p>
<p>
Treat points as:&nbsp;&nbsp;<label for="lg">
<input type="radio" value="p" name="polyline" id="lg" checked>
polygon
</label>&nbsp;
<label for="ll">
<input type="radio" value="l" name="polyline" id="ll">
polyline
</label>
</p>
<p>
<input type="button" value="COMPUTE"
onclick="var t = GeodesicArea(document.area.input.value,
document.area.polyline[1].checked);
document.area.status.value = t.status;
document.area.area.value = t.area;" />
</p>
<p>
status:
<input name="status" size=50 readonly />
</p>
<p>
number perimeter area:
<input name="area" size=55 readonly />
</p>
</form>
<hr>
<address>Charles Karney
<a href="mailto:charles@karney.com">&lt;charles@karney.com&gt;</a>
(2011-08-04)</address>
<br>
<a href="http://geographiclib.sourceforge.net">
<img
src="http://sourceforge.net/sflogo.php?group_id=283628&amp;type=9"
border="0" height="15" width="80" alt="SourceForge.net" />
</a>
</body>
</html>
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