<html><head><title>Math Addendum 7</title><style type="text/css"> p { margin: 0 0 0 0; font-family: Arial, sans-serif; font-size:10pt; font-weight: normal; font-style: normal; vertical-align: baseline; color: black; text-decoration: none; } div { border-width:0px; border-style: solid; border-color: red; } body { background-color: #fff; } p img { vertical-align: middle; } p b, li b { font-weight : bold; } p i, li i { font-style : italic; } p u, li u { text-decoration : underline; } p so, li so { text-decoration : line-through; } p sub, p sub { font-size : 70%; vertical-align: sub; } p sup, p sup { font-size : 70%; vertical-align: super; } ul { padding-left : 2em; /* for mozilla list marker box */ margin-left : 0; /* for IE list marker box */ } ol { padding-left : 2em; /* for mozilla list marker box */ margin-right : -0.5em; margin-left : 0; /* for IE list marker box */ } p.Style1, li.Style1 /*Normal*/ { margin-left: 0pt; margin-right: 0pt; text-align: left; font-family: 'Arial', sans-serif;; 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top: 72.75pt; left: 30pt; width: 353pt; "><a name="" /><p class="Style1"><span style="font-size: 12pt; font-family: 'Arial', sans-serif;"><b><span>Addendum 7: Latitude, Longitude, and distances on a sphere</span></b></span></p></div><div id="mc-region-385" style="position: absolute; top: 104.25pt; left: 30pt; width: 286.25pt; "><a name="" /><p class="Style1"><span style="font-family: 'Arial', sans-serif;"><i><span>First define some constants and dimensional units needed below</span></i></span></p></div><div id="mc-region-736" style="position: absolute; top: 129pt; left: 48pt; width: 56.25pt; "><a name="" /><img border="0" src=".\addendum 7 latitude and longitude calculations_images/IMG0003_208380078.PNG" id="generatedImage736" path=".\addendum 7 latitude and longitude calculations_images/IMG0003_208380078.PNG" style="width: 56.25pt;height: 16.5pt;"></img></div><div id="mc-region-19" style="position: absolute; top: 158.25pt; left: 24pt; width: 408.5pt; "><a name="" /><p class="Style1">1. <span style="font-family: 'Arial', sans-serif;"><b><i><span> </span></i></b></span><span style="font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;">The</span></span><span style="font-family: 'Arial', sans-serif;"><span style="font-style: normal;"><b> latitude </b></span></span><span style="font-weight: normal; font-style: normal;"><span style="font-family: 'Arial', sans-serif;">and </span></span><span style="font-family: 'Arial', sans-serif;"><b><span style="font-style: normal;">longitude </span></b></span><span style="font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;">of a location on a sphere are the angles measured between the location and the equator (latitude) or between the reference longitude line (Greenwich England for the Earth) and the projected distance in the equatorial plane (longitude) These angles are easier to understand by inpsecting a figure, as illustrated for the location 27 deg N latitude and 74 deg W longitude on the Earth's sphere, in the figure below.</span></span></p></div><div id="mc-region-815" style="position: absolute; top: 234pt; left: 108pt; width: 210.75pt; "><a name="" /><img border="0" src=".\addendum 7 latitude and longitude calculations_images/IMG0005_208380093.PNG" id="generatedImage815" path=".\addendum 7 latitude and longitude calculations_images/IMG0005_208380093.PNG" style="width: 210.75pt;height: 211.5pt;"></img></div><div id="mc-region-681" style="position: absolute; top: 464.25pt; left: 18pt; width: 414.5pt; "><a name="" /><p class="Style1">2. The <b>distance</b> (measured on the surface on the sphere) between any two locations is given by first finding the angle between the points, then calculating the chord length by taking the ratio of the included angle and the circumference of the sphere. </p></div><div id="mc-region-787" style="position: absolute; top: 506.25pt; left: 30pt; width: 252.5pt; "><a name="" /><p class="Style1">To do this problem, first we need a little vector arithmetic:</p></div><div id="mc-region-682" style="position: absolute; top: 530.25pt; left: 30pt; width: 414.5pt; "><a name="" /><p class="Style1">(a) On a unit sphere (radius =1), a vector pointing from the origin to a location with coordinates <span style="font-size: 10pt; font-family: 'Symbol', sans-serif;"><span style="font-weight: normal; font-style: normal;">f</span></span><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;"> (l</span></span>atitude) and <span style="font-size: 10pt; font-family: 'Symbol', sans-serif;"><span style="font-weight: normal; font-style: normal;">l</span></span><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;"> (longitude) has x,y,z components:</span></span></p></div><div id="mc-region-788" style="position: absolute; top: 561.75pt; left: 156pt; width: 111.75pt; "><a name="" /><img border="0" src=".\addendum 7 latitude and longitude calculations_images/IMG0006_208380109.PNG" id="generatedImage788" path=".\addendum 7 latitude and longitude calculations_images/IMG0006_208380109.PNG" style="width: 111.75pt;height: 48pt;"></img></div><div id="mc-region-797" style="position: absolute; top: 626.25pt; left: 30pt; width: 216.5pt; "><a name="" /><p class="Style1">(b) Therefore two locations 1 and 2 have vectors: </p></div><div id="mc-region-789" style="position: absolute; top: 650.25pt; left: 144pt; width: 59.25pt; "><a name="" /><img border="0" src=".\addendum 7 latitude and longitude calculations_images/IMG0007_208380109.PNG" id="generatedImage789" path=".\addendum 7 latitude and longitude calculations_images/IMG0007_208380109.PNG" style="width: 59.25pt;height: 17.25pt;"></img></div><div id="mc-region-790" style="position: absolute; top: 650.25pt; left: 240pt; width: 59.25pt; "><a name="" /><img border="0" src=".\addendum 7 latitude and longitude calculations_images/IMG0008_208380109.PNG" id="generatedImage790" path=".\addendum 7 latitude and longitude calculations_images/IMG0008_208380109.PNG" style="width: 59.25pt;height: 17.25pt;"></img></div><div id="mc-region-798" style="position: absolute; top: 674.25pt; left: 30pt; width: 374pt; "><a name="" /><p class="Style1">(c) The inner (or dot) product of any 2 vectors with length 1 is simply the cosine of the included angle <span style="font-size: 10pt; font-family: 'Symbol', sans-serif;"><span style="font-weight: normal; font-style: normal;">q</span></span>:</p></div><div id="mc-region-791" style="position: absolute; top: 704.25pt; left: 186pt; width: 63.75pt; "><a name="" /><img border="0" src=".\addendum 7 latitude and longitude calculations_images/IMG0009_208380109.PNG" id="generatedImage791" path=".\addendum 7 latitude and longitude calculations_images/IMG0009_208380109.PNG" style="width: 63.75pt;height: 13.5pt;"></img></div><div id="mc-region-799" style="position: absolute; top: 728.25pt; left: 30pt; width: 348.5pt; "><a name="" /><p class="Style1">where the dot product is defined as the sum of the products of the components:</p></div><div id="mc-region-803" style="position: absolute; top: 753pt; left: 132pt; width: 157.5pt; "><a name="" /><img border="0" src=".\addendum 7 latitude and longitude calculations_images/IMG0010_208380109.PNG" id="generatedImage803" path=".\addendum 7 latitude and longitude calculations_images/IMG0010_208380109.PNG" style="width: 157.5pt;height: 12.75pt;"></img></div><div id="mc-region-812" style="position: absolute; top: 782.25pt; left: 30pt; width: 139.25pt; "><a name="" /><p class="Style1">(d) Hence the included angle is</p></div><div id="mc-region-811" style="position: absolute; top: 777pt; left: 186pt; width: 75.75pt; "><a name="" /><img border="0" src=".\addendum 7 latitude and longitude calculations_images/IMG0011_208380125.PNG" id="generatedImage811" path=".\addendum 7 latitude and longitude calculations_images/IMG0011_208380125.PNG" style="width: 75.75pt;height: 18.75pt;"></img></div><div id="mc-region-809" style="position: absolute; top: 812.25pt; left: 30pt; width: 365.75pt; "><a name="" /><p class="Style1">(e) Finally, to find the actual distance separating the points on a sphere of radius R, we form the ratio: </p></div><div id="mc-region-805" style="position: absolute; top: 838.5pt; left: 174pt; width: 59.25pt; "><a name="" /><img border="0" src=".\addendum 7 latitude and longitude calculations_images/IMG0012_208380125.PNG" id="generatedImage805" path=".\addendum 7 latitude and longitude calculations_images/IMG0012_208380125.PNG" style="width: 59.25pt;height: 32.25pt;"></img></div><div id="mc-region-717" style="position: absolute; top: 884.25pt; left: 72pt; width: 88.25pt; "><a name="" /><p class="Style1">and solve for S<sub>chord</sub></p></div><div id="mc-region-730" style="position: absolute; top: 884.25pt; left: 174pt; width: 55.5pt; "><a name="" /><img border="0" src=".\addendum 7 latitude and longitude calculations_images/IMG0013_208380125.PNG" id="generatedImage730" path=".\addendum 7 latitude and longitude calculations_images/IMG0013_208380125.PNG" style="width: 55.5pt;height: 17.25pt;"></img></div><div id="mc-region-731" style="position: absolute; top: 884.25pt; left: 246pt; width: 93.5pt; "><a name="" /><p class="Style1">where <span style="font-size: 10pt; font-family: 'Symbol', sans-serif;"><span style="font-weight: normal; font-style: normal;">q</span></span><span style="font-size: 10pt; font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;"> is in radians</span></span></p></div><div id="mc-region-745" style="position: absolute; top: 914.25pt; left: 18pt; width: 397.25pt; "><a name="" /><p class="Style1">3. <b>Converting from ddmmss format to fractional degrees</b>. Typically latitude and longitude are given in deg, min, sec format. The conversion to fractional degress can be done by dividing the minutes by 60 and seconds by 3600, then adding:</p></div><div id="mc-region-762" style="position: absolute; top: 962.25pt; left: 24pt; width: 406.25pt; "><a name="" /><p class="Style1">For example, Iowa City's coordinates are 41 deg 38 min 10 sec, N and 91 deg 33 min 12 sec W, so :</p></div><div id="mc-region-814" style="position: absolute; top: 992.25pt; left: 114pt; width: 114pt; "><a name="" /><img border="0" src=".\addendum 7 latitude and longitude calculations_images/IMG0014_208380140.PNG" id="generatedImage814" path=".\addendum 7 latitude and longitude calculations_images/IMG0014_208380140.PNG" style="width: 114pt;height: 27.75pt;"></img></div><div id="mc-region-763" style="position: absolute; top: 998.25pt; left: 252pt; width: 53.25pt; "><a name="" /><img border="0" src=".\addendum 7 latitude and longitude calculations_images/IMG0015_208380140.PNG" id="generatedImage763" path=".\addendum 7 latitude and longitude calculations_images/IMG0015_208380140.PNG" style="width: 53.25pt;height: 13.5pt;"></img></div><div id="mc-region-813" style="position: absolute; top: 1022.25pt; left: 114pt; width: 114.75pt; "><a name="" /><img border="0" src=".\addendum 7 latitude and longitude calculations_images/IMG0016_208380140.PNG" id="generatedImage813" path=".\addendum 7 latitude and longitude calculations_images/IMG0016_208380140.PNG" style="width: 114.75pt;height: 27.75pt;"></img></div><div id="mc-region-764" style="position: absolute; top: 1028.25pt; left: 258pt; width: 54pt; "><a name="" /><img border="0" src=".\addendum 7 latitude and longitude calculations_images/IMG0017_208380156.PNG" id="generatedImage764" path=".\addendum 7 latitude and longitude calculations_images/IMG0017_208380156.PNG" style="width: 54pt;height: 13.5pt;"></img></div><div id="mc-region-755" style="position: absolute; top: 1058.25pt; left: 42pt; width: 389pt; "><a name="" /><p class="Style1">Alternatively, MathCAD has a nice converter function (DMS) that does this automatically:</p></div><div id="mc-region-758" style="position: absolute; top: 1078.5pt; left: 24pt; width: 81pt; "><a name="" /><img border="0" src=".\addendum 7 latitude and longitude calculations_images/IMG0018_208380156.PNG" id="generatedImage758" path=".\addendum 7 latitude and longitude calculations_images/IMG0018_208380156.PNG" style="width: 81pt;height: 45.75pt;"></img></div><div id="mc-region-759" style="position: absolute; top: 1094.25pt; left: 120pt; width: 60pt; "><a name="" /><img border="0" src=".\addendum 7 latitude and longitude calculations_images/IMG0019_208380156.PNG" id="generatedImage759" path=".\addendum 7 latitude and longitude calculations_images/IMG0019_208380156.PNG" style="width: 60pt;height: 17.25pt;"></img></div><div id="mc-region-760" style="position: absolute; top: 1078.5pt; left: 216pt; width: 81.75pt; "><a name="" /><img border="0" src=".\addendum 7 latitude and longitude calculations_images/IMG0020_208380156.PNG" id="generatedImage760" path=".\addendum 7 latitude and longitude calculations_images/IMG0020_208380156.PNG" style="width: 81.75pt;height: 45.75pt;"></img></div><div id="mc-region-761" style="position: absolute; top: 1094.25pt; left: 324pt; width: 60.75pt; "><a name="" /><img border="0" src=".\addendum 7 latitude and longitude calculations_images/IMG0021_208380156.PNG" id="generatedImage761" path=".\addendum 7 latitude and longitude calculations_images/IMG0021_208380156.PNG" style="width: 60.75pt;height: 17.25pt;"></img></div><div id="mc-region-825" style="position: absolute; top: 1142.25pt; left: 54pt; width: 212.75pt; "><a name="" /><p class="Style1">Note: DMS requires a 3x1 matrix as a argument</p></div><div id="mc-region-744" style="position: absolute; top: 1256.25pt; left: 30pt; width: 375.5pt; "><a name="" /><p class="Style1" style="margin-left: 0pt; margin-right: 0pt; text-align: left;"><b><u>Examples</u></b></p><p class="Style1" style="margin-left: 0pt; margin-right: 0pt; text-align: left;"></p><p class="Style1" style="margin-left: 0pt; margin-right: 0pt; text-align: left;"><b>Note: Latitude and longitude for many cities around the world can be found at</b></p><p class="Style1" style="margin-left: 0pt; margin-right: 0pt; text-align: left;"></p><p class="Style1" style="margin-left: 0pt; margin-right: 0pt; text-align: left;"><b> http://www.bcca.org/misc/qiblih/latlong.html</b></p></div><div id="mc-region-720" style="position: absolute; top: 1346.25pt; left: 54pt; width: 255.5pt; "><a name="" /><p class="Style1">1. How far is it from Iowa City to Chicago (41.8N, 87.7W)?</p></div><div id="mc-region-780" style="position: absolute; top: 1364.25pt; left: 84pt; width: 83.25pt; "><a name="" /><img border="0" src=".\addendum 7 latitude and longitude calculations_images/IMG0022_208380171.PNG" id="generatedImage780" path=".\addendum 7 latitude and longitude calculations_images/IMG0022_208380171.PNG" style="width: 83.25pt;height: 17.25pt;"></img></div><div id="mc-region-781" style="position: absolute; top: 1364.25pt; left: 192pt; width: 84pt; "><a name="" /><img border="0" src=".\addendum 7 latitude and longitude calculations_images/IMG0023_208380171.PNG" id="generatedImage781" path=".\addendum 7 latitude and longitude calculations_images/IMG0023_208380171.PNG" style="width: 84pt;height: 17.25pt;"></img></div><div id="mc-region-721" style="position: absolute; top: 1394.25pt; left: 72pt; width: 65.25pt; "><a name="" /><img border="0" src=".\addendum 7 latitude and longitude calculations_images/IMG0024_208380171.PNG" id="generatedImage721" path=".\addendum 7 latitude and longitude calculations_images/IMG0024_208380171.PNG" style="width: 65.25pt;height: 17.25pt;"></img></div><div id="mc-region-816" style="position: absolute; top: 1394.25pt; left: 192pt; width: 114.75pt; "><a name="" /><img border="0" src=".\addendum 7 latitude and longitude calculations_images/IMG0025_208380171.PNG" id="generatedImage816" path=".\addendum 7 latitude and longitude calculations_images/IMG0025_208380171.PNG" style="width: 114.75pt;height: 17.25pt;"></img></div><div id="mc-region-724" style="position: absolute; top: 1424.25pt; left: 72pt; width: 69.75pt; "><a name="" /><img border="0" src=".\addendum 7 latitude and longitude calculations_images/IMG0026_208380171.PNG" id="generatedImage724" path=".\addendum 7 latitude and longitude calculations_images/IMG0026_208380171.PNG" style="width: 69.75pt;height: 13.5pt;"></img></div><div id="mc-region-817" style="position: absolute; top: 1424.25pt; left: 168pt; width: 61.5pt; "><a name="" /><img border="0" src=".\addendum 7 latitude and longitude calculations_images/IMG0027_208380171.PNG" id="generatedImage817" path=".\addendum 7 latitude and longitude calculations_images/IMG0027_208380171.PNG" style="width: 61.5pt;height: 17.25pt;"></img></div><div id="mc-region-818" style="position: absolute; top: 1425pt; left: 258pt; width: 82.5pt; background-color: inherit; "><a name="" /><img border="0" src=".\addendum 7 latitude and longitude calculations_images/IMG0028_208380171.PNG" id="generatedImage818" path=".\addendum 7 latitude and longitude calculations_images/IMG0028_208380171.PNG" style="width: 82.5pt;height: 16.5pt;"></img></div><div id="mc-region-786" style="position: absolute; top: 1454.25pt; left: 48pt; width: 323pt; "><a name="" /><p class="Style1">2. How far is it from Iowa City to Paris (48deg 49 min N, 2 deg 29 min E)?</p></div><div id="mc-region-769" style="position: absolute; top: 1490.25pt; left: 282pt; width: 102.5pt; "><a name="" /><p class="Style1">Note:Paris is E longitude which switches the sign</p></div><div id="mc-region-770" style="position: absolute; top: 1486.5pt; left: 42pt; width: 93.75pt; "><a name="" /><img border="0" src=".\addendum 7 latitude and longitude calculations_images/IMG0029_208380187.PNG" id="generatedImage770" path=".\addendum 7 latitude and longitude calculations_images/IMG0029_208380187.PNG" style="width: 93.75pt;height: 45.75pt;"></img></div><div id="mc-region-771" style="position: absolute; top: 1486.5pt; left: 168pt; width: 99.75pt; "><a name="" /><img border="0" src=".\addendum 7 latitude and longitude calculations_images/IMG0030_208380187.PNG" id="generatedImage771" path=".\addendum 7 latitude and longitude calculations_images/IMG0030_208380187.PNG" style="width: 99.75pt;height: 45.75pt;"></img></div><div id="mc-region-782" style="position: absolute; top: 1544.25pt; left: 72pt; width: 65.25pt; "><a name="" /><img border="0" src=".\addendum 7 latitude and longitude calculations_images/IMG0031_208380187.PNG" id="generatedImage782" path=".\addendum 7 latitude and longitude calculations_images/IMG0031_208380187.PNG" style="width: 65.25pt;height: 17.25pt;"></img></div><div id="mc-region-784" style="position: absolute; top: 1544.25pt; left: 150pt; width: 90.75pt; "><a name="" /><img border="0" src=".\addendum 7 latitude and longitude calculations_images/IMG0032_208380187.PNG" id="generatedImage784" path=".\addendum 7 latitude and longitude calculations_images/IMG0032_208380187.PNG" style="width: 90.75pt;height: 17.25pt;"></img></div><div id="mc-region-775" style="position: absolute; top: 1574.25pt; left: 60pt; width: 69.75pt; "><a name="" /><img border="0" src=".\addendum 7 latitude and longitude calculations_images/IMG0033_208380187.PNG" id="generatedImage775" path=".\addendum 7 latitude and longitude calculations_images/IMG0033_208380187.PNG" style="width: 69.75pt;height: 13.5pt;"></img></div><div id="mc-region-819" style="position: absolute; top: 1574.25pt; left: 168pt; width: 61.5pt; "><a name="" /><img border="0" src=".\addendum 7 latitude and longitude calculations_images/IMG0034_208380187.PNG" id="generatedImage819" path=".\addendum 7 latitude and longitude calculations_images/IMG0034_208380187.PNG" style="width: 61.5pt;height: 17.25pt;"></img></div><div id="mc-region-820" style="position: absolute; top: 1575pt; left: 258pt; width: 87pt; background-color: inherit; "><a name="" /><img border="0" src=".\addendum 7 latitude and longitude calculations_images/IMG0035_208380203.PNG" id="generatedImage820" path=".\addendum 7 latitude and longitude calculations_images/IMG0035_208380203.PNG" style="width: 87pt;height: 16.5pt;"></img></div><div></div></body></html>