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Reload this page GPS (Global Positioning Systems)

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    Topics this page:

  • Positioning Systems
  • Sundials
  • Star Systems
  • Global Positioning Satellites
  • Waypoints
  • Microsoft Map
  • Google Maps
  • Apple Maps
  • Lattitudes and Longitudes
  • Using NMEA Raw Data
  • Your comments???
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    Set this at top of window. Positioning Systems

    Set this at top of window. Google Maps

      http://maps.google.com enable you to seek locations based on geographic names.

      http://www.google.com/history enables

      iTouchMap displays Latitude/Longitude on Google Maps.

      Latitude sounds like "flat" as in going between East and West. The lat in Latitude reminds us that its goes lateral (sideways). Since the latitutde/longitude coordinate system was developed by the British Navy at Greenwich (London), zero degree 0° latitude starts there, at the Prime Meridian. The 360 degrees to a lateral circle are divided in a positive and negative 180 degrees.

      Each one degree of latitude is just over 69 miles, or 111 km of the earth's 25,000 mile circumference.

      • The Eastern-most point in the US is near Bangor, Maine at 69 W.
      • Honolulu, Hawaii is the Western-most point in the US at 157 W.
      • Anchorage, Alaska is at 149 W.

      Longitude contains an N for North (not in the word latitude). At 45 degrees N or S of the equator, one degree of longitude is about 49 miles. But as you move north or south of the equator, the distance between the lines of longitude gets shorter.

      • The Equator is 0 degrees.
      • The North Pole is 90 degrees North. South Pole is 90 degrees South.
      • Northern Arctic circle (66.5 degree North)
      • Southern Antarctic circle ( 66.5 degree South)
      • Tropic of Cancer ( 23.5 degree North)
      • Tropic of Capricorn ( 23.5 degree South)
      • Greenich is at 51.5° North latitude.
      • The "48th Parallel" in history is a L____ number.

      For additional precision, a coordinate can be broken down into minutes (') and then seconds ("). Each degree has 60 minutes. Each minute has 60 seconds. Each seconds can be further divided into tenths, hundredths, or even thousandths.

    Set this at top of window. Sundials and Sextents

      A GPS is not needed to know which direction is North.

      Yes, you can find the North Star on a cloudless night.

      Yes, you can find the North Star.

    Set this at top of window. Star Systems

      Procession of Earth North Pole because there is a precess of 23.5 degrees from a theoretical eliptic pole, such that a complete rotation takes 25,800 years. This torque is created by a whobble caused by the bulge at Earth's equator.

      What enabled Columbus and other sailors in his time to sail the oceans and "discover" new lands is the ability to navigate using the pattern of stars and the sun.

      Sundials do this.

      The specification of position of stars from Earth is described by Michael Richmond

      This time-lapse photo show how stars in the Northern Hemisphere revolve around the "North Star" Polaris at the handle end of the "Big Dipper" constallation. But where the Celestial Pole points is only temporary

      H. A. Rey's book The Stars: A New Way to See Them (1976: Houghton Mifflin) illustrates shows that people living in 500 BC see stars rovolving around the star Kochab.

      Astronomers measure (from the Celestial Equinox radiating from earth's equator) the proper motions of stars in "arcseconds" or "milliarcseconds" per year vertically in Right Ascension (RA) and horizontally in Declinations (Dec.).

      Jamie Love's description of precession


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    Set this at top of window. Paper Maps and Magnetic Compass

      The compass is based on magnetic fields inside the earth.

      The trouble is that the strength and alignment of magnetic fields differ widely, especially over iron mines.

      Those who use map and compass add lines to Topo maps.


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    Set this at top of window. Microsoft Maps

      Microsoft Streets and Trips (SnT) is the entry-level consumer edition. It covers US and Canada. Microsoft's AutoRoute covers the streets of Europe. Full price for it at Staples is $39, or $69 with a GPS USB device. But I lose the GPS device often enough that I need to buy it for $50 online on Amazon.

      Microsoft's accessories site.

      The $299 Microsoft MapPoint alternative edition provides "business" features like export to Excel. SnT is limited to importing Excel but not export because Microsoft's

      TIP: Unlike maps provided on mobile phones, an internet connection is not needed for these products installed on desktop Windows 7/Vista/XP. Its maps load into the larger hard disk on laptops from the DVD that comes with the product.

      The down side of course is that updates need to be purchased. To keep up with changes, Microsoft publishes annual editions. The 2013 edition was available July 2012.

      On top of the DVD, DOWNLOAD: Get and load waypoint pushpins, such as the 9+ MB Points of Interest Mega file various hobbyists update for free use. It still doesn't match the information shown on Google Maps. But it contains many points of interest that can save you money and hassle:

      • Accident black spots to avoid.
      • Presidential libraries, hot springs, waterfalls
      • campgrounds, RV parking at Walmarts
      • Stores travelers frequent: Home Depot, Northern Tool, Bass Pro Shops, Cabellas, Trader Joes
      • Lower priced fuel at truck stops like Flying J, Kroger, Loves, Murphy as well as fuel at
      • Membership stores with gas: Costo, Frys, Smiths, Walmart.

      BLAH: Churches are missing from the file. The POI file is not easy to use. The procedure for first-time setup:

      1. After download, open Windows Explorer and navigate to where .est files are stored. That is typically My Documents.
      2. In 2009 and earlier versions, there is a Templates folder which the program goes to for the default template file when File > Open is invoked by the user.
      3. But there is no such folder in 2013 version. So from the zip folder drag and drop the template among the .est files.
      4. Open a second instance of the program. Your workstation may not have enough memory for this.
      5. Click on the help flags and icons and right-click to cut them.
      6. Within 2013 editions right-click on specific groups to show, then later hide each group.
        Within prior editions, copy individual icons and paste
      7. The customer icons are rather big, which is good at high levels of zoom. But this requires one to zoom in to read road names under the icon.

      Set this at top of window. Trip Planning using Microsoft Maps

      TIP: The sequence of my approach to (obsessively) planning out a trip:

      • Guidebooks help to provide priority of why I'm there: If it's an event, I would plot the event site first, then surrounding points of interest such as Museums or views or places to eat.
      • Sunsets can be a special time for landscape viewing, so I try to be at a good place to enjoy it, such as a vista, especially in a restaurant or hotel with a view. It's also not a good idea to be driving when the sun can get in people's eyes and when animals are out to feed. QUESTION: To find the time of sunsets?
      • Destination restaurants on Yelp, Zagat, Gayot, etc. may alter my lodging plans since its wise to avoid too much driving after a meal.
      • However, the availability of lodging does have impact. So it's a back-and-forth process.

      BLAH: Scheduling requires a few tricks.

      • To stop for the night, right-click on a route icon (yellow box) to set schedule stop for 12 hours.
      • To avoid changes to my schedule being changed automatically if a default option is changed, I manually set each starting time to a custom time like 9:01.
      • Set the default Start Time each day early (like 5 am) in menu Route > Route Options.
      • Create a duplicate item to set the Starting Time each day.
      • Once I have an inital schedule, I then adjust the sequence of locations.
      • TIP: It's wise to allow extra time to stop at interesting spots on the spur of the moment, or for traffic delays.
      • If I am using a rental car or have to drive long distances, to avoid the stress I figure out ahead where to fill up on gasoline.
      • When crossing time zone boundaries (such as going between Arizona and another state) I click on the road near the border to add a location, then rename it to say change time zone.
      • TIP: Avoid clicking on the name of a town because that usually routes to a weird place. Instead, get a general understanding of the location, then create a location by right-clicking on a road.

      TIP: Tactics I've found useful:

      • BLAH: The program does't show the direction of travel. So to avoid confusion, for a complex itinery I use separate files for going into and out of a destination.
      • QUESTION: To transfer a route item from one est file to another?
      • I keep a scraps.est file to hold locations that are possible alternatives not on my planned route, but helps me save time from looking them up.
      • Unlike other Microsoft programs, pressing F2 does not allow editing. Instead, right-click and select Rename. Copy the text before dragging yellow squre icons across the map. After the move rename and paste the text.

      Even though my mobile device can't hold the whole DVD, supposedly I can transfer a limited number of specific maps to my mobile phone's SD card. BLAH: I wonder why there is a "Send to Mobile" button if its greyed because upgrade to MapPoint is needed to use it.

      REFERENCE: msstreets.com is the Streets & Trips team blog.

      Set this at top of window. Quick Keys

      Zoom In = Ctrl+Z
      Zoom Out = PgUp
      Zoom In Max = End
      GPS Pane = Ctrl+A
      Driving Guidance = Ctrl+B
      Nearby Places = Ctrl+D
      Import Data Wizard = Alt+V, Ctrl+I
      Legend = Ctrl+G
      Show in MSN Virtual Earth = Ctrl+H
      Search Internet = Ctrl+K
      Toggle Route Planner = Ctrl+G
      Map Style Toggle = Ctrl+M
      Map Style Toggle = Ctrl+M
      Save = Ctrl+N
      Open = Ctrl+O
      Print = Ctrl+P
      Start GPS Tracking = Ctrl+Q
      Route Planner = Ctrl+R
      Save (no confirm) = Ctrl+S
      Directions = Ctrl+U


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    Set this at top of window. Global Positioning Satellite Technology

      Each major superpower each has their own set of satellites.

      • China has their BeiDou Satellite Navigation System
      • Russia has ...
      • US has the NAVSTAR (NAVigation Satellite Timing and Ranging) satellites

      The Samsung Electronics' Galaxy Note 3 was the first smartphone to incorporate all 3, using the IZat location tools for Qualcomm chips inside them.


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    Set this at top of window. Global Positioning Satellite Internals

      GPS receivers display a waypoint -- a specific position on the earth -- by interpreting signals from a constellation of 24 NAVSTAR (NAVigation Satellite Timing and Ranging) satellites put up by the US Dept. of Defense. The satellites circle the earth every 12 hours.

      Waypoints (WPS) are based on the UTM (Universal Transverse Mercator) projection of the world defined by the U.S. National Imagery and Mapping Agency (NIMA).

      For example, where actress Eva Gabor is buried in Westwood Cemetary in Los Angeles: Longitude 34.05.880 N, Lattitude -118.44.145 W is equivalent to this GPS coordinate:

        11S 0123456 3772456

      11S" is equivalent to a 100,000-meter square area between Longitudes -120 and -114 degrees West and Lattitudes between xxx and xxx degrees North.

      11" specifies a horizontal East-West location on this grid which horizontally divides the 360 degree circumference of our world into increments of 6 degree each for 60 Zone Numbers, beginning half way around the world from the “prime meridian" near the British Royal Naval Observatory near London suburb of Greenwich, England.

      S" specifies a North-South location among grid increments equivalent to 8 degrees above and below the equator. The 20 zone Designations go alphabetically from “C" at the South Pole (-80 degree lattitude), through “N" at the equator (zero degree lattitude), to “X" (+80 degree lattitude) at “True North". “I" and “O" are excluded to avoid confusion.

      0123456" is the Easting value of 12.3456 meters. Using the NAD 27 datum, a central meridian at the middle of each zone is assigned an easting value of 500,000 meters. Easting values to the west of this central meridian are less than 500,000 meters (yards); to the east, more than 500,000 meters.

      3772456" is the sample Northing value of 3777.456 meters. Positive northing values measure the distance Northward from the equator. Negative northing values measure the distance Southward from the equator.

      To avoid confusion, UTM maps do not display digits after the decimal point and highlight the last two significant digits. An Easting of 3772.456 appears as 3772.




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      Set this at top of window. Waypoints from other famous spots:
      LocationParallels of Lattitude DegreesEasting Zone NumberNorthing Zone DesignationMeridians Longitude Degrees
      Royal Naval Observatory, England031U48 N
      Los Angeles, CA, USA-118.44.145 W11S34.05.880 N
      Cairns, Queensland, Australia144 E55K-24 S

      GNIS (Geographic Name Information Server) Search the GNIS database of all names USGS uses on maps.

      To calculate the exact distance between two coordinates (LAT1,LG1 to LAT2,LG2): The simplest but least accurate approximation method is (in BASIC):

        D = 69.1 * SQR[ (LAT2-LAT1)^2 + 0.6 * (LG2-LG1)^2 ]

      One degree of longitude or latitude is equal to 69.1 miles.

      The most accurate calculation uses spherical geometry's arccosine function after converting numbers to radians by dividing 180/pi — the constant C=57.3:

        A = ( sin(LAT1/C) * sin(LAT2/C) ) + (cos(LAT1/C) * cos(LAT2/C) * cos(LG2/C - LG1/C) D = 3959 * atn( sqr(1-A^2) / A )


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      Set this at top of window. Map Grid Overlays

      “7.5 x 15 Minute Quads" (1:25,000 scale) maps created by the USGS (US Geological Survey) with UTM grid markings.

      download Download this PostScript file and print the overlay on acetate film

      A large region of several states are shown at scales of around 1:580,000. Metro areas are shown at around 1:200,000. City maps are shown at views of around 1:150,000, with some historic and downtown areas shown at around 1:14,000.

      Set this at top of window. Accuracy

      When civilians first used GPS “Standard Positioning Service" signals, DoD imposed Selective Availability (SA) accuracy of about 100 meters (110 yards) radius. This means that the receiver could actually be anywhere within 200 meters of a given GPS coordinate. As of May 2, 2000 SA IS OFF, improving average EPE (estimated Position Error) to 5-15 metres.

      GPS Elevation readings are several times less accurate.

      Also, each GPS satellite broadcasts a channel at frequency L1 (1575.42MHz) and L2 (1227.6 MHz) . The more channels a GPS receiver can read, the more accurate its readings.

      Readings may take several minutes, require a clear view of the sky with adequate Mask Angle -- the minimum acceptable satellite elevation above the horizon to avoid blockage of line-of-sight.

      A receiver that reads 12 chennels in parallel is quicker than a receiver which reads 8 channels in sequence (i.e. multiplex mode).

      Differential GPS (DGPS) receivers use fixed land-based beacons maintained by the Coast Guard for 3-5 meter accuracy.

      Accuracy within 3 meters comes from receivers which read differential signals from FAA's WAAS (Wide Area Augmentation System) satellites over the equator, which broadcast adjustments needed to compensate for GPS satellite orbit and clock drift plus signal delays caused by the atmosphere and ionosphere. These adjustments are calculated by 25 ground reference stations across the United States. Similarly, the Japanese have their Multi-Functional Satellite Augmentation System (MSAS), Russians have their Global Navigation Satellite System (GLONASS). The European Space Agency (ESA) plans to have, by 2008, the Euro Geostationary Navigation Overlay Service (EGNOS) project known as Galileo, 30 satellites orbiting at 24,000 kilometers about the earth, guaranteeing accuracy within 4 meters.

      The Differential “Precision Positioning System" used by the military is most accurate. The BlackJack system, delivers accuracy of 2 to 3 centimeters.

      The accuracy of GSP can be improved by using a grid of them. USC is using this technique to detect earthquakes in Los Angeles and elsewhere.


    A website external to this site Campmor.

    Get Your Bearings

    Many GPS receivers can log waypoints to determine the direction you've been traveling. But this eats up batteries. So many people also use a regular magnetic compass ( adjusted to magnetic decliations) to compliment the GPS. Learn to use a compass at Kjetil's Compass Lesson.

    Trivia Reminder

    Earth and Mars have virtually identical spin axis tilts at about 23.5 degrees.


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    Set this at top of window. Using GPS NMEA Raw Data

      Raw GPS data issued by modern GPS devices follow a standard format defined by the NMEA (National Marine Electronics Association).

      Among the over fifty kinds of comma-delimited "sentence" issued by a GPS device, the "Recommended Minimum" sentence is the most commonly used. Here is an example from Jon Person's tutorial on how Visual Studio.NET program can intrepret GPS data such as:

        $GPRMC,040302.663,A,3939.7,N,10506.6,W,0.27,358.86,200804,,*1A

      This one sentence contains nearly everything a GPS application needs: latitude, longitude, speed, bearing, satellite-derived time, fix status and magnetic variation.


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      Set this at top of window. Map Grid Overlays

      Country StateCity Lat. Long. Lat. Long. Time
      ° ' ° '
      USA NY Albany 42 40 73 45     12:00 noon
      USA NM Albuquerque 35 05 106 39     10:00 a.m.
      USA TX Amarillo 35 11 101 50     11:00 a.m.
      USA AK Anchorage 61 13 149 54     8:00 a.m.
      USA GA Atlanta 33 45 84 23     12:00 noon
      USA TX Austin 30 16 97 44     11:00 a.m.
      USA OR Baker 44 47 117 50     9:00 a.m.
      USA MD Baltimore 39 18 76 38     12:00 noon
      USA ME Bangor 44 48 68 47     12:00 noon
      USA AL Birmingham 33 30 86 50     11:00 a.m.
      USA ND Bismarck 46 48 100 47     11:00 a.m.
      USA ID Boise 43 36 116 13     10:00 a.m.
      USA MA Boston 42 21 71 5     12:00 noon
      USA NY Buffalo 42 55 78 50     12:00 noon
      CDN AB Calgary 51 1 114 1     10:00 a.m.
      USA NM Carlsbad 32 26 104 15     10:00 a.m.
      USA SC Charleston 32 47 79 56     12:00 noon
      USA WV Charleston 38 21 81 38     12:00 noon
      USA NC Charlotte 35 14 80 50     12:00 noon
      USA WY Cheyenne 41 9 104 52     10:00 a.m.
      USA IL Chicago 41 50 87 37     11:00 a.m.
      USA OH Cincinnati 39 8 84 30     12:00 noon
      USA OH Cleveland 41 28 81 37     12:00 noon
      USA SC Columbia 34 0 81 2     12:00 noon
      USA OH Columbus 40 0 83 1     12:00 noon
      USA TX Dallas 32 46 96 46     11:00 a.m.
      USA CO Denver 39 45 105 0     10:00 a.m.
      USA IA Des Moines 41 35 93 37     11:00 a.m.
      USA MI Detroit 42 20 83 3     12:00 noon
      USA IA Dubuque 42 31 90 40     11:00 a.m.
      USA MN Duluth 46 49 92 5     11:00 a.m.
      USA ME Eastport 44 54 67 0     12:00 noon
      CDN AB Edmonton 53 34 113 28     10:00 a.m.
      USA CA El Centro 32 38 115 33     9:00 a.m.
      USA TX El Paso 31 46 106 29     10:00 a.m.
      USA OR Eugene 44 3 123 5     9:00 a.m.
      USA ND Fargo 46 52 96 48     11:00 a.m.
      USA AZ Flagstaff 35 13 111 41     10:00 a.m.
      USA TX Fort Worth 32 43 97 19     11:00 a.m.
      USA CA Fresno 36 44 119 48     9:00 a.m.
      USA CO Grand Junction 39 5 108 33     10:00 a.m.
      USA MI Grand Rapids 42 58 85 40     12:00 noon
      USA MT Havre 48 33 109 43     10:00 a.m.
      USA MT Helena 46 35 112 2     10:00 a.m.
      USA HI Honolulu 21 18 157 50     7:00 a.m.
      USA AK Hot Springs 34 31 93 3     11:00 a.m.
      USA TX Houston 29 45 95 21     11:00 a.m.
      USA ID Idaho Falls 43 30 112 1     10:00 a.m.
      USA IN Indianapolis 39 46 86 10     12:00 noon
      USA MS Jackson 32 20 90 12     11:00 a.m.
      USA FL Jacksonville 30 22 81 40     12:00 noon
      USA AK Juneau 58 18 134 24     8:00 a.m.
      USA MO Kansas City 39 6 94 35     11:00 a.m.
      USA FL Key West 24 33 81 48     12:00 noon
      CDN ON Kingston 44 15 76 30     12:00 noon
      USA OR Klamath Falls 42 10 121 44     9:00 a.m.
      USA TN Knoxville 35 57 83 56     12:00 noon
      USA NV Las Vegas 36 10 115 12     9:00 a.m.
      USA ID Lewiston 46 24 117 2     9:00 a.m.
      USA NE Lincoln 40 50 96 40     11:00 a.m.
      CDN ON London 43 2 81 34     12:00 noon
      USA CA Long Beach 33 46 118 11     9:00 a.m.
      USA CA Los Angeles 34 3 118 15     9:00 a.m.
      USA KY Louisville 38 15 85 46     12:00 noon
      USA NH Manchester 43 0 71 30     12:00 noon
      USA TN Memphis 35 9 90 3     11:00 a.m.
      USA FL Miami 25 46 80 12     12:00 noon
      USA WI Milwaukee 43 2 87 55     11:00 a.m.
      USA MN Minneapolis 44 59 93 14     11:00 a.m.
      USA AL Mobile 30 42 88 3     11:00 a.m.
      USA AL Montgomery 32 21 86 18     11:00 a.m.
      USA VT Montpelier 44 15 72 32     12:00 noon
      CDN QC Montreal 45 30 73 35     12:00 noon
      CDN SK Moose Jaw 50 37 105 31     11:00 a.m.
      USA TN Nashville 36 10 86 47     11:00 a.m.
      CDN BC Nelson 49 30 117 17     9:00 a.m.
      USA NJ Newark 40 44 74 10     12:00 noon
      USA CT New Haven 41 19 72 55     12:00 noon
      USA LA New Orleans 29 57 90 4     11:00 a.m.
      USA NY New York 40 47 73 58     12:00 noon
      USA AK Nome 64 25 165 30     8:00 a.m.
      USA CA Oakland 37 48 122 16     9:00 a.m.
      USA OK Oklahoma City 35 26 97 28     11:00 a.m.
      USA NB Omaha 41 15 95 56     11:00 a.m.
      CDN ON Ottawa 45 24 75 43     12:00 noon
      USA PA Philadelphia 39 57 75 10     12:00 noon
      USA AZ Phoenix 33 29 112 4     10:00 a.m.
      USA SD Pierre 44 22 100 21     11:00 a.m.
      USA PA Pittsburgh 40 27 79 57     12:00 noon
      USA ME Portland 43 40 70 15     12:00 noon
      USA OR Portland 45 31 122 41     9:00 a.m.
      USA RI Providence 41 50 71 24     12:00 noon
      CDN QC Quebec 46 49 71 11     12:00 noon
      USA NC Raleigh 35 46 78 39     12:00 noon
      USA NV Reno 39 30 119 49     9:00 a.m.
      USA UT Richfield 38 46 112 5     10:00 a.m.
      USA VA Richmond 37 33 77 29     12:00 noon
      USA VA Roanoke 37 17 79 57     12:00 noon
      USA CA Sacramento 38 35 121 30     9:00 a.m.
      USA NB St. John 45 18 66 10     1:00 p.m.
      USA MO St. Louis 38 35 90 12     11:00 a.m.
      USA UT Salt Lake City 40 46 111 54     10:00 a.m.
      USA TX San Antonio 29 23 98 33     11:00 a.m.
      USA CA San Diego 32 42 117 10     9:00 a.m.
      USA CA San Francisco 37 47 122 26     9:00 a.m.
      USA SJ San Jose 37 20 121 53     9:00 a.m.
      USA PR San Juan 18 30 66 10     1:00 p.m.
      USA NM Santa Fe 35 41 105 57     10:00 a.m.
      USA GA Savannah 32 5 81 5     12:00 noon
      USA WA Seattle 47 37 122 20     9:00 a.m.
      USA LA Shreveport 32 28 93 42     11:00 a.m.
      USA SD Sioux Falls 43 33 96 44     11:00 a.m.
      USA AK Sitka 57 10 135 15     8:00 a.m.
      USA WA Spokane 47 40 117 26     9:00 a.m.
      USA IL Springfield 39 48 89 38     11:00 a.m.
      USA MA Springfield 42 6 72 34     12:00 noon
      USA MO Springfield 37 13 93 17     11:00 a.m.
      USA NY Syracuse 43 2 76 8     12:00 noon
      USA FL Tampa 27 57 82 27     12:00 noon
      USA OH Toledo 41 39 83 33     12:00 noon
      CDN ON Toronto 43 40 79 24     12:00 noon
      USA OK Tulsa 36 09 95 59     11:00 a.m.
      CDN BC Vancouver 49 13 123 06     9:00 a.m.
      CDN BC Victoria 48 25 123 21     9:00 a.m.
      USA VA Virginia Beach 36 51 75 58     12:00 noon
      USA DC Washington 38 53 77 02     12:00 noon
      USA KS Wichita 37 43 97 17     11:00 a.m.
      USA NC Wilmington 34 14 77 57     12:00 noon
      CDN MB Winnipeg 49 54 97 7     11:00 a.m.


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    Set this at top of window. Decimal versus Radian Coordinates

      Databases generally store coordinates in a decimal degrees floating point number rather than separate degrees/minutes/seconds integers.

      To convert from decimal degrees such as 45.63248 to degrees/minutes/seconds:

      1. Subtract 45, leaving only the decimal .63248. Keep 45 for later reference.
      2. Multiply by 60, to obtain 37.94880 – 37 is the number of minutes; 37 then will follow 45 degrees.
      3. Subtract 37 to leave only .94880.
      4. Multiply by 60 once again to obtain 56.92800, and round to 57, which represents the seconds.
      5. This yields 45 degrees, 37 minutes, 57 seconds.

      To convert to decimal degrees from degrees-minutes-seconds such as 45 degrees, 37 minutes, 57 seconds:

      1. Begin with 57 seconds and divide by 60 to obtain .95000.
      2. Add the 37 minutes to yield 37.95000.
      3. Divide by 60 once again to obtain .63250.
      4. Add the 45 degrees to obtain 45.63250.
      5. CAUTION: Notice that rounding less than one-tenth of a second changes the conversion by .00002 degrees.


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