Section Four  Latitude and Longitude  One Example of a Geographic Grid
For as long as humans have had the goal to explore our planet, they have also had the need to devise a navigation method to make this process easier, and more importantly, a method which was reliable and repeatable. As these exploration and discovery missions took explorers over longer distances and away from land, which is super handy for sighting back to and knowing where you are, a worldwide system that could be used day or night was needed. With a rather long history of discovery, trail, and error, latitude also known as 'parallels' the eastwest portion of a geographic grid measured with angles between 0 and 90° and longitude became a worldwide grid system, giving an explorer the ability to know where they were at all times, with a general error of about 10 feet. Not too bad without a GPS receiver!
Geographic grids are created when a grid is applied to the Earth (initially a sphere for easy measuring, as we will see in this section, then later, that sphere is transformed into an ellipsoid) and the intersections are labeled via a Cartesian Coordinate system method. In this section, we will look at the specifics of creating that grid, by first by creating a spherebased Earthcentered, Earthfixed coordinate system (EFEC coordinate system) Earthcentered, Earthfixed coordinate systems use the center of the Earth as a start point for measurements, while localnorth systems use a smaller area affixed to the Earth’s surface as the start point). ECEF coordinate system require us to select a principle meridian (the northsouth line from which to establish the X origin point, aka the place to start counting) and an before laying out the grid. Since the goal of an ECEF coordinate systems is to label locations on a 3D sphere, we can't efficiently use a linear unit of measure, like feet or meters. As we will see when the lines are drawn and the labels placed, using angles to establish measurements will be a boatload easier than using linear units, which would only be accomplished by wrapping an imaginary cloth tape measure around the Earth, which wouldn't lay flat, and really only measure the circumference, and what would the circumference even tell us .... you get it ... or you will when we look at establishing the line of latitude also known as 'parallels' the eastwest portion of a geographic grid measured with angles between 0 and 90° and longitude.
longitude is an and is just one example of a geographic grid the result of using an established angular unit of measure to label the intersections of northsouth and eastwest lines on the surface of the Earth starting the labels at a principal meridian the northsouth line from which the labeling begins. Eastwest lines have a very obvious start point: the equator. Northsouth lines must start somewhere, so when it is established for a particular geographic grid, it can be considered the principal meridian. , utilizing a specific principle meridian, the Prime Median, and degrees measurement of plane angle, representing 1⁄360 of a full rotation (circle). In full, a degree of arc or arc degree. Usually denoted by ° as the angular unit of measure the selected units for measuring angles. Choices include degree and radians. to label and later be able to navigate to locations on the Earth’s surface.
Geographic grids, it should be noted, are not a complete geographic coordinate system nor are they a datum. Latitude and longitude, albeit the most common  is not the only geographic grid the result of using an established angular unit of measure to label the intersections of northsouth and eastwest lines on the surface of the Earth starting the labels at a principal meridian the northsouth line from which the labeling begins. Eastwest lines have a very obvious start point: the equator. Northsouth lines must start somewhere, so when it is established for a particular geographic grid, it can be considered the principal meridian. out there. All of this reading is working towards the established goal of creating a tied down, worldwide "address system" we call a geographic coordinate system. The GCS is made up of two parts, one of which is a datum (which we will look at in the next section) and the other of which is geographic grid the result of using an established angular unit of measure to label the intersections of northsouth and eastwest lines on the surface of the Earth starting the labels at a principal meridian the northsouth line from which the labeling begins. Eastwest lines have a very obvious start point: the equator. Northsouth lines must start somewhere, so when it is established for a particular geographic grid, it can be considered the principal meridian. (which we are about to see requires an ECEF Coordinate System to be transformation into a reference ellipsoid an ellipsoid that is drawn to bestfit an area. World reference ellipsoids are drawn to bestfit the entire geoid; local ellipsoids are best fit on one side to a single place of the geoid ).
Concept Quiz
2.4.1: Cartesian Coordinate Systems
A Cartesian coordinate system An equally spaced grid that exists in a single geometric plane where intersections of perpendicular lines are labeled with the count of units from a specified origin (0,0) point , created by René Descartes in 1637, is defined as an equally spaced grid that exists in a single geometric plane where intersections of perpendicular lines are labeled with the count of units from a specified (0,0) point. You most likely learned all about Cartesian Coordinate Systems in grade school or jr. high. Each grid has a 0,0 point at the origin of the system, and each intersection of the X and Y lines are labeled according to how far said intersection is from the origin.
FIGURE 2.16: Cartesian Coordinate System 

In the example, the red point is defined (labeled) as (3,1) after counting left (negative) three spaces and up (positive) one space from the purple (0,0) point. The green point is defined as (2,3) by counting two spaces on the X axis and 3 spaces on the Y axis. Finally, the blue point is defined as (1.5, 2.5) utilizing the same method. 
In cartography and GIS, each datum has an origin and labeled points where the X and Y (and since datums are 3D, the Z) lines intersect. In the next section we will look at one method of labeling those intersections using latitude also known as 'parallels' the eastwest portion of a geographic grid measured with angles between 0 and 90° and longitude, but for now we will just look at how the 2D XY values reference ellipsoid an ellipsoid that is drawn to bestfit an area. World reference ellipsoids are drawn to bestfit the entire geoid; local ellipsoids are best fit on one side to a single place of the geoid is connected to the 3D Z value geoid a model of the variation between global mean (average) sea level and local mean sea level the measurement above or below the global average at a single point on the Earth's surface used for recording the elevation of topographic surface a detailed map of the surface features of land. It includes the mountains, hills, creeks, and other bumps and lumps on a particular hunk of earth. The word is a Greekrooted combo of topos meaning "place" and graphein "to write." 's relief the difference between the highest and lowest point within a particular area while landforms are the descriptive words for individual features , which is used to measure precise elevations on the topographic surface a detailed map of the surface features of land. It includes the mountains, hills, creeks, and other bumps and lumps on a particular hunk of earth. The word is a Greekrooted combo of topos meaning "place" and graphein "to write." .
Concept Quiz
2.4.2: Latitude
Lines of Latitude ( latitude also known as 'parallels' the eastwest portion of a geographic grid measured with angles between 0 and 90° (n.) late 14c., "breadth," from Old French latitude also known as 'parallels' the eastwest portion of a geographic grid measured with angles between 0 and 90° (13c.) and directly from Latin latitudo "breadth, width, extent, size," from latus "wide,") or parallels are the eastwest portion of the grid, running horizontally  along the horizon. Using the Equator as a starting point for the geographic grid the result of using an established angular unit of measure to label the intersections of northsouth and eastwest lines on the surface of the Earth starting the labels at a principal meridian the northsouth line from which the labeling begins. Eastwest lines have a very obvious start point: the equator. Northsouth lines must start somewhere, so when it is established for a particular geographic grid, it can be considered the principal meridian. , historically, latitude also known as 'parallels' the eastwest portion of a geographic grid measured with angles between 0 and 90° was easier to compute and comprehend than longitude (northsouth lines), as splitting the Earth in half in the middle was simple to understand and easy to start with. Starting at the Equator, lines of latitude also known as 'parallels' the eastwest portion of a geographic grid measured with angles between 0 and 90° use an angle system based upon a right triangle, encircling the earth at each of the 90 degrees measurement of plane angle, representing 1⁄360 of a full rotation (circle). In full, a degree of arc or arc degree. Usually denoted by ° North and South. The best way to understand how lines of latitude also known as 'parallels' the eastwest portion of a geographic grid measured with angles between 0 and 90° are labeled is to look at the process it takes to create the lines from scratch.
Step 4. Find the distance between lines of latitude also known as 'parallels' the eastwest portion of a geographic grid measured with angles between 0 and 90° 

When the circumference of the Earth is divided by 360 (total degrees measurement of plane angle, representing 1⁄360 of a full rotation (circle). In full, a degree of arc or arc degree. Usually denoted by ° in a circle), each line of latitude also known as 'parallels' the eastwest portion of a geographic grid measured with angles between 0 and 90° is 111 kilometers or 68.972 miles (we will just round that up to 69 miles) from the next. Which by navigation standards is like getting lost on the way to Denver and ending up in Fort Collins.

Step 5. Invent minutes "arc minute" 1/60th of a degree. Usually denoted by " and seconds arc minute. 1/60th of an arc minute; 1/3600 of a degree. Usually denoted by ' to better divide up the Earth’s circumference 

To reduce the error created using a system with an accuracy of 69 miles, each degree was divided into 60 equal parts, or arc
minutes
"arc minute" 1/60th of a degree. Usually denoted by "
. Just as there are sixty
minutes
"arc minute" 1/60th of a degree. Usually denoted by "
on the clock, and it takes all sixty
minutes
"arc minute" 1/60th of a degree. Usually denoted by "
for the hour hand to travel from the 12 back to the 12, there are 60 arc
minutes
"arc minute" 1/60th of a degree. Usually denoted by "
in a degree. By dividing each degree into sixty equal parts and doing a bit of math, we can determine each arc minute is equal to 1.15 statute miles (or 1 nautical mile, since nautical and statute miles were based upon two different measurement systems)^{4}. Breaking down each degree into arc minutes "arc minute" 1/60th of a degree. Usually denoted by " (which in GIS is simply shortened to ‘ minutes "arc minute" 1/60th of a degree. Usually denoted by " ’), navigation accuracy was dramatically improved. Yet, one nautical mile was still a bit course to navigate by, so the arc minute was broken into sixty equal parts to create the arc second. Arc seconds arc minute. 1/60th of an arc minute; 1/3600 of a degree. Usually denoted by ' (or just ‘ seconds arc minute. 1/60th of an arc minute; 1/3600 of a degree. Usually denoted by ' ’ in GIS) measure in at roughly twotenths of a mile, or about 121 feet (nautical) or 105 feet (statute). In modern terms, where even a smart phone is capable of being accurate down to about a foot, 100 feet seems like a pretty big margin of error, but when you think of latitude also known as 'parallels' the eastwest portion of a geographic grid measured with angles between 0 and 90° and longitude in navigational terms, if Siri is telling you your destination is on the right but that Taco Bell is still 100 feet away, its pretty darn good 
Concept Quiz
2.4.3: Longitude
Lines of longitude (late Middle English (also denoting length and tallness): from Latin longitudo, from longus ‘long.’) or meridians are similar to latitude also known as 'parallels' the eastwest portion of a geographic grid measured with angles between 0 and 90° in the fact they divide the Earth into equal parts, but a different due to the fact that all lines of longitude all pass through the North and South poles, creating what looks like a spider web when viewed from the top down. The convergence of lines at the North and South poles results in the distance between lines of longitude not being an equal measure of 69 miles for the entire distance like lines of latitude also known as 'parallels' the eastwest portion of a geographic grid measured with angles between 0 and 90° , but a variation between 0 miles at the poles and 69 miles at the Equator.
To solve for the variation, multiply either 68.972 miles or 111 kilometers by the cosine of the latitude also known as 'parallels' the eastwest portion of a geographic grid measured with angles between 0 and 90° for which the longitude measurement is being taken. For example, Denver, CO is at 39° 44" 31.3548' N, 104° 59 29.5116" W. To find the distance between the lines of longitude (in this case, 104° and 103° or 105° along the 39the parallel), first find the cosine of the latitude also known as 'parallels' the eastwest portion of a geographic grid measured with angles between 0 and 90° (cosine of 39° = 0.777), then multiply it by 68.972 miles (68.972 * 0.777) which would equal 54.38 miles. In other words, if you walked north along 104° W from the 38the to the 39the parallel, you would walk 68.972 miles, and if you continued the journey on the the 40the parallel, you would walk another 68.972 miles. However, if you were to walk west along the 39the parallel from 104°W to 105°W, you would walk 54.38 miles. If you moved that journey north one degree of latitude also known as 'parallels' the eastwest portion of a geographic grid measured with angles between 0 and 90° and walk west along the 38the parallel from 104°W to 105°W, you would only walk 54.33 miles (68.972 * 0.788), a difference of only 0.03 miles, but indeed a shorter distance.
With lines of latitude also known as 'parallels' the eastwest portion of a geographic grid measured with angles between 0 and 90° , there is a logical start point: divide the sphere in half creating the equator. But when it comes to lines of longitude, the part where the divisions are made by drawing a line from the North to the South pole is logical, deciding where to put the zero line, or the prime meridian, is a bit less obvious. So a place was chosen  Greenwich, England^{5}.
While lines of latitude also known as 'parallels' the eastwest portion of a geographic grid measured with angles between 0 and 90° are measured as angles between the North (or South) pole and the equator, angles which measure lines of longitude start at the Prime Meridian the name of the principal meridian the northsouth line from which the labeling begins. Eastwest lines have a very obvious start point: the equator. Northsouth lines must start somewhere, so when it is established for a particular geographic grid, it can be considered the principal meridian. in the latitude/longitude system and travel along the Equator marking each degree to the west until the reaching the other half of the Prime Meridian the name of the principal meridian the northsouth line from which the labeling begins. Eastwest lines have a very obvious start point: the equator. Northsouth lines must start somewhere, so when it is established for a particular geographic grid, it can be considered the principal meridian. in the latitude/longitude system , this creates 180 marks (and moving from the Prime Meridian the name of the principal meridian the northsouth line from which the labeling begins. Eastwest lines have a very obvious start point: the equator. Northsouth lines must start somewhere, so when it is established for a particular geographic grid, it can be considered the principal meridian. in the latitude/longitude system east makes a maximum of 180 marks east, for a total of 360 degrees measurement of plane angle, representing 1⁄360 of a full rotation (circle). In full, a degree of arc or arc degree. Usually denoted by ° , a full circle)
Concept Quiz
2.4.4: Creating a Complete Geographic Grid
Now that we have looked at how the Earthfixed, Earthcentered coordinate system is established for latitude also known as 'parallels' the eastwest portion of a geographic grid measured with angles between 0 and 90° and longtiude, we need turn the spherebased system in to an ellipsoid based system to complete the geographic grid the result of using an established angular unit of measure to label the intersections of northsouth and eastwest lines on the surface of the Earth starting the labels at a principal meridian the northsouth line from which the labeling begins. Eastwest lines have a very obvious start point: the equator. Northsouth lines must start somewhere, so when it is established for a particular geographic grid, it can be considered the principal meridian. . We learned in Section Three that the Earth is best represented by an ellipsoid of revolution, more specifically, an oblate spheroid a spherelike object which is wider than it is tall and the geoid a model of the variation between global mean (average) sea level and local mean sea level the measurement above or below the global average at a single point on the Earth's surface used for recording the elevation of topographic surface a detailed map of the surface features of land. It includes the mountains, hills, creeks, and other bumps and lumps on a particular hunk of earth. The word is a Greekrooted combo of topos meaning "place" and graphein "to write." 's relief the difference between the highest and lowest point within a particular area while landforms are the descriptive words for individual features , which is used to measure precise elevations on the topographic surface a detailed map of the surface features of land. It includes the mountains, hills, creeks, and other bumps and lumps on a particular hunk of earth. The word is a Greekrooted combo of topos meaning "place" and graphein "to write." is best represented by an ellipsoid. In Section Four, we learned the definiton of a geodetic datum the result of attaching a "freefloating" reference ellipsoid an ellipsoid that is drawn to bestfit an area. World reference ellipsoids are drawn to bestfit the entire geoid; local ellipsoids are best fit on one side to a single place of the geoid to a specifically measured geoid a model of the variation between global mean (average) sea level and local mean sea level the measurement above or below the global average at a single point on the Earth's surface used for recording the elevation of topographic surface a detailed map of the surface features of land. It includes the mountains, hills, creeks, and other bumps and lumps on a particular hunk of earth. The word is a Greekrooted combo of topos meaning "place" and graphein "to write." 's relief the difference between the highest and lowest point within a particular area while landforms are the descriptive words for individual features , which is used to measure precise elevations on the topographic surface a detailed map of the surface features of land. It includes the mountains, hills, creeks, and other bumps and lumps on a particular hunk of earth. The word is a Greekrooted combo of topos meaning "place" and graphein "to write." via control points and benchmarks. is combining a geoid a model of the variation between global mean (average) sea level and local mean sea level the measurement above or below the global average at a single point on the Earth's surface used for recording the elevation of topographic surface a detailed map of the surface features of land. It includes the mountains, hills, creeks, and other bumps and lumps on a particular hunk of earth. The word is a Greekrooted combo of topos meaning "place" and graphein "to write." 's relief the difference between the highest and lowest point within a particular area while landforms are the descriptive words for individual features , which is used to measure precise elevations on the topographic surface a detailed map of the surface features of land. It includes the mountains, hills, creeks, and other bumps and lumps on a particular hunk of earth. The word is a Greekrooted combo of topos meaning "place" and graphein "to write." with a reference ellipsoid an ellipsoid that is drawn to bestfit an area. World reference ellipsoids are drawn to bestfit the entire geoid; local ellipsoids are best fit on one side to a single place of the geoid , connecting them via control points. In order to keep ontrack with our definitions, we need to transform the sphere used to create the EFEC latitude also known as 'parallels' the eastwest portion of a geographic grid measured with angles between 0 and 90° /longitude coordinate system into not an oblate spheroid a spherelike object which is wider than it is tall , as we are not trying to represent the Earth, but instead into an ellipsoid to bestfit inside the geoid a model of the variation between global mean (average) sea level and local mean sea level the measurement above or below the global average at a single point on the Earth's surface used for recording the elevation of topographic surface a detailed map of the surface features of land. It includes the mountains, hills, creeks, and other bumps and lumps on a particular hunk of earth. The word is a Greekrooted combo of topos meaning "place" and graphein "to write." 's relief the difference between the highest and lowest point within a particular area while landforms are the descriptive words for individual features , which is used to measure precise elevations on the topographic surface a detailed map of the surface features of land. It includes the mountains, hills, creeks, and other bumps and lumps on a particular hunk of earth. The word is a Greekrooted combo of topos meaning "place" and graphein "to write." via a method called affine transformation. Affine transformation is a means of changing a sphere into an ellipsoid while maintaining straight lines.
After we took all the time to carefully create the lines of latiude and longitude and we understand they are used to gridout and label locations on the Earth's surface, we'd like to take that hard work and preserve it over in the reference ellipsoid an ellipsoid that is drawn to bestfit an area. World reference ellipsoids are drawn to bestfit the entire geoid; local ellipsoids are best fit on one side to a single place of the geoid . All we are really doing is linearly streching the measurement of the radius into a semiminor and a different semimajor axis. By increasing the measurements of the radius, we are able to create an ellipsoid without curving the straight lines.
The Main Point... 

Our overall goal is to create a geographic coordinate system. To accomplish that, we need to first create a datum, which combines a
geoid
a model of the variation between global mean (average) sea level and
local mean sea level
the measurement above or below the global average at a single point on the Earth's surface used for recording the elevation of
topographic surface
a detailed map of the surface features of land. It includes the mountains, hills, creeks, and other bumps and lumps on a particular hunk of earth. The word is a Greekrooted combo of topos meaning "place" and graphein "to write."
's
relief
the difference between the highest and lowest point within a particular area while landforms are the descriptive words for individual features
, which is used to measure precise elevations on the
topographic surface
a detailed map of the surface features of land. It includes the mountains, hills, creeks, and other bumps and lumps on a particular hunk of earth. The word is a Greekrooted combo of topos meaning "place" and graphein "to write."
and a
reference ellipsoid
an ellipsoid that is drawn to bestfit an area. World reference ellipsoids are drawn to bestfit the entire geoid; local ellipsoids are best fit on one side to a single place of the geoid
, since an ellipsoid is a better representation of a
geoid
a model of the variation between global mean (average) sea level and
local mean sea level
the measurement above or below the global average at a single point on the Earth's surface used for recording the elevation of
topographic surface
a detailed map of the surface features of land. It includes the mountains, hills, creeks, and other bumps and lumps on a particular hunk of earth. The word is a Greekrooted combo of topos meaning "place" and graphein "to write."
's
relief
the difference between the highest and lowest point within a particular area while landforms are the descriptive words for individual features
, which is used to measure precise elevations on the
topographic surface
a detailed map of the surface features of land. It includes the mountains, hills, creeks, and other bumps and lumps on a particular hunk of earth. The word is a Greekrooted combo of topos meaning "place" and graphein "to write."
. A
reference ellipsoid
an ellipsoid that is drawn to bestfit an area. World reference ellipsoids are drawn to bestfit the entire geoid; local ellipsoids are best fit on one side to a single place of the geoid
is really just an plain old ellipsoid until a
geographic grid
the result of using an established angular unit of measure to label the intersections of northsouth and eastwest lines on the surface of the Earth starting the labels at a
principal meridian
the northsouth line from which the labeling begins. Eastwest lines have a very obvious start point: the equator. Northsouth lines must start somewhere, so when it is established for a particular geographic grid, it can be considered the principal meridian.
has been applied to it, and that
geographic grid
the result of using an established angular unit of measure to label the intersections of northsouth and eastwest lines on the surface of the Earth starting the labels at a
principal meridian
the northsouth line from which the labeling begins. Eastwest lines have a very obvious start point: the equator. Northsouth lines must start somewhere, so when it is established for a particular geographic grid, it can be considered the principal meridian.
starts out as an Earthfixed, Earthcentered coordinate system, such as the
latitude
also known as 'parallels'
the eastwest portion of a geographic grid measured with angles between 0 and 90°
and longitude system. After the EFEC system is complete, it is necessary to use and affine transformation to "convert" the sphere into an ellipsoid. An affine transformation maintains the straight lines of the
latitude
also known as 'parallels'
the eastwest portion of a geographic grid measured with angles between 0 and 90°
/longitude system between the sphere and the ellipsoid. 
Concept Quiz
2.4.5: Null Island: A Real Fake Place
2.4.6: Additional Resources for Latitude and Longitude
YouTube Break: The Clock That Changed the World
An explanation of using time for navigation (the longitude problem) and satellite navigation ( trilateration the process of determining absolute or relative locations of points by measurement of distances, using the geometry of circles, spheres or triangles. ... In contrast to triangulation, it does not involve the measurement of angles ).
Run Time: 29 minutes "arc minute" 1/60th of a degree. Usually denoted by " , but really well done and quite interesting. Highly recommended to better understand the geographic grid the result of using an established angular unit of measure to label the intersections of northsouth and eastwest lines on the surface of the Earth starting the labels at a principal meridian the northsouth line from which the labeling begins. Eastwest lines have a very obvious start point: the equator. Northsouth lines must start somewhere, so when it is established for a particular geographic grid, it can be considered the principal meridian. that is longitude and latitude also known as 'parallels' the eastwest portion of a geographic grid measured with angles between 0 and 90° .
 Note: SatNav = GPS Receiver
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