# 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 world-wide system that could be used day or night was needed. With a rather long history of discovery, trail, and error, latitude and longitude became a world-wide 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 sphere-based Earth-centered, Earth-fixed coordinate system (EFEC coordinate system)  (Earth-centered, Earth-fixed coordinate systems use the center of the Earth as a start point for measurements, while local-north 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 north-south line from which to establish the X origin point, aka the place to start counting) and an angular unit of measure, 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 and longitude.

Latitude and longitude is an and is just one example of a geographic grid, utilizing a specific principle meridian, the Prime Median, and degrees as the angular unit of measure 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 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 (which we are about to see requires an ECEF Coordinate System to be transformation into a reference ellipsoid).

### 2.4.1: Cartesian Coordinate Systems

Cartesian coordinate system, 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 and longitude, but for now we will just look at how the 2D XY values reference ellipsoid is connected to the 3D Z value geoid.

### 2.4.2: Latitude

Lines of Latitude (latitude (n.) late 14c., "breadth," from Old French latitude (13c.) and directly from Latin latitudo "breadth, width, extent, size," from latus "wide,") or parallels are the east-west portion of the grid, running horizontally - along the horizon.  Using the Equator as a starting point for the geographic grid, historically, latitude was easier to compute and comprehend than longitude (north-south 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 use an angle system based upon a right triangle, encircling the earth at each of the 90 degrees North and South.  The best way to understand how lines of latitude are labeled is to look at the process it takes to create the lines from scratch.

Creating Lines of Latitude from Scratch, Step 1. Draw the Earth as a Circle and Cut the Circle in Half
If we draw the Earth as a circle, then bisect that circle (cut in half in the exact middle), we have defined three things already:
1. The Equator (equa = equal; tor = a suffix to make an adjective do an action; thus: the equal maker)
2. The Northern Hemisphere (hemi = half; thus: half a sphere)
3. The Southern Hemisphere
Sidebar: Let’s state a few more facts before we move on.
1. We know the Earth rotates around a central axis, a single line that connects the North and South Poles right through the center of the sphere.
• Let’s mark the North and South poles on our drawing.
2. Since we are dealing with circles, it’s best to define sections with angles, which are measured in degrees (I promise this will be a very short visit back to 10the grade geometry).

Great. We have drawn:

1. A sphere (the Earth)
2. A single point at the top and bottom of that sphere (the North and South poles)
3. a line bisecting that sphere right in the middle (the Equator)
4. And we’ve decided to use a method of measuring parts of that sphere Angles measured in degrees
Creating Lines of Latitude from Scratch, Step 2. Establish the Equator as 0°
Lines of latitude are labeled by the angle which is created from the North Pole to the center of the circle representing the Earth, then from the center of the circle back to the edge somewhere between the North Pole and the Equator. To determine the latitude of the Equator, draw a straight line from the North Pole to the center of the circle and out to the edge right at the Equator (which in our drawing, already exists as the Equator itself). The angle created, according to that 10the grade geometry, is 90° - thus the latitude of the Equator is: 0° latitude; and by default, the North Pole is at 90° latitude.

Step 3. Mark all the angles between 0° and 89° and connect them from east to west

If we add in each angle from 1° to 89°, then from that point, draw a straight line to the other side of our circle, we see we’ve created a series of 90 parallel lines running from the North Pole to the Equator, or the lines of Northern Latitude. If we mark a point representing Denver, we would need to start at the Equator and count up 39 lines of latitude, then drop the point. To create the lines of Southern Latitude, we simple repeat this process starting at the South Pole and moving up.

• When labeling lines of latitude, all lines in the northern hemisphere have an ‘N’ tacked on, while all the lines in the southern hemisphere have an ‘S’ tacked on. The Equator is in neither the northern nor the southern hemisphere, thus gets no label of N or S.
Step 4. Find the distance between lines of latitude

When the circumference of the Earth is divided by 360 (total degrees in a circle), each line of latitude 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.

• While there are 360 degrees in a circle and that is the distance between lines of latitude, we only mark 90 in the north and 90 in the south. This is because latitude sits like a crown placed upon one’s head, encircling the Earth. When we drew our two dimensional representation, we drew all the angles between each pole and the equator, of which there are only 90 in each quadrant. When we connect each degree all the way around, it creates 90 lines of latitude in the north and 90 lines of latitude in the south.
Step 5. Invent minutes and seconds 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. Just as there are sixty minutes on the clock, and it takes all sixty minutes for the hour hand to travel from the 12 back to the 12, there are 60 arc minutes 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 (which in GIS is simply shortened to ‘minutes’), 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 (or just ‘seconds’ in GIS) measure in at roughly two-tenths 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 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

### 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 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, 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 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 (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 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, 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, England5.

While lines of latitude are measured as angles between the North (or South) pole and the equator, angles which measure lines of longitude start at the Prime Meridian and travel along the Equator marking each degree to the west until the reaching the other half of the Prime Meridian, this creates 180 marks (and moving from the Prime Meridian east makes a maximum of 180 marks east, for a total of 360 degrees, a full circle)

Creating Lines of Longitude from Scratch, Step 1. Draw the Earth as Circle.
First, we are going to draw a new circle to represent the Earth. Since we are looking down at the North Pole from space, we are going to place a single dot right in the middle of the circle to represent the North Pole. Since we know the Prime Meridian passes through Greenwich, we will mark the location with a dot. At this point, the circle can be bisected with a line passing through Greenwich.
Step 2. Fly around the globe along the 51st Parallel to create a 90° angle
When you hop into your plane in Greenwich, England and begin to fly east, if you maintain that bearing, you will be flying along at 51.4° N, or approximately the 51st parallel. If you complete a 90° angle, you will land in Russia, just north of Mongolia . Mark this point, and bisect the circle with a line passing through the point in Russia. Note the other half of the circle passes through a very remote part of Ontario, Canada. Label it 90° W between the North and South Poles on the left side of the circle, and 90°E on the Russian side.

Step 3. Continue your flight to make a 180° angle
If you continue your flight to complete a half circle, you will land somewhere in the middle of the Bearing Sea. You will be on the opposite side of the planet from Greenwich, at 180° No need for W or E, as it is simply labeled 180° from the North to the South pole. Between the North and South poles on the Greenwich side, the line of longitude is labeled 0°. Again, no need for W or E.

Step 4. Add all of the angular measurements between 0° and 180°, then bisect the circle
At this point, we need to mark all of the angular measurements between 0° and 180°, remembering we are still looking down at the North Pole from space. Bisect the circle at each of the remaining 178 locations (since we started with 0° and 180°, and marked 90° in step 2)

Step 5. Add all of the labels using the idea of supplementary angles
In geometry, supplementary angles add up to 180. Lines of longitude are supplementary angles in reference to Greenwich.
If we are looking down on our labeled circle and we choose any point we've marked, then draw a straight line through the circle, we see the angle in the west to be supplementary to angle in the east. In this same fashion, lines of longitude are labeled with one measurement in the west (between the North and South poles) and the supplement of that angle in the east (also, between the North and South poles).

### 2.4.4: Creating a Complete Geographic Grid

Now that we have looked at how the Earth-fixed, Earth-centered coordinate system is established for latitude and longtiude, we need turn the sphere-based system in to an ellipsoid based system to complete the geographic grid.  We learned in Section Three that the Earth is best represented by an ellipsoid of revolution, more specifically, an oblate spheroid and the geoid is best represented by an ellipsoid.  In Section Four, we learned the definiton of a geodetic datum is combining a geoid with a reference ellipsoid, connecting them via control points.  In order to keep on-track with our definitions, we need to transform the sphere used to create the EFEC latitude/longitude coordinate system into not an oblate spheroid, as we are not trying to represent the Earth, but instead into an ellipsoid to best-fit inside the geoid 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 grid-out and label locations on the Earth's surface, we'd like to take that hard work and preserve it over in the reference ellipsoid.  All we are really doing is linearly streching the measurement of the radius into a semi-minor and a different semi-major 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 and a reference ellipsoid, since an ellipsoid is a better representation of a geoid.  A reference ellipsoid is really just an plain old ellipsoid until a geographic grid has been applied to it, and that geographic grid starts out as an Earth-fixed, Earth-centered coordinate system, such as the latitude 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/longitude system between the sphere and the ellipsoid.

### 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).

Run Time: 29 minutes, but really well done and quite interesting. Highly recommended to better understand the geographic grid that is longitude and latitude.

• Note: SatNav = GPS Receiver