Distance and Direction

DISTANCE is defined as a spatial interval. For all practical purposes, distance is the length of the shortest interval which connects two points. Keeping in mind that the earth is a sphere, the shortest line which connects two points follows the curvature of the Earth thus, it is a great circle line. Distances may be taken by either measurement or by mathematical calculations.

The nautical mile, which equals one minute arc of latitude, is the most common unit used for navigational purposes. Several countries adopted the metric system and are using the kilometer as a unit of distance. The statute mile is occasionally used for navigation. The following are the conversion factors, one nautical mile equals to:

• 6076.10 feet
• 2027 yards
• 1.852 kilometers
• 1.151 statute mile

The ratio between a nautical and statute miles is given by:

Statute miles         76
–––––––––––––  =  –––
Nautical miles        66

Measuring distances can be accomplished in several ways. Aeronautical chart are scaled, that means that each inch on a chart represents a certain number of inches on the surface. The most common method is using a navigation plotter which is calibrated for various scales. When a plotter is not available, two other options may be used. First, the distance is measured with a string or a sheet of paper. Then either calculating the distance by multiplying the measured distance by the scaling factor or by counting degrees and minutes of latitude. Each degree of latitude equals 60 Nautical miles and each minute equals 1 Nautical mile. When using the scale method, the result must be converted from inches to nautical miles.

A direction can be obtained by either measurement or by mathematical calculation. An ordinary protractor can be used to measure the direction, but it is rarely used. Several styles of plotters have been designed specifically to accommodate the navigator. Long range segments are more difficult to measure than short ones. Long range navigation involves following a great circle which is not a straight line. This problem is solved by breaking the curved line into numerous straight segments. The following are the mathematical formulas for calculating great circle distance and bearing.

Distance

Where: D = Distance (in Nautical Miles) L1 = Original Latitude L2 = Destination Latitude = Origin Longitude
= Destination Longitude Latitude

Direction

Where: C = Initial Bearing (in degrees) D = Distance L1 = Original Latitude L2 = Destination Latitude

Example:
The distance and initial bearing from New York JFK, N 40° 38' 24"    W 74° 46' 42" and London LHR, N 51° 28' 39"   W 00° 27' 41"  are:

Rhumb Line

Ideally, straight line course segments are sought because they are easy to follow compared to curved lines. Since the lines of longitude are approximately parallel, a straight line would be a line that crosses all the lines of longitude at the same angle. Straight lines make the plotting and tracking simple. A rhumb line is a line on the surface of Earth which cross every meridian (line of longitude) at the same angle. On a sphere, where the meridians are converging at the poles, a rhumb line will form a spiraling curve that eventually ends at either of the Earth's poles. The adjacent illustration demonstrates the spiral created by a rhumb line. The spiral that is created by a rhumb line is a Loxodromic Curve or a Loxodrome. Since a loxodrome is not a great circle, it follows that by tracking a loxodrome a longer distance must be traveled compared to a great circle line.

Great Circle Line

A great circle is a circle on a sphere's surface whose plane is passing exactly through the center of the sphere. An arc on a great circle represents the shortest distance between two points on a sphere. Because a great line follows the curvature of the Earth, it forms a curved line rather than a straight one. The line between New York and London as shown in the adjacent illustration lays on a great circle.

In long range navigation, great circle routes are desired. Since the great circle is not a straight line and therefore difficult to follow, it is divided into a sequence of shorter rhumb lines segments.
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