# distance

EMPLOYING THE GEOGRAPHIC sense of the term, distance can be defined as the amount of separation between two points or objects on the surface of the Earth. Geographic distance is usually expressed as a linear measurement between locations using one of several commonly accepted metrics (a metric is a standard of measurement using associated units).

Historical distance metrics were often based on the lengths of human body parts. Among these was the cubit, defined as the distance from the tip of the elbow to the end of the middle finger, and the foot (whose corporal association is obvious). These measuring instruments were extraordinarily useful in that they were always available and could not be misplaced. Unfortunately such metrics differed based on the size of person making the measurements, sometimes by several inches or more.

Although attempts at standardizing distance metrics have appeared throughout history, only a few have persisted in wide use to the present day. One of these originated with King Edward I of England when he commissioned the Iron Ulna, or master yardstick, in the early 14th century. One-third of the yard was decreed to be a foot, and one-36th was termed the inch. This system of linear measurement has persisted as two related systems: the British Imperial System and the U.S. Customary System. In the U.S. Customary System the yard is the base unit while a rod equals 5.5 yards, a furlong equals 220 yards, and a mile equals 1,760 yards. The distances related to the depth of bodies of water are often given in fathoms, each fathom being equivalent to 2 yards. The UNITED STATES is the only major country widely employing this system for linear measurement, and efforts have been made to replace it with the more widely accepted metric system.

The metric system became standardized in the late 18th century in France when several proposals were made for defining the standard length of a meter. The successful proposal used the size of the Earth as the ultimate standard of measurement. More specifically, a meter was defined as one ten-millionth of the distance between the North Pole and the equator along a meridian traveling through Paris. Although error associated with making this measurement (due to miscalculations of the shape of the Earth) resulted in the standard meter being slightly shorter than it ought to be given its definition, this distance became the standard nevertheless. A prototype bar of platinum-iridium was constructed as the standard meter and was kept at standard atmospheric pressure to avoid changes in its length. A subsequent definition of the meter was made based on the length of the path traveled by light in a vacuum over a very small fraction of a second. With the meter established as the basis for the metric system of linear measurement other units were computed as decimal ratios of the meter. The metric system is now the most widely used system of measurement and is more accurately termed the International System of Units (SI).

Given that there are several well-accepted units for linear measure, one can use these metrics to determine geographic distances. In many cases, a geographic distance to be measured is small enough that the curvature of the earth does not alter the measurement within the precision capabilities of the measuring instruments being used.

When this is true, one can assume that the surface on which the measurement is being made is a plane, and the calculation of distance between two points can be made using the Pythagorean theorem. (a²+b²=c²) where a and b are the lengths of two sides of a right triangle, and c is the length of the hypotenuse of that triangle. If the two points are given by pairs of coordinates (x1, y1) and (x2, y2), then a² = (x1 – x2)² and b² = (y1 – y2)². Solving for the value of c allows one to calculate the straight line distance between the two points across the plane.

When longer distances across the surface of the earth are being measured, one must compute the spherical distance between the locations. Spherical distance calculations consider that the distances across the surface of a curved object (like the Earth) are longer than the planar distance between points.