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Distance

Distance is often defined as the extent between two objects or positions in space and/or time (e.g., “the distance from Los Angeles to San Francisco”). Often, it is seen to represent “the shortest straight line between two points,” but this covers only one small part of the distance equation, as distance is not always covered in a straight line. Spatially, it can be measured in many units, such as meters, kilometers, feet, yards, nautical miles, furlongs, microns, light-years, degrees of arc, and so on, and temporally from eons to nanoseconds. Offset is a related term referring to displacement from a reference point, named place, or other feature, without the heading (for example, it refers to the “10 miles” portion of “10 miles from Albuquerque”).

In a GIS context, distance can be seen as one of several types: distance in a straight line (Euclidean distance), distance along a path (e.g., by road, river), or weighted distance, which takes terrain effects into account. Joseph Berry recently introduced the concept of proximity to describe three types of distance: “simple proximity,” for a straight-line distance; “effective proximity,” for a not-necessarily-straight line (such as driving distance); and “weighted proximity,” which is related to the characteristics of the mover (including speed).

Euclidean Distance

Euclidean distance is the straight-line distance between two points, usually calculated on a single plane and in n-dimensional space by using Pythagoras's theorem. It is often equated with using a ruler to measure the distance on a map.

It is very important to note that while Euclidian distance can be easily calculated in a GIS from any set of (x, y) coordinates, not all coordinates are rectangular. For example, longitude and latitude coordinates (often referred to as “geographic coordinates”) are spherical, so that “straight-line” distances calculated using Pythagoras's theorem and geographic coordinates do not produce an accurate measure of the distance between two points on the earth's surface. Thus, Euclidean distance is valid only when calculated in rectangular coordinate systems.

Distance along a Path

The distance along a path (such as the distance a car may take between two places or the distance along a river, etc.) is much more complicated to calculate, as it must take into account distances along curves and irregularly shaped linear features. If using a map to make the calculation, the resulting distance will depend on the scale of the map, since the representation of the sinuosity and angularity of the path will become more generalized as the map scale becomes smaller (i.e., the map covers a larger area). In fact, the concept of fractals was developed to formalize this relationship between the length of a measured line and the scale at which it is measured.

The simplest method of calculating a distance along a path is to use an arc length formula. In this case, the line is divided into a series of short straightline segments, and the straight-line distance along each segment is summed. The greater the number of nodes on the line, the more accurate the resultant calculation will be.

Weighted Distance

Weighted distance takes into account both vertical and horizontal terrain features, so the actual distance traveled by a hiker walking up and down a mountain will be greater than the horizontal distance shown on a planimetric map. Weighted distance can also apply to relative distances, such as a place being “a 2-hours'drive” away.

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