GIS For Dummies

Chapter 12

Measuring Distance

In This Chapter

Measuring absolute distance

Calculating relative measurements

Figuring out distance based on function

You can measure anything you put on a map. You use map measurements to figure out how far you are from one place or another, how much land

you own, how much ore you might expect to get from a new mine, and many other measurements. You can measure heights, widths, depths, lengths, areas, and volumes. You can also compare measurements as ratios — for example, ratios of length to width, perimeter to area, or height to distance.

Measuring geographic features is one of a GIS’s strengths, and it does the measuring very quickly! Because you can measure many features quickly, you can use those measurements to help you make decisions about what route to take (by comparing distances between features), whether one place is better than another for mining (by calculating the volume of a specific feature), or where you’re likely to find the best ski slopes (by evaluating a feature’s height against its length).

In this chapter, I show how GIS gives you the power to measure all sorts of features. First, I explain the measurements that GIS software generates by default (such as grid cell size), and then I describe how the GIS uses these numbers to calculate other measurements (such as distance between objects). I also tell you about the difference between relative and absolute measurements, and when you might want to use each. I explain how you can use measurement tools to describe how features are distributed relative to each other and to other surrounding features. And finally, I describe how to apply distance to make entirely new polygons based on the distances you measure.

Taking Absolute Measurement

When you enter data into a GIS, the software keeps records of those points, lines, and polygons. More important, it keeps a record of the X and Y locations

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of your vector or raster objects. It stores the size of the grid cells and the lengths of the lines. Having these records saves a lot of time when you calculate distances.

The measurement of absolute distances, whether between objects or along lines, is computed differently for raster and vector data structures. The following sections describe these measurement methods and how the computer implements them for both data types.

Finding the shortest straight-line path

Finding the shortest straight-line path between any two points is the simplest method to measure distance. Often called “as the crow flies” distance, this Euclidean method makes two assumptions:

1.No obstructions block the path between the two points.

2.You don’t need to travel along established paths and roads.

The following sections describe how the GIS software enables you to find the shortest straight-line path between two points that exist under a variety of circumstances.

Finding the shortest distance on a flat surface

If you’re working on a flat surface, the shortest distance between any two points (a straight line) is based on a well-known mathematical equation

(c2 = a2 + b2), the Pythagorean Theorem (named after the ancient Greek mathematician Pythagoras). The Pythagorean Theorem is based on the use of a geometric form called a right triangle — any triangle that has one corner that makes a right-angle turn (90 degrees). The longest side of this triangle, the one opposite the right-angle bend, is called the hypotenuse. Figure 12-1 shows an example of a right triangle.

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Remembering sines and cosines

Each of the two points (that you want to measure the distance between) is defined by an X coordinate and a Y coordinate along number lines. To locate each point, you have to move horizontally and vertically along the number lines from the origin point (where the right angle forms) to the values that define your points. These movements along the number lines define the straight legs of your right triangle. You can think of the distance between those two points as the hypotenuse of your right triangle.

When measuring real Earth distances in a GIS, the software calculates the hypotenuse distance using the X and Y values of two points in a modification of the Pythagorean Theorem called the distance formula. You’ll encounter this distance formula many times in your GIS work.

When you need a quick online refresher on the distance formula, visit www. purplemath.com/modules/distform.htm.

Finding the shortest distance on the spherical Earth

The GIS can also calculate distance between a set of latitude and longitude coordinates on a spherical Earth. You may need to find the shortest distance between two points on the globe. This calculation is called a great circle distance and uses a geographic grid. The calculation of latitude and longitude distances has some very strong similarities to calculating distance by using

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flat (planar) coordinates and the distance equation. (In Chapter 2, I describe a bit about converting a spherical Earth to its representation on a flat piece of paper.) This process, called map projection, involves mathematics because it converts spherical coordinates to planar coordinates.

Conversely, no space on the Earth is totally flat, so calculating the distance between any two points requires some determination of the spherical coordinates. Your GIS is already set up to perform these calculations for you. Using trigonometric formulas and the X and Y (planar) coordinates of geographic features, your software can easily change back and forth from spherical geometry to planar geometry. If you really want a peek at the math behind this conversion, check out the sidebar “Remembering sines and cosines.”

Measuring distances in grid cells

Measuring distances in raster systems (using grid cells) is a bit different than measuring distance in vector systems, both conceptually and computationally. Many raster GIS packages allow you to calculate measurements in both flat coordinate systems and the geographic grid (latitude and longitude coordinates). The alternative to measuring by coordinates is counting up grid cells.

If all the grid cells in the path you’re measuring are edge-to-edge (orthogonal), you can easily figure the distance the path covers. Just count up the grid cells and multiply that number by the width of the grid cell. For example, if you count 150 grid cells from one point to another and each grid cell is 20 meters across, the total distance is 20 × 150, or 300 meters.

This very simple approach gives you pretty accurate results, but only if all the grid cells are orthogonal or all distances are exactly lined up by the cardinal directions. In real distances, you seldom get such an exact configuration. If you’re working with a raster system in which you want to measure distances that are diagonal, you need to measure grid cells that are diagonal to one another, at least for some portion of the distance (as shown in Figure 12-2).

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To calculate Manhattan distance, you use a modification of the distance equation, but conceptually, the calculation is really much simpler than that. Because each of the edges for each block is a straight line segment, you need to know only where the X and Y coordinates are for the ends of each line segment. Most GIS software automatically stores that information when you first input the data. You can easily calculate each line segment’s length using the distance equation included in the GIS software. To determine the total Manhattan distance, you simply let the software add together each individual line segment’s length.

Calculating distance along networks

You encounter many types of routes in GIS work — including footpaths, animal tracks, rail lines, intercity roads, and street patterns — that look nothing like Manhattan. Each of these networks typically indicates an imposed structure on distance traveled. To calculate distance along these networks, the GIS measures vector segment lengths or grid-cell lengths for vector and raster, respectively.

Vector: The GIS first retrieves the X and Y coordinates of each line segment’s endpoints (stored at input) and then uses the distance equation to determine each line segment’s length. Finally, the GIS adds together all the individual line segment lengths.

Raster: The GIS finds each orthogonal grid cell’s width and then adds those to get the total length. When grid cells are diagonal to each other, the lengths are measured as the width multiplied by 1.414 (the diagonal distance of a grid cell), as illustrated in Figure 12-2.

You can measure linear features in raster, but you can both represent and measure features more accurately in vector.

In Manhattan distance, you don’t gain anything by choosing a different route. With most irregular networks, however, you can choose many routes from point A to point B, and these routes often have wildly different distances. Your software can measure all these possible routes and find the one that has the shortest distance by using the shortest path algorithm (an algorithm is like a recipe to solve a problem). Chapter 16 covers the shortest path algorithm in more detail.

Working with buffers

Nearly all GIS software has a set of special distance functions designed to selectively calculate distances around existing geographic features. These operations are called buffering (no, not like the aspirin), and the result is

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Bidirectional buffer: Most buffers measure distance outward. Setback buffers can measure distance inward. One advantage of doing a buffer around an area feature is that you can have a buffer going both directions at once. Most GIS software allows you to select this option so if you want to measure a buffer distance both inward and outward from the outside perimeter of an area feature, you can.

Point

Figure 12-4:

You can create buffers around points.