Any map you will ever see has a scale. It may be only implicit, as in a graphic artist’s rendering of a summer festival site, or a city’s advertising map, but more often you’ll find explicit scales. An important question for the use and creation of geographic information and maps is: What is the appropriate scale? A scale too small, that shows a large area, will require that small spe- cific things and events be removed, whereas a large scale may lead to impor- tant contextual information being left out. To work well with scale it is criti- cal to familiarize yourself with different ways of representing the relationship between a distance unit of geographic information on a map and the corre- sponding distance unit on the ground.
Scale is shown for geographic information and maps in three ways:
Two-dimensional polar coordinate systems.
• Representative fraction
• Scale bar
• Statement
The three types are equivalents, but have different representations. A representative fraction provides a ratio between the same units of measure on a page and on the ground. A scale bar graphically represents distinct dis- tances at the scale of the geographic information or map. A statement describes the scale in words. The most important thing for representing scale is that the measurement units on the page (or for the geographic infor- mation) and on the ground must be kept the same. For example, the repre- sentative fraction scale 1:24,000 indicates that 1 inch on the map corre- sponds to 24,000 inches on the ground. Divide by 12 (the number of inches in a foot) and you’ll have the basis for the statement of scale: “1 inch equals 2,000 feet.” Using metric units, the calculations are even easier: the represen- tative fraction scale 1:25,000 indicates that 1 cm on the map corresponds to
Global tessellation.
Fromwww.spatial-effects.com. Reprinted by permission of Geoff Dutton.
Representative scale and scale bars from a USGS map.
25,000 cm on the ground. Divide by 100,000 (the number of cm in a km) to determine the statement of scale “1 cm equals 250 m or a quarter km.”
Scale Transformations
GI, whether collected in the field, collected from existing geographic infor- mation, or digitized from existing maps, can be readily transformed to other scales. The scaling of geographic information may be helpful for many rea- sons. Most often, scale transformations allow the association of any arbitrary coordinates from known places—for example, building corners or street intersections—to be associated with coordinates of the same places in other coordinate systems. In this way, locations of things and events drawn on a piece of paper can be transformed into geographic information using a coor- dinate system.
Scale transformations allow for an infinite number of alterations to shapes and changes. They can change all axes by the same factor, each axis by different factors, locally vary the transformation values, or use logarithmic factors. These different types of scale transformations are necessary to sup- port the different type of changes to coordinates required when working with geographic information from different sources.
Several things need to be considered for working with scale transforma- tions. First, it is important to remember to keep using the same units throughout the transformation. Geographic information locations stored in
TABLE 5.2. Representative Scale and Equivalent Ground Distances
Scale Ground Distance
Standard (inches)
1:2,400 200 ft
1:20,000 1,667 ft
1:24,000 2,000 ft
1:62,500 approximately 1 mile 1:63,360 5,280 feet (exactly 1 mile) 1:125,000 approximately 2 miles 1:800,000 approximately 8 miles
Metric (centimeters)
1:1,000 10 m
1:2,500 25 m
1:10,000 100 m
1:25,000 250 m
1:50,000 500 m
1:100,000 1,000 m (1 km)
1:250,000 4,000 m
1:500,000 50,000 m (5 km) 1:1,000,000 100,000 m (10 km) 1:2,000,000 200,000 m (20 km)
metric units should be kept in metric units. If a transformation is made between metric and standard units, be sure that all geographic information was converted using the same constants. The transformations can also alter geographic representations and cartographic representations, leading to geographic information that is not only inaccurate but also incorrect. A com- mon example is scaling small-scale maps to match large-scale maps of the same area. Because the small-scale maps lack accuracy in comparison to a large scale map, differences between the two maps can be the results of changes made during the generalization process—for example, when a road is displaced to fit the railroad track symbol in next to a bend in a river.
A Sample Scale Transformation
The simplest type of sale transformation is an affine transformation. Even an affine transformation makes it possible to scale, rotate, skew, and trans- late geographic information coordinates.
Affine transformations use two equations for the x and y coordinates of two-dimensional geographic information.
x′= Ax + By + C y′= Dx + Ey + F
The values x and y stand for the coordinates of the input geographic information; x′ and y′ stand for the coordinate values of the transformed geographic information. A, B, C, D, E, and F are the six geometric parame- ters for transforming the geographic information coordinate values. Some GIS require the entry of these parameters; others will calculate them for you based on common reference points in the input geographic information and in the output geographic information. A linear transformation simply multi-
Affine scale transformation operations (generalized).
plies the coordinate values by the scale factor to obtain the scaled geo- graphic information.
Summary
This chapter turns to location systems and coordinate systems, both of which often involve a projection, but may be developed without any reference to the earth’s size or shape. Location systems are more likely to be locally devel- oped ways for describing location using a grid of letters and numbers. Coor- dinates without a reference to the earth’s size and shape are a type of loca- tion system. Coordinate systems may have a reference to the earth’s size and shape through a projection, normally described as a datum. Location sys- tems are important because they are very common and can be used to coor- dinate activities. Land subdivision is an important activity involving both location systems and coordinate systems. It establishes the divisions of land used in determining ownership. The U.S. Public Land Survey is possibly the most widely used systematic survey for subdividing land. More common in the rest of the world are unsystematic surveys that use metes-and-bounds approaches to recording the boundaries of land parcels. Because of their importance, law often specifies coordinate systems; usually these are called national grids. In the United States, the State–Plane Coordinate System is the best example. The newer U.S. National Grid is another example of a
Map showing counties of Minnesota before (left) and after (right) scale transfor- mation.