Chapter 12: Managing Scale

Exercise 12a--establishing view scale

We will idealize the world's shape and assume it is exactly described by some spheroid.  Consider two points P1 and P2 on this spheroid.  The distance between P1 and P2 is (by definition) the length of the shortest path from P1 to P2 that lies entirely on the surface of the spheroid.  (This shortest path is a geodesic containing P1 and P2.)

This picture shows P1 and P2 at the centers of crosses in white circles.  P1 is on Long Island, New York, and P2 is near Rome, Italy.  The thick blue line shows the shortest path between P1 and P2; it is a portion of the great circle containing P1 and P2.

The spheroid shown is the ArcView "sphere."  

On a map there will be two points M1 and M2 corresponding to P1 and P2, respectively.

The thick blue line still shows the geodesic.

This map uses a stereographic projection centered at 74 degrees west, 41 degrees north--almost exactly at point P1.  Therefore the thick blue line is very nearly straight on the map.  Its apparent upwards bowing is an optical illusion caused by the curved graticule of latitude and longitude lines used a map reference.

The scale of the map, relative to points P1 and P2, is the map distance from M1 to M2 divided by the geodesic distance from P1 to P2.

Scale really does depend on which two points are used for its computation.   The Mercator projection provides an extreme example.

The vertical dark blue line shows a geodesic extending due north from the Long Island point (P1).  It has the same length as the geodesic between the points--which means its other end extends beyond the north pole.  The Mercator projection places the north pole infinitely high up on the map, so the map distance along this geodesic is effectively infinite.

The problem apparently is that projections distort distances differently at different locations.  The stretching of the regular latitude graticule in this image is testimony to the Mercator's huge distortion at northern latitudes.

Consider some of the implications of this definition of scale:

Scale is a ratio of distances, and therefore is unitless.
However, correctly computing the scale often requires a conversion from the units of measurement on the earth (kilometers or nautical miles, for example) to the units of measurement on the map (inches or centimeters, typically).
Scales are almost always much less than 1.0, because usually the map is much smaller than the area being mapped.  (This is not always the case: consider a map of an integrated circuit, for example.)
Of two maps that are otherwise the same size, the one with the larger scale shows the smaller portion of the earth.
Scale may (and can) vary across a map.  Even locally--that is, even when you consider only pairs of points M1 and M2 very close to each other on a map--scale may vary considerably.  The map can distort distances more in some directions than in others.  For example, the Mercator projection above has a scale of about 1:173,000,000 in the east-west direction at each point, but a scale of around 1:300,000,000 at those points in the north-south direction.
Because scale can vary, map scales are usually reported in terms of the scale in one direction (often left to right) at a single point on a map.  It often happens that the scale (in a fixed direction) is constant along a special line, such as a meridian (north-south line on the spheroid) or line of latitude, in which case such as line is designated the "reference" line for the map projection.

Things to watch out for

The computer must have enough information to measure distances on the earth and on the map.  In ArcView this means you must specify both the "map units" and the "distance units" in the View|Properties dialog.  Until you do this, you will see no scale shown in the scale box (in the upper right portion of the tool bar).
ArcView is confused about scale.  It frequently uses the reciprocal of the scale in its dialogs instead of the scale itself.
To estimate scale on a video screen, the computer needs to know the size of the screen.  This will depend on the monitor.  Think about the difference between output to a 17" monitor and output on a video projector (often ten feet across or more).  Usually the computer has no information whatsoever about this device.  ArcView makes an intelligent guess (something like one pixel = 1/96 inches) to estimate video scale.

Exercise 12b--managing themes by scale

Things to watch out for

The the "1 : " text in the Theme Properties dialog reminds you that the numbers you type are the reciprocals of the scales.  However, "Minimum Scale" does not mean what it says!  It refers to the minimum reciprocal.  Likewise, "Maximum Scale" really refers to the maximum reciprocal.  Note how the GTKAV text is careful to refer to these reciprocals as "the value in the scale box," not as the scale.  The text and the software clearly distinguish scale from reciprocal, but the interface is not at all clear.
The View's table of contents does not indicate which themes have scale thresholds set.  This can be confusing, because themes may fail to appear for many reasons: their features may be outside the view, too small to see, internally corrupted, drawn with invisible colors, obscured by another theme, or the same color as the background.  You simply have to remember to inspect the scale threshold settings whenever you expect a theme to appear but it does not.
You or another ArcView operator using your project may become confused by themes simply disappearing or reappearing as you zoom in and out.  Therefore use scale thresholds sparingly and judiciously, reserving them for situations where you really need to include themes shown at widely varying scales within one view.

Laboratory Exercises

  1. Write down what the following tools do.  Use each until you are comfortable with it.

  1. The text explains that the and buttons zoom in and zoom out, respectively.  They do so by multiplying the scale by a fixed amount.  What is that amount?  Based on your answer, approximately how many times would you have to push the zoom in button to change from a scale of 1:100,000,000 to a scale of 1:100?  (Figure this out and then check your answer, if you like, by actually doing it.)
  2. How many times in a row can you use Zoom to Previous Extent ?
  3. Press and hold the right mouse button when the cursor is in a view.  What happens?  Find four more ways to change the view scale.
  4. When you change view scale, the view's extent changes and so do the sizes of many of its features.  But not everything responds to changes of scale.  Using the view in GTKAV exercise 12a, determine which of the following do not change when the scale is changed:
Thickness of polygon outlines
Thickness of lines (for this you may need to open a different exercise, such as 12b, that contains a polyline theme)
Sizes of point symbols
Spacing of hatched fills (use the Legend Editor with one of the polygon themes to specify a fill pattern, then watch how it varies with scale)
Spacing of line patterns (again, use the Legend Editor to experiment)
Font sizes of text labels
  1. You can change the default scaling behavior for some of the items in the preceding question.  Find out how.  Hint: explore the "Advanced" buttons in the Legend Editor.
  2. The GTKAV exercise notes that "when the [Maximum] Scale box contains a value smaller than 150,000, the Streets theme draws."  What happens when the value exactly equals 150,000?  Now answer the same question about the Minimum Scale box.  (Understanding these details is important when you want one of two themes to draw no matter what the scale, for example.)
  3. Using the world and U.S. data that come installed with ArcView (usually in C:/ESRI/Esridata), create a view showing data from the world level down to the level of the county in which you live.  Include themes of world features, countries, states, and counties.  Set theme scale thresholds appropriately so that as you zoom into your home, only appropriate themes appear.  How could you use this technique to make it appear that your computer has detailed information for every place on the globe?
  4. Obtain shapefiles of the U. S. Census' TIGER data for your county.  Select some features of interest, such as detailed roads, and add them to the view created in the preceding exercise.
  5. (This is a follow-up to the "thought question" at the end of exercise 8.)  How much disk space, approximately, would you need in order to create a project showing similar data for every place in the United States?  (Assume you were to use all the available TIGER data for every county.)  If you were to create such a project, would there be any performance penalties (such as long data search times) compared to the performance you are seeing for your county alone?  Why or why not?