msuheader2.gif (15254 bytes)Topographic Map
Interpretation

TECHNIQUE NUMBER 4 hdistance.gif (2134 bytes)

HORIZONTAL DISTANCE

X2Y2 - X1Y1

The horizontal distance is also referred to as the "map distance", and is the distance between two points as scaled off of the map. It does not take into account the relief between two points (see Technique Number 5 - Slope Distance). There are two common types of horizontal distance: straight-line distance and distance of travel (road, stream, etc.).

Straight-line Distance. Straight-line distances are measured in several ways. The least accurate is to align a paper across the map between the two points, make tics to indicate their separation, and compare this distance to the bar scales at the bottom of the page. Because you must interpolate on the bars, and because many distances will be longer than the bars, thus be measured in two or more steps, this procedure is not recommended.

Within a map sheet, the best way to measure distance is to do so with an accurate ruler (or "scale"). The distance should be interpolated to the nearest 0.1 mm, then multiplied by the scale factor to define a distance. When this method is used, you must keep in mind the horizontal uncertainty of + 0.02 inches. On a map at a scale of 1:24000,an uncertainty of 0.02 inches on the map indicates an uncertainty of 0.02 x 24,000 / 12 = 40 feet on the ground. It is unnecessary, and in fact misleading and WRONG to give a distance as "1252.7 feet" when you know that each point may be in error by 40 feet or more.  As with elevation uncertainty, error is combined as the square root of the sum of the squared errors.  The distance in such a case should be given as 1252.7 + (402+402)1/2, or  1252.7 + 56.7' or "1250 + 60 feet". Another way to word this warning is to say that the hundred's digit is the last SIGNIFICANT FIGURE (the ten's digit is approximate, and the unit's digit is insignificant).

NOTE:

map_sheet.gif (3215 bytes)If a distance exceeds the length of your scale, you can best measure it by placing two scales next to each other such that one of the points to be measured is on the inside of one scale and the other is on the inside of the other, setting the zero points of the scales on the points to be measured, reading the distances on each scale to a point common to both, adding those distances, and then using the scale factor to compute actual horizontal distance.

long-distance.gif (5868 bytes)When it is necessary to calculate straight-line distances between points on different map sheets, the easiest way is NOT to cut the margins off the maps (or fold them), stick them together, and then measure. The best way is to determine the UTM coordinates of the points and use the Pythagorean Theorem:

L = ((E2 - E1)2 + (N2 - N1)2)1/2,

where L is the distance, E1 and E2 are the Eastings of the points, and N1 and N2 are the Northings. This will only work, of course, within a single gore of the UTM grid.

For example: The distance between the Fowler School SW of Bozeman on the Bozeman 15' quad and the Bozeman benchmark (4810', just east of the post office [P.O.] and to the left of the crossbar of the red "7" in section 7, T2S, R6E) is 90.0 mm, x 62,500 = 5625 m. This answer is not correct, because the uncertainty has not been taken into consideration. The uncertainty is + 0.02 inches on each end of the line, or + 0.028 inches.) 0.028 x 62500 = 1770 inches, which, divided by 39.37 in/m = 44.9 m. So the actual distance between those points is 5625 + 45 m (to 4 significant figures).

NOTE: My downloaded map is about 1% larger than the original.  To check yours, measure the distance across the map between T1S and T2S - it should be 189.0 mm.

EXERCISE: Find the distance from the Bozeman benchmark (above) to the Beck Ranch, in the NE corner of the map.

Distance (in mm) ______ x 62,500 = Distance (in m) _________

Uncertainty (in m) = ______, so the measured distance, to 4 significant figures, is ___________m.

EXERCISE: Find the distance between the intersection of US 191 and US 10 in Bozeman (96445824, from the Location page) and the point in UTM Grid Zone 12 with Grid coordinates 81117022.
  1. The easting of BM 5355 is ________m.
  2. The easting of the other point is _______m.
  3. The northing of BM 5355 is ________m.
  4. The northing of the other point is ________m.
  5. The difference in eastings is _________m.
  6. The difference in northings is _________m.
  7. 5) squared is ___________meters squared.
  8. 6) squared is ___________meters squared.
  9. 7) + 8) = ____________meters squared.
  10. The square root of 9) is ___________m. This is the distance!
  11. What is the uncertainty? ___________m.
  12. What is the distance, to four significant figures? __________ + ____m.

Distance of Travel. With the measurement of distance of travel we come to the first (of many!) measurement technique which carries with it a significant amount of operator judgement. From here on, you must use your own judgement in making measurements. How accurate must you be? The answer is not a simple one, as it will depend on the uses of your data and the time you have available, among other things.

The measurement of distance of travel can be done in several ways, with varying degrees of ease, accuracy, and of course, cost. The simplest way is to use approximation by straight-line segments. Place the corner of a piece of paper at the starting point, and lie the edge of the paper along the first nearly straight segment. Make a light tic with a soft, sharp pencil on the edge of the paper and on the map at the end of that segment. Then rotate the paper so that the tics are still aligned and the edge of the paper lies parallel to the next nearly straight line segment. Make another light tic at the end of that segment, rotate the paper, and continue until you reach the end of the distance to be mapped. You can then treat the edge of the paper as a straight-line distance, comparing it against the bar scales or measuring its length and multiplying by the appropriate scale factor.

Another way to measure a non-linear distance is simply to lay a string along the feature to be measured. A fine piece of twine or heavy thread works best, being both flexible and manageable. It takes a little effort to match the string with a sinuous feature like a river, but it doesn't have to be perfect. If the errors are random, they can be expected to cancel, and to have no effect on the end results. If the errors are non-random, as, for example, when you cut each corner shorter than it actually is, they result in a bias, which hurts the accuracy of the measurement. You should not worry about random error, but strive at all times to eliminate bias. Once the twine has been laid along the line, the ends can be marked, the twine straightened and measured, and the distance computed.

NOTE:

Non-linear distances can also be measured using tools invented for that purpose. The simplest is a small wheel attached through a gearing mechanism to an indicator dial - each rotation or partial rotation of the wheel is indicated on the dial as map distance (in cm or in) or, by conversion scales, as ground distance. These instruments, called "map measurers" or "opisometers", can be purchased in most stationary stores for under $5. Their accuracy may be suspect, so it is important to calibrate them against the bar scales prior to using them. They require some practice at following the lines (remember random error and bias!), but are very rapid once you work with them for a while.

Finally, there are various brands of "digitizing tablets". These gizmos are available as accessories to micro-computers and will feed grid information constantly or on signal into the processor. Available software will compute the distance before your hand leaves the page, at any scale you define. The only problem? Can YOU afford one?

EXERCISE: On the Bozeman MT 15' sheet, find the distance along the former Chicago, Milwaukee, St. paul and Pacific railbed from the north edge of the map to Blackwood (SW corner). Don't forget to estimate the uncertainty of your measurement and use the correct number of significant figures.

______________________m
Copyright 1998 by W. Locke.  Commercial use prohibited.  Educational use permitted under condition of a courtesy notification to Dr. Locke.  Return to the Table of Contents.  Return to Dr. Locke's home page.