3.5 Software for Using the UTM Grid with Topographic Maps
4.1.2 Calculation Using Plane Geometry
The two points P1 and P2 share the same local system of easting and northing coordinates
because they lie within the same UTM longitude zone and the same hemisphere. In this case, the horizontal distance and bearing between the two points are calculated using plane analytic geometry.
The horizontal distance (D) between the points is calculated using the two-point distance formula [Eves, 1984]:
D =p(x2− x1)2+ (y2− y1)2 (4.1.1)
where x1 and x2 are the UTM easting coordinates, and y1 and y2 are the UTM northing coor-
dinates, respectively, of points P1 and P2. The distance formula is based on the Pythagorean
theorem. If the easting and northing coordinates are given in meters, then the distance will be in meters.
The bearing (β) from point P1 to point P2 is calculated as follows [Langley, 1998]:
β = arctan x2− x1 y2− y1
!
(4.1.2)
Equation (4.1.2) is based on the two-point slope formula [see Eves, 1984]. Here the angle β is measured clockwise from the positive y (northing) axis, and is measured in units of radians (rad). To obtain the bearing in units of degrees, multiply by the conversion factor (180◦/π rad).
The UTM easting and northing coordinates of the two points must be expressed relative to a common horizontal datum, or the use of equations (4.1.1) and (4.1.2) will give erroneous results.
Equations (4.1.1) and (4.1.2) are convenient to use because of their simplicity. Suppose one has multiple pairs of points for which the horizontal distances and bearings must be calculated. If the number of point pairs is relatively small, then one can easily perform the calculations using a handheld electronic calculator. On the other hand, it’s also possible to write a computer program or to configure a computer spreadsheet to perform the calculations quickly and accurately for thousands of point pairs, if necessary.
Calculating the horizontal distance and bearing using equations (4.1.1) and (4.1.2) is more versatile than measuring the distance and bearing on a paper map, in that it doesn’t require that the two points plot on the same quadrangle map. For additional discussion of horizontal distance and bearing calculations, see Langley [1998].
Table 4.1: Data for Distance and Bearing Calculation Example – 1st of 2
Point Description Easting Northing x (m) y (m)
P1 Spring 458 576 5 294 756
P2 Gaging Station 459 288 5 291 192
Notes:
(1) Horizontal datum: NAD 27
(2) Both points lie in UTM lon. zone 11, in N. hemis.
Example: Horizontal Distance and Bearing between Points in Same Zone and Hemisphere
Problem: Determine the horizontal distance, and bearing, between the unnamed spring near the mouth of Sandy Canyon, Washington and the gaging station at Nine Mile Falls, Washington. Assume that the points have the UTM coordinates listed in Table4.1.
Solution: Note that the UTM coordinates of the two points are expressed relative to the same horizontal datum. Also, both points are known to reside in UTM longitude zone 11, in the northern hemisphere. Therefore, one can use equations (4.1.1) and (4.1.2) to calculate the horizontal distance and bearing, respectively, between the points. Substituting the numerical values tabulated above for the corresponding variables in equation (4.1.1) gives
D = p(9288 − 8576)2+ (1192 − 4756)2 m
= 3634 m
Substituting the numerical values tabulated above for the corresponding variables in equation (4.1.2) gives β = 180 ◦ π rad ! arctan 9288 − 8576 1192 − 4756 ! = 168.7◦
Example: Verification by Direct Map Measurement
Problem: How can one verify the horizontal distance calculated in the previous example?
Solution: In this particular example, both points lie within the coverage area for the Nine Mile Falls Quadrangle, Washington (7.5-minute) topographic map [USGS, 1973b]. Therefore, one can use a paper map to physically measure the map distance between the two points, and then use the map scale to convert the map distance to the equiv- alent horizontal ground distance. Finally, one can compare the results from the two methods to see if they’re consistent.
I obtain the following result for the measured map distance:
Dmap = 151 mm
Converting to the equivalent horizontal ground distance,
D = Dmap u = 151 mm 24 m mm = 3624 m
Here u denotes the map scale. This result (3624 m) differs from that obtained in the previous example (3634 m) by only 10 m. Suppose one can directly measure map distance with a precision of about 1 mm. For a map scale of 1:24 000 the equivalent measurement precision of horizontal ground distance is then about 24 m. The difference one obtains using the two methods (i.e., computation versus direct measurement), 10 m, is well within this precision. Therefore, the distance results from the two methods are consistent.
Table 4.2: Data for Distance and Bearing Calculation Example – 2nd of 2
Point Description Easting Northing Quadrangle Name x (m) y (m) (7.5-minute series)
P1 Summit 448 496 5 298 673
Four Mound Prairie, WA [USGS, 1973a]
P2 Gaging Station 459 288 5 291 192
Nine Mile Falls, WA [USGS, 1973b]
Notes:
(1) Horizontal datum: NAD 27
(2) Both points lie in UTM lon. zone 11, in N. hemis.
Example: Determining Horizontal Distance and Bearing Between Points in Same Zone and Hemisphere
Problem: Determine the horizontal distance, and bearing, from the summit of Eagle Rock (Stevens County, Washington) to the gaging station at Nine Mile Falls, Washington. Assume that the points have the UTM coordinates listed in Table4.2.
Solution: The points lie in areas covered by two different USGS quadrangle maps, but are in the same UTM longitude zone and hemisphere, and their UTM coordinates are ex- pressed relative to the same horizontal datum. Therefore one can use equations (4.1.1) and (4.1.2) to calculate the horizontal distance and bearing, respectively, between the points.
Substituting the numerical values tabulated above for the corresponding variables in equation (4.1.1) gives
D = p(59 288 − 48 496)2+ (1192 − 8673)2m
= 13 130 m
Substituting the numerical values tabulated above for the corresponding variables in equation (4.1.2) gives β = 180 ◦ π rad arctan 59 288 − 48 496 1192 − 8673 = 124.7◦
That is, from the summit, the gaging station is on a bearing of approximately 125◦ east of north.