The New York Botanical
© 2003 The New York Botanical Garden
Wayne Cahilly is Manager of the Institutional Mapping Department at The New York Botanical Garden. Please feel free to contact him with any questions at firstname.lastname@example.org
Using an Inverse Traverse Calculation to Locate a Lost Control Point
Here’s a technique for finding a lost control point if you know its coordinates and can locate another nearby control point.
Determining polar coordinates from known rectangular coordinates is a fairly straightforward process. This process requires three steps, deriving the rectangular difference between the two points, determining the polar distance, and determining the azimuth.
To derive the rectangular difference between two points, subtract the Northing (Y) of the known location from the Northing (Y) of the lost location. In this example that returns a product of +175.00.
Secondly, subtract the Easting (X) of the known location from the Easting (X) of the lost location returning a product of +155.00.
Thus, the rectangular difference between the known location and the lost location is a difference of North 175 and East 155 feet. These distances form two know sides of a right triangle and are shown in dashed lines on the diagram.
Step Two: Determine the distance from the known point to the lost point. To accomplish this the following equation is used:
The result is 233.7734 or the polar distance from the known to the lost point.
Step 3: Determining the azimuth of the line from the known to the lost point.
The equation used is:
This results in the following solution: (Y/X) or 175.00/155.00 = 1.1290
The ArcTangent, or inverse tangent of 1.1290 is 48.4682, the decimal azimuth for the line from the known to the lost location. Convert to degrees, minutes, and seconds with a calculator, set up your total station on the known location, level and backsight, then rotate the scope to an azimuth of 48*28'06" and measure a distance of 233.7734 to the lost point.
In calculating the difference in Northing and Easting (Y and X) be sure to keep track of positive and negative numbers. If the Northing of the lost location is a smaller number than the Northing of your known location then your lost point will have a negative Northing. If the lost point has a Easting that is smaller than your known location, then the lost point is said to have a negative Easting.
For example, if the lost point had coordinates of Northing = 30, Easting = 50 the results of calculating the rectangular difference would be X -155.00 Y -175.00.
which is the same distance as we saw previously.
The azimuth results of the above variation on the data would be ArcTangent -175.00/-155.00 or a decimal azimuth of 48.4682 but because our Rectangular difference in the North/South(Y = -175.00) direction was negative, this azimuth is added to 180 resulting in an actual decimal azimuth of 228.4682 which can be converted to 228*28'06.
When the Northing (Y) is a positive number, the azimuth will be north of the east-west line, or between 270 degrees and 90 degrees. If the resulting easting is positive as well, the azimuth will be in the northeast quadrant between 0 and 90 degrees and the calculation returns the actual azimuth. However, if the easting is negative, the azimuth will be in the northwest quadrant and the result of the calculation must be subtracted from 360 degrees.
When the results of the rectangular difference calculations return a negative northing then the resulting azimuth will be below the east-west line, or between 90 degrees and 270 degrees. The azimuth is added to 180 degrees. A product of the azimuth calculation that is positive (45 degrees for example) will be added to 180 resulting in 225 degrees or an azimuth in the southwest quadrant. If the result had been negative (-45 degrees for example) then when this was added to 180 the result would be 135 degrees, or an azimuth in the southeast quadrant.
Updated October 15, 2003
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