Plotting Horizontal Control
Computations for horizontal control become greatly clarified when you can see a plot (that is, a graphic representation to scale) of the traverse on which you are working. A glance at the plot of a closed traverse, for instance, tells you whether you should add or subtract the departure or the latitude of a traverse line in computing the departure or latitude of an adjacent line or in computing the coordinates of a station. For linear distances that are given in feet and decimals of feet, you use the correct scale on an engineer’s scale for laying off linear distances on a plot. For plotting traverses, there are three common methods: by protractor and scale, by tangents, and by coordinates.
PLOTTING ANGLES BY PROTRACTORAND SCALE.— For the traverse on which you have been working, the adjusted bearings and distances are as follows:
Traverse Line Bearing Distance
ABN26°09'E 285.14 feet
BCS75°26'E 610.26 feet
CDS15°31'W 720.28 feet
DAN41°31'W 789.96 feet
Figure 7-35.—Traverse plotted by protractor-and-scale method.
Figure 7-35 shows the method of how to plot thistraverse with a scale and protractor. First select a scale that will make the plot fit on the size of your paper. Select a convenient point on the paper for stations A and draw a light line NS, representing the meridian through the station.
ABbears N26°9'E. Set the protractor with the central hole on A and the 00 line at NS, and lay off 26°09'E. You will have to estimate the minutes as best you can. Draw a line in this direction from A, and on the line measure off the length of AB (285. 14 feet) to scale. This locates station B on the plot. Draw a light line NS through B parallel to NS through A, and representing the meridian through station B. BC bears S75°26'E. Set the protractor with the central hole on B and the 00 line on NS, lay off 75°26' from the S leg of NS to the E, and measure off the length of BC (610.26 feet) to scale to
Figure 7-36.—Plotting traverse lines by parallel method froma single meridian.
locate C. Proceed to locate D in the same manner. Thisprocedure leaves you with a number of light meridian lines through stations on the plot. A procedure that eliminates these lines is shown in figure 7-36. Here you draw a single meridian NS, well clear of the area of the paper on which you intend to plot the traverse. From a convenient point O, you layoff each of the traverse lines in the proper direction. You can then transfer these directions to the plot by one of the methods for drawing parallel lines.
PLOTTING ANGLES FROM TANGENTS.—Sometimes instead of having bearing angles to plot from, you might want to plot the traverse from deflection angles turned in the field. The deflection angles for the traverse you are working on are as follows:
ABto BC 78°25'R
BCto CD 90°57'R
CDto DA 122°58'R
DAto AB 67°40'R
Figure 7-37.—Plotting by tangent-offset method from deflectionangles larger than 45°.
You could plot from these angles by protractor. Lay off one of the traverse lines to scale; then lay off the direction of the next line by turning the deflection angle to the right of the firt line extension by protractor and soon.
However, the fact that you can read a protractordirectly to only the nearest 30 minutes presents a problem. When you plot from bearings, your error in estimation of minutes applies only to a single traverse line. When you plot from deflection angles, however, the error carries on cumulatively all the way around. For this reason, you should use the tangent method when you are plotting deflection angles.
Figure 7-37 shows the procedure of plotting deflection angles larger than 45°. The direction of the starting line is called the meridian, following a conventional procedure, that the north side of the figure being plotted is situated toward the top of the drawing paper. In doing this, you might have to plot the appropriate traverse to a small scale using a protractor and an engineer’s scale, just to have a general idea of where to start. Make sure that the figure will fit proportionately on the paper of the desired size. Starting at point A, you draw the meridian line lightly. Then you lay off AO, 10 inches (or any convenient round-figure length) along the referenced meridian. Now, from O you draw a line OP perpendicular to AO. Draw a light lineOP as shown. In a trigonometric table, look for the natural tangent of the bearing angle 26°90', which equals to 0.49098. Find the distance OP as follows:
OP= AO tan 26°09' = 4.9098, or 4.91 inches.
You know thatOP is equal to 4.91 inches. Draw AP extended; then you lay off the distance AB to scale along AP. Remember that unless you are plotting a closed traverse, it is always advantageous to start your offsets from the referenced meridian. The reason is that, after you have plotted three or more lines, you can always use this referenced meridian line for checking the bearing of the last line plotted to find any discrepancy. The bearing angle, used as a check should also be found by the same method (tangent-offset method).
Now to plot the directions of lines from deflection angles larger than 45°, you have to use the complementary angle (90° minus the deflection angle). To plot the direction of line BC in figure 7-37, draw a light perpendicular line towards the right from point B.
Measure off again a convenient round-figured length,say 10 inches, representing BOBOThe complement of the deflection angle of BC is 90° – 78°25' = 11°35'. The natural tangent value of 11°35’ is equal to 0.20497. From O1 draw OIP1 perpendicular to BOI Solving for OIPl, you will have
O1P1 = BO2 tan 11°35' = 2.0497, or 2.05 inches.
Now lay off the distance O1P2 Draw a line from B through P1 extended; lay off the distance BC to scale along this line. The remaining sides, CD and DA, are plotted the same way. Make sure that the angles used for your computations are the correct ones. A rough sketch of your next line will always help to avoid major mistakes.
When the deflection angle is less than 45°, the procedure of plotting by tangent is as shown in figure 7-38. Here you measure off a convenient round-figure length (say 500.00 feet) on the extension of the initial traverse line to locate point O, and from O, draw OP perpendicular to AO. The angle between BO and BC is,
Figure7-38.—Plotting by tangent-offset method from deflectionangle smaller than 45°.
Figure 7-39.—Plotting by coordnates.
in this case, the deflection angle. Assume that23°21'. The formula for the length of OP is this is OP = BO tan 23°21' = 500 x 0.43170= 215.85 feet.