Page 291 



HYDROGRAPHY 



3741 



will fall. The intersection of the arcs from each group of three points (A, B, and C) will be the center 

 for the arc through those points. A straight line drawn through the four intersections will define the 

 Line of Centers. An additional check on the accuracy of the construction is to compare the distances 

 between the four centers graphically determined with the computed d values (see 37416(3)). Any 

 discrepancy should be adjusted in relation to the four points established .by the construction work. 



The centers for all arcs required on the sheet are then plotted on the Line of Centers from the 

 computed d values, each measurement being made from the nearest plotted center. It will be found 

 convenient to establish an initial point of some even thousand meters near each of the centers laid 

 down by construction, to facilitate subtracting between adjacent distances between centers. 



From the centers thus determined arcs are drawn on the survey sheet, using the computed c 

 values as radii. A check is obtained if the arcs pass through the 12 computed points on the survey 

 sheet. A further check is obtained if the arc corresponding to a given angle passes through the center 

 of the arc of half that angle. (See equation (3) in 371.) 



In using a long beam compass best results are obtained by applying light pressure at the fixture 

 directly over the pen, the other fixture being lightly held in place at the circle center. Pressure applied 

 at any other part of the beam may cause variable flexure while drawing the arcs. 



For quick identification in the field, it will be found helpful to use colored inks for the arcs. In 

 some cases it may be desirable to use diff'erent colors for the different systems, while in others it may 

 be more helpful to use colors to designate the angular values, such as black for whole degrees, blue for 

 half degrees, and other colors for smaller intervals if used. (See 735 for instructions for inking arcs 

 on smooth sheets.) 



2. By computation. — There may be cases where it is preferable to compute the rectangular co- 

 ordinates of points on the arcs and then draw them in with the aid of standard curves. Such a method 

 will be found useful where the radii of the circles are too long for the beam compass at hand or where 

 not too many circles are required. 



When this method is used, the procedure up to the point of constructing the arcs is the same as 

 heretofore described, except that all the arcs that will be required on the survey sheet must be drawn 

 on the layout sheet, rather than only four. Three points are selected on each arc as before, but the 

 geographic position of only the middle point {B in fig. 63) need be computed, only the azimuths (scaled) 

 of points A and C being used. The procedure is as follows: 



(a) Compute the geographic positions of the intersections of the radii passing through the selected 

 points at each end of an arc (A and C in fig. 63) with the tangent to the arc at the selected middle 

 point B. This is readily done by using the scaled azimuths of the radii and the computed geographic 

 position of the point B (see 37416(5)). Plot the tangent line on the sheet. 



(fa) Considering the radius to the selected middle point on the arc as the F-axis and the tangent 

 line as the A^-axis, the rectangular coordinates of points on the circle passing through the origin of the 

 system are derived as follows: 



D 



In figure 64, 



Then, 



and 

 and 



DB= azimuth line from center of circle 

 to selected middle point, or the 

 radius c of the circular arc BR. 

 BP=th.e normal line. 

 B»S=.T-coordinate of point R. 

 SR = y-coordinate of point R. 

 RP=Sin extension of radius DR. 

 = the angle RDB or PRS. 



RP 



Tan (9 = — 

 c 



BP-SP=BP- 



-c — c cos d. 



y tan d. 



Figure 64. — Construction of the arcs from 

 rectangular coordinates. 



Assigned values of BP are entered in the above formulas and the corresponding values of x and y 

 are solved by logarithms. It is unnecessary to find the angular value of d, but simply the value of the 

 cosine corresponding to the tangent of the angle. 



3. With three-arm protractor. — This method of drawing the arcs is not strictly a part of the "Gradu- 

 ated Perpendicular Method" since all the steps included in that method are not necessary for its use, 

 and as will be seen it is also part of the "Auxiliary Straight-Line Method" (see 3742). 



