6826 HYDROGRAPHIC MANUAL PaGE 644 



form of plotting. This method is not practicable in R.A.R. surveys because the three series of closely 

 spaced concentric circles would obscure soundings, positions, and other data on the sheet. 



B. GRAPHIC-APPROXIMATE METHOD 



In this method the assumption is made that the hyperbolic lines of position in the vicinity of P 

 are straight lines within graphic limits, which is usually the case where P is not close to any of the 

 R.A.R. stations. There are three variations of this method, the first two of which are modifications 

 of (6) above, and which can be used on an R.A.R. sheet with conventional distance circles spaced about 

 4 inches apart. 



(c) Select that R.A.R. distance circle from station A on the base sheet, which is as close to 

 position P as can be estimated, and call its radius x' . Prepare a transparent overlay as described 

 in (6) above and place it on the base sheet so that its center is on the selected distance circle from 

 A and at the same time the p circle is tangent to the distance circle of x' radius from B. Then the 

 center of the transparency is one point on the hyperbolic line of position referred to A and B. Now 

 unless this happens to be the desired point P, the (p + q) circle will miss tangency with the C dis- 

 tance circle of %' radius by a distance which mist be accurately scaled and given its correct sign, 

 depending on whether the (p + q) circle falls toward or away from C with reference to the C distance 

 circle of x' radius. Mark the center of the transparency on the base sheet P' . Now select another 

 A distance circle with radius x" on the base sheet, which will make the (p + q) circle fall in the oppo- 

 site direction from C, and again place the transparency so that its center is on the second A distance 

 circle and its p circle is tangent to the B distance circle of x" radius. This center is another point 

 on the hyperbolic line of position referred to A and B. Mark this center P" and again accurately 

 scale the distance the (p + q) circle misses tangency with the C distance circle of x" radius. The 

 two scaled distances must be opposite in sign, but P' and P" should be as close to each other as 

 practicable. Then connect P' and P" with a straight line which is a chord of the hyperbolic line of 

 position and find point P on it by the proportion PP' : PP" : : first scaled distance : second scaled 

 distance. 



{d) As in (c) above, plot two points, P' and P", and connect them with a straight line which is 

 a chord of the hyperbolic line of position with reference to stations A and B. Then with another 

 transparent oveilay on which a circle with radius q is scribed, similarly find points Q' and Q" with 

 reference to stations B and C, and connect Q' and Q" with a straight line which is a chord of the other 

 hyperbolic line of position with reference to stations B and C The intersection of the two chords 

 is the desired point P if the conditions have been selected so they intersect; if not, other points must 

 be plotted until two pairs of points have been found which give intersecting lines, with each pair of 

 points as close together as practicable. 



(e) This method may be found especially useful on the boat sheet. Knowing the approximate 

 position of P, set off on a beam compass a distance x' , slightly less than PA, and with A as a center, 

 scribe a small arc through P'; then with a distance {x' -{- p) set on the beam compass and with B 

 as a center, scribe a second arc intersecting the first arc at P' — this is one point on the hyperbola 

 with reference to stations A and B. Repeat the above operation with another distance x", slightly 

 longer than PA. The point P" thus found is a second point on the hyperbola, and the straight line 

 joining points P' and P" is a chord of the hyperbola. Repeat the operation with reference to stations 

 B and C and obtain a chord of the second hyperbola; the intersection of the two chords is the desired 

 point P. 



C. MECHANOGRAPHIC METHOD 



In this method a special three-arm device is used. Three metal arms are pivoted at a center, 

 with one edge of each arm serving as a graduated scale from the pivot. The graduations may be on 

 an arbitrary scale, but so marked that equal distances from the pivot can be easily identified on the 

 three arms. Each arm is broken near the pivot and a supplemental adjustable metal strip attached, 

 in such a manner that each arm can be extended by an accurate amount. These supplemental metal 

 strips must be graduated in terms of the base sheet upon which the instrument is to be used. 



In plotting a position the scale of the supplemental metal strip on arm A is set at zero, that on 

 arm B at the difference of distance p, and that on arm C Sit the difference of distance (p + q). It is 

 apparent that any graduation x on the three arms is at x distance from the pivot on the A arm, at 

 (x+p) distance on the B arm, and at (x+p + q) distance on the C arm. The device is then manipu 

 lated until the readings on the arms at stations A, B, and C are equal; the center of the device is then 



