Page 681 



THE SMOOTH SHEET 



7371 



Whenever a smooth sheet is prepared m advance of the position computations, two 

 distances not less than one-half meter in length should be plotted on the sheet at right 

 angles to each other, along the bottom and along the right-hand edge. The ends of these 

 distances should be marked with small inked circles and the distances at the scale of the 

 survey indicated. This will permit checking the distortion at any time and simplify 

 the future application of a projection. 



There are two methods of reconstructing a pro- ,^^ 



jection on a survey sheet. One is a rigid method 

 applicable to small-scale surveys and the other is a 

 graphic method applicable to large-scale surveys. 

 In both methods the essential problem is the deter- 

 mination of the cardinal lines of the projection, 

 namely, the central meridian and the central con- 

 struction line. 



,'''d 



AA' 



f / lExaRgerated) 

 Cofistfuct'on / f 



AC 



Figure I50.--Construction of a polyconic pro- 

 jection on a completed survey sheet— for small- 

 scale surveys. 



7371. On Small-Scale Surveys 



For surveys on scales smaller than 1:10,000 the 

 following method is used: 



(1) Select three triangulation stations, A, B, and C (fig. 

 150) so situated with respect to the central meridian that the 

 distance de will cover more than half the latitude extent of 

 the sheet. Two of the stations should be selected, if possible, 

 in about the same latitude (see (6) below). From the scaled 

 lengths of BA and CA and the corresponding computed 

 lengths (make inverse computations for these if they are not in the triangulation data), determine 

 the distortion factor along each line (see 7361). 



(2) Select the central meridian (as near to the middle of the sheet as possible) and compute the 

 distances Bd and Ce and the latitudes of the points d and e. 



This is accomplished on Form 27, Position Computation, Third-Order Triangulation. Since both the distance and latitude are 

 involved in the position computation formulas, a trial-and-error method is used. A value is assumed for the required latitude (<t>') 

 and from the known longitudes (X) and (X') of station C and the central meridian, respectively, and the known azimuth (a) of the 

 line Ce (obtained either directly from the triangulation data or from the inverse computation), the distance (s) is computed from the 

 longitude formula. This distance is then used in the latitude formula and the latitude increment or decrement (A^) is obtained. 

 This computed A0 may not agree with that derived from the assumed latitude i<t>'), but by repeated assumptions a value for A0 will 

 be obtained that is in absolute agreement with the assumed latitude. 



To obtain a close first approximation, a sketch should be drawn at about one-fifth the scale of the sheet, using a rectangular grid 

 and using as units of measurement the values of a minute of latitude and longitude at the center of the sheet. From this sketch the 

 latitude where a given line crosses the central meridian can be scaled and this value used as the first assumed value in the compu- 

 tation. 



(3) Plot points d and e along lines BA and CA correcting each computed distance for the distor- 

 tion determined along the respective lines. Through d and e draw a straight line for the full length of 

 the sheet. This is the central jneridian of the projection. 



(4) Scale the distance de on the sheet and from the computed latitudes obtain from the polyconic 

 projection tables the true distance. From these two values the distortion in a north-south direction is 

 determined. 



(5) On the central meridian lay off the tabular distance (corrected for distortion) from point 

 d or e to the central parallel and at this point construct a perpendicular to the central meridian extend- 

 ing the full east-west length of the sheet. This is t\ie-central construction line. 



(6) From here the problem is the ordinary one of constructing a polyconic projection as given in 

 7324. The distortion factor to be applied to east-west measurements may sometimes be obtained 

 from the disposition of the selected triangulation stations (see (1) above). 



The projection should be checked against every triangulation station on the sheet. Small 

 differences in latitude or longitude may be due to unequal distortion of the sheet. In such case the 

 projection should be made consistent with the triangulation even though this results in a slightly 

 skewed projection. 



