and marked point when rays from A and B will givea strong intersection (angle ADB is greater than 300).First set up and level the plane table at point D(first setup, fig. 9-4). Using plotted points a and b,draw resection rays from A and B. These rays intersectat d which is the tentative position of D. Draw a rayfrom d’ toward C. Plot c’ on this line at the estimateddistance from D to C.Next, set up the plane table at C (second setup, fig.9-4) and orient by backlighting on D. Sight on A anddraw a ray through c’ intersecting line ad’ at a’. In alike manner, sight on B to establish b’. You now havea quadrilateral a’b’d’c’ that is similar to ABDC. Since,in these similar quadrilaterals, line a’b’ should alwaysbe parallel to line AB, the error in orientation isindicated by the angle between ab and a’b’.To correct the orientation, place the alidade online a’b’ and sight on a distinctive distant point. Thenmove the alidade to line ab and rotate the table to sighton the same distant point. The plane table is noworiented, and resection lines from A and B through aand b plot the position of point C.THREE-POINT METHOD.— The three-pointmethod involves orienting the plane table and plottinga station when three known plotted stations can beseen but not conveniently occupied.Set up the plane table at the unknown point P(fig. 9-5) and approximately orient the table by eye orcompass. Draw rays to the known points A, B, and C.The point ab denotes the intersection of the ray to Awith the ray to B. Points bc and ac are similar in theirnotation. If the plane table is oriented properly, theBFigure 9-5.—Three-point method of resection.three rays will intersect at a single point. Usually,however, the first orientation is not accurate, and therays intersect at three points (ab, bc, and ac) forminga triangle, known as the triangle of error.From the geometry involved, the location of thedesired point, P, must fulfill the following threeconditions with respect to the triangle:1. It will fall to the same side of all three rays; thatis, either to the right or to the left of all three rays.2. It will be proportionately as far from each ray asthe distance from the triangle to the respective plottedpoint.3. It will be inside the triangle of error if thetriangle of error is inside of the main plotted triangleand outside the triangle of error if it is outside the maintriangle.In figure 9-5, notice that the triangle of error isoutside the main triangle, and almost twice as far fromB as from A, and about equally as far from C as fromB. The desired point, P, must be about equidistantfrom the rays to B, and to C, and about one half as farfrom the ray to A, and the three measurements must bemade to the same side of the respective rays. As drawn,only one location will fulfill all these conditions andthat is near P’. This is assumed as the desired location.The plane table is reoriented using P’ and back-lighting on one of the farther points (B). The new rays(a’,b’,and c’)are drawn. Another (smaller) triangle oferror results. This means that the selected position, P’,was not quite far enough. Another point, P, is selectedusing the above conditions, the table is reoriented, andthe new rays are drawn. If the tri- angle had becomelarger, a mistake was made and the selected point wason the wrong side of one of the rays. The directionsshould be rechecked and the point reselected in theproper direction.The new point, P, shows no triangle of error whenthe rays are drawn. It can be assumed to be the desiredlocation of the point over which the plane table is set.In addition, the orientation is correct. Using a fourthknown and plotted point as a check, a ray drawn fromthat point should also pass through P. If not, an errorhas been made and the process must be repeated.Normally the second or third try should bring thetriangle of error down to a point. If, after the third try,the triangle has not decreased to a point, you shoulddraw a circular arc through one set of intersections (ab,a’b’)and another arc through either of the other sets(bc, b’c’,or ac, a’c’). The intersections of the two arcs9-4