Bessel’s Graphical Solution of Three-Point Problem in Plane-Table Surveying—Limitation and Alternative Solutions

The orientation of a plane table at a new station is carried out with the help of three well-defined visible points available on the ground along with their corresponding plot on the plan using Bessel’s graphical solution. There are seven regions $(R1–R7)$ in and around the area bounded by the three well-defined ground points and the plane table may be stationed in any one of them for orientation. While carrying out such field work, a completely opposite orientation was obtained for the plane-table station in reference to the ground points and corresponding plotted plan points for some of the regions, hinting at possible limitation of the method. This paper makes an effort to understand and analyze the limitation of Bessel’s graphical solution. The orientations were carried out on an AutoCAD worksheet and cross checked through field works. It was observed that the position of the plane table station and the third point used to draw the ray in reference to the line joining the first two points used for sighting played a critical role in obtaining the correct orientation of the plane table. When the position of the plane-table station and the third point were on the same side of the line joining the first two points in $R2$ , $R3$ , and $R4$ regions, it resulted in a completely opposite orientation of the table. This necessitated modification in Bessel’s method to obtain correct orientations, but the orientations obtained had a very high variation in the magnitude of errors in the distances at a lower scale of 1:500. The field orientations carried out at a higher scale (1:100) yielded a lower magnitude of errors in the distances. Alternatively, the limitation could be overcome by interchanging the third point with any of the first two points used for sighting such that the plane-table station point and the third points were on the opposite side of the line joining the first two points used for sighting. This alternative also yielded correct orientations with much lesser errors in distances. However, when the plane table was positioned within the area bounded by the three well-defined ground points (i.e., $R1$ region) and orientation was carried out as per the method, it yielded a correct orientation even though the plane-table station and the third point were on the same side of the line joining the first two points used for sighting. When the position of the plane-table station and the third points were on the opposite side of the line joining the first two points in $R5$ , $R6$ , and $R7$ regions, Bessel’s method yielded correct orientations.