Solving the Three-Point Problem in Surveying

The three-point problem, often referred to as the resection problem in plane table surveying, is a fundamental task. It involves determining the location of the plane table station based on bearings taken from that station to three known, visible points. Several methods exist to solve this problem accurately.

Tracing Paper Method Explained

The tracing paper method is a popular graphical technique for solving the three-point problem. Here's how it generally works:

• First, the bearings (or orientations) to the three known stations (A, B, and C) are measured from the unknown station P.
• These bearings are carefully drawn on a sheet of tracing paper, aligning the lines of sight with the corresponding points A, B, and C, which are already plotted on the main map sheet.
• The tracing paper is then rotated over the map sheet until the lines of sight drawn on the tracing paper simultaneously pass through the plotted points A, B, and C.
• The point P on the tracing paper that is directly above the actual station P is the calculated position.

Bessel's Graphical Method

Bessel's method is another graphical approach that provides a precise solution. It relies on geometric principles:

• It involves plotting the triangle formed by the three known points (A, B, C).
• From the unknown station, bearings are taken to A, B, and C.
• Using these bearings and the known coordinates of A, B, and C, a graphical construction is performed.
• This often involves constructing specific lines and arcs on the map.
• The method typically accounts for potential errors by constructing a small "triangle of error" and refining the position.

Lehmann's Method (Stereographic Method)

Lehmann's method is considered highly accurate and is based on the principles of stereographic projection. It's sometimes called the 'two-trace' method:

• It utilizes the property that the observer's station lies on the intersection of two specific circles, each defined by two of the known points and their corresponding bearings.
• The method involves plotting the triangle ABC.
• A tracing paper overlay is used. On this overlay, the bearings to two points (e.g., A and B) are drawn. The station is somewhere on the line of sight to A.
• The tracing paper is then repositioned and the bearings to another pair of points (e.g., B and C) are drawn. The station must lie on the line of sight to B in this orientation.
• By rotating the tracing paper on which the bearings are drawn, the correct orientation is found when the lines of sight align with the corresponding known points.

Conclusion: All Methods are Valid

Since the tracing paper method, Bessel's graphical method, and Lehmann's stereographic method are all established and recognized techniques for solving the three-point problem in surveying, the option encompassing all of them is the correct choice.