Three Point Problem Solving Methods

The three-point problem in surveying is a fundamental task that involves determining the exact position of an unknown instrument station by observing three well-defined points whose coordinates or positions are already known. This problem is particularly crucial in plane table surveying for orienting the table and locating the station accurately. Several methods have been developed to solve this problem, each with its own advantages and applications.

Tracing Paper Method for Three Point Problem

The tracing paper method is a popular graphical technique widely used for solving the three-point problem, especially in plane table surveying. It is known for its simplicity and quick application in the field.

• A piece of tracing paper is placed over the plane table sheet.
• From an arbitrary point chosen on the tracing paper, rays are drawn towards the three known ground points (let's call them A, B, and C) using an alidade.
• The tracing paper is then carefully moved and rotated over the main plane table sheet until the three rays drawn on the tracing paper simultaneously pass through the corresponding plotted positions of points A, B, and C on the main sheet.
• The arbitrary point on the tracing paper from which the rays were drawn now coincides with the exact position of the instrument station on the main sheet. This point is then pricked through to mark the station on the survey map.

Bessel's Method for Three Point Problem

Bessel's method is another effective graphical solution for the three-point problem. It is based on geometrical construction principles involving circles.

• In this method, two circles are constructed. Each circle is drawn such that it passes through the instrument station and two of the three known points.
• For instance, one circle might pass through the instrument station and points A and B, while another circle passes through the instrument station and points B and C.
• The intersection of these two constructed circles then precisely yields the location of the instrument station.
• This method relies on specific geometric properties to accurately determine the unknown station's position without direct measurement of the station's coordinates.

Lehmann's Method for Three Point Problem

Lehmann's method offers both a graphical and an analytical approach to solving the three-point problem. It is highly regarded for its precision and versatility in various surveying conditions.

• Graphical Approach: Graphically, Lehmann's method involves an iterative process. The surveyor makes trial orientations of the plane table and draws rays to the known points. If these rays do not intersect at a single point, they form a small triangle, known as the "triangle of error". Through systematic adjustments and observations of this triangle, the table is oriented, and the station is located. Rules are applied to move the plane table to reduce the triangle of error until it converges to a point, indicating the true station position.
• Analytical Approach: Analytically, Lehmann's method uses trigonometric calculations. By measuring angles from the unknown station to the three known points and using their known coordinates, a set of equations can be formed and solved to determine the coordinates (\(x, y\)) of the instrument station. This approach provides a high degree of accuracy and is suitable for detailed survey work.

Conclusion on Three Point Problem Methods

All the methods discussed—the tracing paper method, Bessel's method, and Lehmann's method—are valid and widely accepted techniques for solving the three-point problem in surveying. The choice of which method to use often depends on factors such as the required level of accuracy, the specific equipment available, field conditions, and the surveyor's expertise and preference. Each method offers a reliable way to accurately determine the location of an unknown station using three known control points.