Understanding Terms in Plane Table Surveying

Plane table surveying is a graphical method of surveying where the field work and plotting are done simultaneously. This method is particularly useful for filling in details and for small-scale mapping. Several techniques are used in plane table surveying to determine the position and orientation of the table and to locate points.

The Three-Point Problem and Lehmann Method

The question refers to terms like ‘triangle of error’, ‘great circle’, and ‘great triangle’ in the context of plane table surveying. These terms are specifically associated with the method used to solve the three-point problem. The three-point problem is about determining the location of the plane table station (where the surveyor is standing) and simultaneously orienting the plane table when three well-defined points (whose positions are already plotted on the paper) are visible in the field.

Several graphical methods exist to solve the three-point problem, and the Lehmann method is one such prominent graphical technique. Other methods include the Bessel method and tracing paper method.

Triangle of Error in Lehmann Method

In the Lehmann method, the process starts with assuming an approximate orientation of the plane table. Rays are then drawn from the instrument station towards the three visible ground points (say, A, B, and C) and made to pass through their corresponding plotted positions (a', b', and c') on the paper. If the assumed orientation was incorrect, these three rays drawn from the instrument station on the paper will not intersect at a single point. Instead, they will form a small triangle. This triangle is known as the triangle of error or triangle of uncertainty.

The size and position of the triangle of error provide clues about the amount and direction of the error in orientation. The Lehmann method provides graphical rules or procedures based on the triangle of error to adjust the orientation of the plane table iteratively until the three rays converge to a single point, indicating the correct station location and orientation.

Great Circle and Great Triangle

While the triangle of error is the most directly linked term to the process of correcting orientation in the Lehmann method, the terms ‘great circle’ and ‘great triangle’ are also sometimes associated with the graphical solution of the three-point problem. The 'great triangle' often refers to the triangle formed by the three plotted points (a', b', c') on the paper or the corresponding points A, B, C in the field. The geometry related to these points and the instrument station involves circles (sometimes referred to in the context of 'great circle' principles or related geometrical properties) that help in the graphical determination of the station's position.

Considering the options provided, the terms ‘triangle of error’, ‘great circle’, and ‘great triangle’ are most appropriately related to the Lehmann method, which is a specific graphical approach to solving the three-point problem in plane table surveying, where the triangle of error is a central element for orientation correction.

Examining Other Options

• Bessel method: This is another graphical method for solving the three-point problem, conceptually similar to the Lehmann method in purpose but potentially using different graphical constructions. It also deals with finding the correct station.
• Two-point problem: This method is used to orient the plane table when only two points whose positions are plotted on the map are visible. It does not involve a 'triangle of error' formed by three rays.
• Graphic triangulation: This is a general technique in plane table surveying where points are located by drawing rays from known stations and finding their intersections. While it's part of plane table surveying, the specific terms ‘triangle of error’, ‘great circle’, and ‘great triangle’ are primarily linked to the three-point problem's graphical solutions like the Lehmann method.

Therefore, based on the specific terms mentioned, the Lehmann method is the most fitting answer as it directly involves the formation and use of the triangle of error for orientation in the three-point problem, with the other terms also being associated geometrical concepts in this context.

Conclusion

The terms ‘triangle of error’, ‘great circle’, and ‘great triangle’ are key concepts encountered when applying the Lehmann method to solve the three-point problem in plane table surveying. The triangle of error is central to the graphical procedure for correcting the orientation of the table.

Revision Table: Plane Table Surveying Terms

Additional Information on Plane Table Surveying and Resection

Resection is a method in plane table surveying used to determine the location of the plane table station on the drawing sheet by drawing rays from the unknown station towards points whose positions are already plotted on the sheet. The three-point problem is the most common scenario for resection, where three known, visible points are used.

The graphical methods for solving the three-point problem, like the Lehmann method, are iterative. They involve making an initial estimate of the station's position or the table's orientation, checking the accuracy (e.g., by forming the triangle of error), and then adjusting the setup based on specific rules derived from geometrical principles until the desired accuracy is achieved.

The terms 'great circle' and 'great triangle' highlight the geometric principles underlying the solution, often related to the circumscribing circle of the triangle formed by the three known points or the geometry involving the instrument station and these points on the paper and ground.