Understanding Fix Strength in Plane Table Surveying: The Three-Point Problem

The three-point problem is a fundamental method in plane table surveying used to determine the location of the plane table station by observing three well-defined control points (often denoted as P, Q, and R) whose positions are already plotted on the drawing sheet.

Solving the three-point problem involves finding the exact location (fix) of the instrument station on the map. The "fix strength" refers to how well-determined and reliable this location is. A strong fix means the determined position is accurate and unambiguous, while a weak fix means it's less certain or potentially indeterminate.

Analyzing Conditions for Fix Strength

Let's examine the given options and relate them to the strength of the fix in the three-point problem:

• The angle subtended by the two lines obtained by joining the three points is very small: This refers to the geometry of the control points themselves. While the arrangement of P, Q, and R affects how easy or difficult it is to obtain a solution and can influence the clarity of the fix, a very small angle between control points (e.g., they are nearly collinear) can make the fix weaker, especially if the station is far away. This option describes the control points' geometry, not the station's optimal location relative to them for a strong fix.
• The station occupied by the plane table lies on the circle passing through the three points: This describes a critical condition known as the "danger circle" or "great circle". If the station lies exactly on the circle that passes through the three control points (P, Q, R) and the station itself, the three-point problem becomes indeterminate. Any number of points on this circle could satisfy the angular conditions observed from the station. Therefore, this condition results in a very weak or indeterminate fix, not a good one.
• The station occupied by the plane table lies near the circle passing through the three points: If the station is close to the danger circle, the fix is still weak. Small errors in observation can lead to large errors in the determined position. The geometrical intersection used to find the station's location becomes very sensitive to errors when near this circle. This condition results in a poor fix.
• The station occupied by the plane table lies within the triangle formed by the three points: This condition generally leads to a strong fix. When the station is located inside the triangle PQR, it is typically far from the danger circle (unless the triangle itself is very large and contains a segment of the danger circle, but in typical surveying setups with well-chosen points, being inside PQR gives a good fix). The intersections used in solving the problem are well-defined, and the effect of observational errors on the final station position is minimized. This is considered a favorable position for a good fix.

Conclusion: Optimal Station Location for a Good Fix

Based on the analysis, the strength of the fix in the three-point problem is directly related to the location of the plane table station relative to the three control points and the danger circle. A good or strong fix is obtained when the station is situated in a position where the solution is stable and less susceptible to observational errors.

The best location among the given options for a good fix using the trial and error method (Lehmann's method, which is often used for trial and error) is when the plane table station is inside the triangle formed by the three control points.

Therefore, the fix strength is good when the station occupied by the plane table lies within the triangle formed by the three points.