Understanding the concept of 'failure of fix' in plane tabling requires a look at the resection method.

Plane Tabling and Resection Explained

Plane tabling is a surveying method where both field observation and plotting are done simultaneously. The plane table itself serves as the drawing board. Resection is one of the methods used in plane tabling to determine the position of the instrument station (where the plane table is set up) on the sheet by sighting known objects whose positions are already plotted on the map.

In the resection method using three points, the surveyor selects three prominent, well-defined objects (say A, B, and C) whose corresponding points (a, b, and c) are already plotted on the plane table sheet. From the unknown station (say P), sights are taken to A, B, and C using the alidade. Rays are drawn from the plotted points a, b, and c towards the sighted objects A, B, and C respectively. Ideally, these three rays should intersect at a single point, which represents the location of the station P on the map.

What is Failure of Fix in Plane Tabling?

When the three rays drawn during resection do not intersect at a single point, they usually form a small triangle, known as the 'triangle of error'. This triangle indicates errors in observation, plotting, or setting up. The surveyor uses methods to reduce this triangle to find the most probable location of the station. However, in certain geometric conditions, the determination of the station's position becomes ambiguous or impossible, leading to what is called a 'failure of fix'.

The Danger Circle or Great Circle in Resection

The geometric condition that causes a failure of fix in the three-point resection problem involves the location of the instrument station relative to the three known points. If the station P lies on the same circle that passes through the three known points A, B, and C, the fix fails. This specific circle is often referred to as the 'danger circle' or, in some contexts, the 'great circle' related to the problem.

When the station P is located exactly on this danger circle, the rays from the plotted points a, b, and c towards A, B, and C respectively do not form a distinct triangle of error or a clear intersection point that can be easily resolved. The geometry becomes such that the intersection point is indeterminate.

Analyzing the Options for Failure of Fix

Let's consider the given options based on our understanding of the danger circle:

• The plane table is inside the great triangle: The 'great triangle' is usually formed by connecting the three known points. Being inside this triangle does not inherently cause a failure of fix; it's a region where the fix is generally stable.
• The plane table is inside the great circle: Being inside the danger circle (great circle) typically results in a triangle of error that can be resolved to find the station's position.
• The plane table is outside the great circle: Similar to being inside, being outside the danger circle generally results in a solvable triangle of error.
• The plane table is on the great circle: When the plane table station is exactly on the danger circle passing through the three reference points, the geometric conditions lead to an indeterminate fix, meaning a failure of fix occurs.

Therefore, the failure of fix in plane tabling during resection occurs when the plane table station is located on the great circle (danger circle) defined by the three reference points.

Revision Table: Plane Tabling Resection

Additional Information on Plane Tabling Fix

While the three-point problem is common, other methods for fixing a station exist, like the two-point problem or using multiple points. However, the concept of a 'danger locus' where the fix is indeterminate is particularly relevant in the three-point problem. Surveyors are taught to avoid setting up their plane table near or on this danger circle to ensure a reliable fix.

Practical tips to avoid failure of fix or improve the accuracy of the fix include:

• Choosing well-defined and easily identifiable reference points.
• Selecting points that provide a strong geometric intersection (avoiding points that are nearly collinear or points that place the station on the danger circle).
• Taking observations carefully and checking measurements.
• Using more than three points if available, which can help in resolving ambiguity or verifying the station's position.

The great circle or danger circle phenomenon is a critical consideration for surveyors using the three-point resection method in plane tabling.