Question

In the triangulation network shown in the figure, \(b\) and \(g\) are the observed lengths of the baselines, while 1 to 18 are the observed interior angles. The number of redundant observations is __________ (Answer in integer).

Hint:

Count the total observations (angles plus baselines) and compare with the minimum number needed to fix the shape and size of an 8-station triangulation network (\(2s-3\)).

Answer 1

Correct Answer: 7

Step 1: List the total number of observations made.
The network has 18 interior angles observed (numbered 1 to 18) and 2 baseline lengths observed (\(b\) and \(g\)). So the total number of observations is \(N = 18 + 2 = 20\).
Step 2: Count the number of triangulation stations from the figure.
The figure is a chain of single (unbraced) triangles running from baseline \(b\) to baseline \(g\). Since each triangle contributes 3 angles and \(18/3 = 6\), the chain has 6 triangles. A chain of \(t\) single triangles connecting two end baselines has \(t + 2\) triangulation stations, so the number of stations is \(s = 6 + 2 = 8\).
Step 3: Find the minimum number of independent quantities needed to fix the network.
Each of the 8 stations has 2 unknown coordinates, giving \(2s = 16\) coordinate unknowns. Since the network is free (no fixed control), the position and orientation of the whole figure are arbitrary and remove 3 degrees of freedom (2 for translation, 1 for rotation), but the size (scale) of the network must still be determined from the observations. So the minimum number of independent observations required to determine the shape and size of the network is \(2s - 3 = 2(8) - 3 = 13\).
Step 4: Compute the redundant observations.
Redundant observations are the observations in excess of the minimum required, that is, \[ r = N - (2s - 3) = 20 - 13 = 7 \] \[ \boxed{r = 7} \]