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).
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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 (\(2s-3\)).
Step 1: Identify the two independent sources of redundancy. In a triangulation network there are two separate types of condition (redundant) equations: (a) angle/shape conditions coming from the requirement that a triangle's angles sum to $180^\circ$, and (b) side/scale conditions coming from having more than one baseline to fix the scale. Step 2: Compute the angle (shape) redundancy. With $s = 8$ stations (6 triangles, as counted from the 18 interior angles), the minimum number of angles needed to fix the shape (not the size) of the network is $2s - 4 = 2(8) - 4 = 12$, the number of independent angles that fix a set of $s$ points up to a similarity transformation. Since 18 angles were actually observed, the shape (angle) redundancy is \[ r_1 = 18 - 12 = 6 \] Step 3: Compute the scale (baseline) redundancy. Only one baseline length is needed to fix the absolute scale of the network once its shape is known from the angles; the second baseline is then a check observation. With 2 baselines measured, the scale redundancy is \[ r_2 = 2 - 1 = 1 \] Step 4: Add the two redundancies. The total number of redundant observations is the sum of the shape redundancy and the scale redundancy, \[ r = r_1 + r_2 = 6 + 1 = 7 \] \[ \boxed{r = 7} \]
