Question

In a two-dimensional coordinate system, it is proposed to determine the size and shape of a triangle ABC in addition to its location and orientation. For this, all the internal angles and sides of the triangle were observed. Further, the planar coordinates of point A and bearing/azimuth of line AB were known. The redundancy (\( r \)) for the above system will be equal to _________ (Answer in integer).

Hint:

Redundancy in geodetic problems is computed by subtracting the number of unknowns from the number of independent observations. Known control points reduce the number of unknowns.

Answer 1

We are given:
- A triangle \( \triangle ABC \)
- All 3 sides and all 3 internal angles observed \( \Rightarrow \) 6 observations
- Known: planar coordinates of point \( A \) (2 values), and azimuth of line \( AB \) (1 value)

Let’s analyze the unknowns and constraints:

Unknowns:
- Coordinates of \( B \) and \( C \): 2 points × 2 coordinates = 4 unknowns

Additional unknown:
- Orientation (bearing of AB already given, so orientation is fixed)

Total unknowns: 4

Observations:
- 3 sides + 3 angles = 6

Redundancy (r) is given by:
\[ r = \text{Number of observations} - \text{Number of unknowns} = 6 - 4 = 2 \] (Note: Coordinates of A and bearing of AB are known, so we use them to fix the triangle in the coordinate system and do not count them as unknowns.)