Well-conditioned triangle; importance of tie line

Why a well-conditioned triangle is necessary

A triangle with all angles between $30^\circ$ and $120^\circ$ (ideally $60^\circ$) is called well-conditioned. The triangle is plotted from its three measured sides by intersecting arcs. In an ill-conditioned triangle (with an angle very small or very large), the arcs cut at a flat angle, and a small error in a side moves the plotted point a lot:

$$\text{displacement} \approx \frac{\delta}{\sin\theta}$$

With $\theta = 15^\circ$, a 1 mm error becomes 3.9 mm; with $\theta = 60^\circ$ it becomes only 1.15 mm. Hence the framework should be formed of well-conditioned triangles to keep plotting errors small.

Importance of the tie line

A tie line joins two fixed points (tie stations) on the main survey lines. Its importance:

1. Check on the framework. The scaled length of the tie line on the plan is compared with its measured length; a difference reveals errors in the main lines.
2. Location of details. It is used to take offsets to details that are far from the main lines, and to fix interior objects.
3. Fixing the position of the survey lines. It keeps the triangles from distorting and so controls the shape.
4. Avoiding long offsets. A tie line can be run through the middle of a big triangle where details are far from the sides, so that offsets remain short.
5. Joining the stations of different lines, giving a way to start a new triangle.

          C
         / \
        /   \
       D-----E      tie line DE (D on CA, E on CB)
      /       \
     A---------B