Chain stations and lines; well-conditioned triangle

Factors in selecting stations and chain lines

1. Intervisibility. Stations should be visible from each other so that lines can be ranged and chained easily.
2. Well-conditioned triangles. The lines should make triangles with all angles between $30^\circ$ and $120^\circ$.
3. Base line. Choose the longest line on level ground through the middle of the area; it carries the main framework.
4. Short offsets. Lines should pass close to the details so that offsets are not longer than 15 m.
5. Minimum number of stations and lines, to reduce work and errors.
6. Ground. Avoid steep slopes, ponds, buildings and thick bushes on lines.
7. Check and tie lines must be provided.
8. Stations should be on firm, permanent ground and tied to objects.

Why a well-conditioned triangle is necessary

A triangle is plotted from the lengths of its three sides by intersecting arcs. If an angle is very small or very large (an ill-conditioned triangle), the arcs intersect at a flat angle, and a small error in a measured length moves the plotted point by a large amount:

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

where $\delta$ is the error in the side and $\theta$ the intersection angle. For $\theta = 15^\circ$, an error of 1 mm shifts the point 3.9 mm; for $\theta = 60^\circ$ only 1.15 mm. An equilateral triangle is the best shape; angles from $30^\circ$ to $120^\circ$ are acceptable. This keeps the plotting error small and the framework accurate.