Principle of chain surveying; well-conditioned triangle

Chain surveying is the type of surveying in which only linear measurements (with a chain or tape) are made in the field, without angular measurements. It is suitable for small, fairly flat areas with few details.

Principle

The area to be surveyed is covered by a network of well-conditioned triangles, because a triangle is the only simple figure that can be plotted from the lengths of its three sides alone. The sides (chain lines) are measured in the field. The details (buildings, roads, boundaries) are then located by measuring perpendicular or oblique offsets from the chain lines. On paper, the triangles are plotted to scale by intersecting arcs, and the details are plotted from the offsets.

        C
       /|\ 
      / | \       triangle ABC: sides measured
     /  |  \      tie line / check line shown
    A---+---B     offsets to details from AB, BC, CA

The work follows the principle of working from whole to part: the big triangles (framework) are made first and the details are surveyed from them. To avoid accumulation of errors, check lines are measured.

Well-conditioned triangle and plotting error

A well-conditioned triangle has no angle smaller than $30^\circ$ and none greater than $120^\circ$, with an ideal shape close to equilateral ($60^\circ$ each).

The location of a plotted point depends on the intersection of two arcs. If there is an error $\delta$ in a side, the apex shifts by

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

where $\theta$ is the angle of intersection of the arcs.

For example, with $\theta = 15^\circ$, an error of 1 mm in a side shifts the point by $1/\sin 15^\circ = 3.9$ mm, while at $60^\circ$ it shifts only 1.15 mm. Hence well-conditioned triangles keep the plotting errors small.