Fundamentals & Classification of Surveying Notes

Last reviewed 16 Sept 2026 · 8 min read

What is surveying?

Surveying is the art and science of determining the relative positions of points on, above or below the surface of the earth by measuring horizontal distances, vertical distances (elevations), directions and angles, and of establishing points on the ground from given data (setting out).

Levelling is the branch dealing with measurements in the vertical plane.

Objects of surveying

Preparing maps and plans (topographic, cadastral, engineering) showing natural and artificial features.
Providing data for planning and design of engineering works — roads, railways, canals, dams, bridges, buildings.
Setting out works on the ground as per design.
Determining areas, volumes and boundaries of property.

Plane and geodetic surveying

Plane surveying	Geodetic surveying
Curvature of the earth is neglected — the surface is treated as a plane	Curvature of the earth is considered
Lines are straight; triangles are plane triangles; plumb lines are parallel	Lines are arcs; triangles are spherical; plumb lines converge
Suitable for small areas — commonly up to about 250 km²	Large areas — states, countries (Survey of India's triangulation)
Most engineering surveys	Control networks, precise national mapping

The difference between an arc along the earth's surface and its chord is only about 1 cm in 18.2 km, which is why the earth can be treated as flat for small areas.

Fundamental principles of surveying

Working from the whole to the part — first establish a framework of main control points with high accuracy over the whole area, then fill in details by less precise methods. Errors are thereby localised and not accumulated.
Locating a point by at least two independent measurements from two reference points already fixed:
- Two distances (as in chain surveying);
- A distance and a perpendicular offset;
- A distance and an angle (radiation/traversing);
- Two angles (intersection/triangulation).
Independent checks (check lines, closing a traverse on a known point) and consistency of accuracy with the purpose.

Classification of surveys

Basis	Types
Nature of the field	Land surveying — topographical, cadastral (boundaries, ownership), city surveys, engineering surveys; marine or hydrographic surveying (water bodies, depths, shorelines); astronomical surveying (absolute positions using celestial bodies)
Object/purpose	Engineering, military, mine, geological, archaeological, route surveys
Instruments used	Chain, compass, plane table, theodolite, tacheometric, EDM/total station, GPS/GNSS, photogrammetric (aerial and terrestrial), LiDAR and remote sensing surveys
Method	Triangulation (network of triangles), traversing (series of connected lines with measured lengths and angles), trilateration

Units of measurement

Linear: metre (km, cm, mm).
Area: m², hectare (1 ha = 10 000 m²), km² (1 km² = 100 ha).
Volume: m³.
Angular: sexagesimal (degree, minute, second — 360°), centesimal (grade — 400ᵍ in a circle), radian ( $2 π$ in a circle; 1 rad ≈ 57.2958° ≈ 206 265″).

Scales

The scale of a map is the ratio of a distance on the map to the corresponding distance on the ground.

Engineer's scale (numerical scale): e.g. 1 cm = 10 m.
Representative fraction (RF): both distances in the same units, e.g. 1 cm = 10 m → RF = 1/1000 (1 : 1000).
Large-scale maps (e.g. 1 : 500, 1 : 1000) show small areas in detail; small-scale maps (e.g. 1 : 250 000) cover large areas.

Types of scales

Scale	Description
Plain scale	Divided into units and sub-units — read to two dimensions (e.g. metres and decimetres)
Diagonal scale	Uses diagonals to read to three dimensions (e.g. metre, decimetre, centimetre) — small divisions obtained using similar triangles
Vernier scale	A short auxiliary scale sliding along the main scale to read fractions of the smallest main-scale division
Comparative scale	Two scales with the same RF in different units (e.g. metres and feet)
Scale of chords	For setting out and measuring angles on paper
Shrunk scale	The original scale corrected for shrinkage of an old map sheet

Vernier

FormulaLeast count of a vernier

Least count = s - v = \frac{s}{n}

$s$ = value of the smallest division on the main scale; $v$ = value of one vernier division; $n$ = number of divisions on the vernier ( $n$ vernier divisions = $n - 1$ main scale divisions for a direct vernier).

Direct vernier — $n$ vernier divisions equal $(n - 1)$ main scale divisions; vernier and main scale graduated in the same direction.
Retrograde vernier — $n$ vernier divisions equal $(n + 1)$ main scale divisions; graduated in the opposite direction.
Double vernier (theodolites) and extended vernier (e.g. in compasses and some instruments).

Shrunk scale

FormulaShrinkage correction

Shrinkage factor (SF) = \frac{shrunk length}{actual (original) length}

Shrunk scale = S F \times original scale; correct distance = \frac{measured distance on map}{S F} \times original scale

Errors in surveying

No measurement is exact; the true value is never known. The difference between a measured value and the true value is the error.

Type	Description	Treatment
Mistakes (blunders)	Due to carelessness, inexperience, confusion (e.g. wrong reading, wrong booking)	Detected and eliminated by checks and independent measurements
Systematic (cumulative) errors	Follow a definite law; same sign and magnitude under the same conditions — due to instrument defects (wrong length of tape), natural causes (temperature, sag, refraction), personal bias	Computed and corrected, or eliminated by proper procedure (e.g. reciprocal observations, face left/face right)
Random (accidental) errors	Small, unpredictable, both signs, due to limits of human senses and instruments	Cannot be eliminated; treated by theory of probability — minimised by repeated observations and adjustment

Accuracy — closeness of a measurement to the true value. Precision — closeness of repeated measurements to each other (degree of refinement). Measurements can be precise but not accurate (due to systematic error).

FormulaTheory of errors

Most probable value of repeated equal-weight observations = arithmetic mean $\overset{x}{ˉ}$
Residual $v = x - \overset{x}{ˉ}$
Standard deviation of a single observation: $σ = \frac{\sum v ^{2}}{n - 1}$
Probable error of a single observation: $P E_{s} = \pm 0.6745 σ$
Probable error of the mean: $P E_{m} = \frac{P E _{s}}{n}$
Weight of an observation ∝ $\frac{1}{σ ^{2}}$ ; weighted mean $= \frac{\sum w x}{\sum w}$

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Fundamentals & Classification of Surveying