Tacheometric Surveying Notes

Last reviewed 16 Sept 2026 · 8 min read

Principle and uses

Tacheometry (tachymetry) is a branch of angular surveying in which horizontal distances and vertical distances (elevations) are determined optically from instrumental observations, without chaining.

Uses:

Contouring and topographic surveys, especially in rough, hilly or broken terrain and across rivers, swamps and valleys where chaining is difficult or impossible.
Filling in details in large-scale surveys; preliminary route surveys (roads, railways, canals).
Checking distances measured by other methods.

Accuracy is lower than good chaining (typically about 1 in 500 to 1 in 1000 for stadia distances) but sufficient for many topographic purposes; the method is fast.

Instruments

Tacheometer — a transit theodolite whose diaphragm carries two stadia hairs (equidistant above and below the central horizontal hair) in addition to the cross hairs. Features: high magnifying power, bright image, and an anallactic lens (in external focusing telescopes) — or an internal focusing telescope — so that the additive constant is zero.
Stadia rods — long staves (3–4 m) with bold graduations for long sights; ordinary levelling staves for short sights.
Subtense bar — a horizontal bar (commonly 2 m long between targets, made of invar) set at a known length perpendicular to the line of sight.

Anallactic lens

An additional convex lens placed between the eyepiece and objective of an external focusing telescope so that the stadia distance is measured from the centre of the instrument — the additive constant becomes zero and the distance equation becomes $D = K s$ . (Introduced by Porro.) Internal focusing telescopes achieve an additive constant close to zero without it.

Methods of tacheometry

Method	Principle
Stadia method — fixed hair	Stadia hairs fixed; the staff intercept varies with distance — most common
Stadia method — movable hair (subtense)	Stadia hairs movable by micrometer screws to fixed targets on the staff; distance from the hair movement
Tangential method	No stadia hairs; two vertical angles are observed to two targets a known distance apart on the staff
Subtense bar method	Horizontal angle subtended by a bar of known length
Electronic (EDM/total station)	Modern replacement for optical tacheometry

Stadia method — distance equation

FormulaHorizontal line of sight, staff vertical

D = \frac{f}{i} s + (f + d) = K s + C

$s$ = staff intercept (difference between upper and lower stadia hair readings)
$K = \frac{f}{i}$ = multiplying constant (usually made 100)
$C = f + d$ = additive constant ( $f$ = focal length of objective, $d$ = distance from objective to the vertical axis; $i$ = stadia hair interval)
With an anallactic lens (or internal focusing): $C = 0$ → $D = K s = 100 s$

Determination of constants

Measure staff intercepts $s_{1}$ and $s_{2}$ at two accurately measured distances $D_{1}$ and $D_{2}$ :

D_{1} = K s_{1} + C, D_{2} = K s_{2} + C \Rightarrow K = \frac{D _{2} - D _{1}}{s _{2} - s _{1}}, C = D_{1} - K s_{1}

Inclined sights

FormulaStaff held vertical (most common)

For vertical angle $θ$ (elevation or depression):

D = K s cos^{2} θ + C cos θ

V = K s \frac{sin 2 θ}{2} + C sin θ

$D$ = horizontal distance; $V$ = vertical distance between the horizontal axis of the instrument and the point on the staff cut by the central hair.

FormulaStaff held normal to the line of sight

D = (K s + C) cos θ \pm h sin θ, V = (K s + C) sin θ

( $h$ = central hair reading; + for angle of elevation, − for depression, in $D$ .)

Reduced levels

Angle of elevation:

R L_{s t a f f s t a t i o n} = R L_{in s t r u m e n t s t a t i o n} + h_{i} + V - h

Angle of depression:

R L_{s t a f f s t a t i o n} = R L_{in s t r u m e n t s t a t i o n} + h_{i} - V - h

( $h_{i}$ = height of instrument axis above its station; $h$ = central hair reading.) When the instrument is referred to a benchmark, $R L = H I \pm V - h$ , where HI = RL of the instrument axis.

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Tacheometric Surveying