Tacheometry: measuring distance with stadia hairs

Tacheometry finds horizontal distance and height difference optically, without a tape. It was the standard way to survey rough or hilly ground before total stations, and it still appears in every surveying exam.

Stadia principle

A tacheometer has two extra horizontal hairs. The difference between the top and bottom hair readings on the staff is the staff intercept, s.

For a horizontal line of sight:

D = fi s + (f + d) = k s + C

• k = f/i is the multiplying constant, usually 100.
• C = f + d is the additive constant, zero for an anallactic lens.

Inclined sights, staff vertical

With vertical angle θ:

D = k s cos² θ + C cos θ

V = k s sin 2θ2 + C sin θ

Then the RL of the staff station is

RL = HI ± V − h

where h is the central hair reading and HI the height of instrument.

Worked example

k = 100, C = 0, θ = +8°, hair readings 1.155, 1.765 and 2.375. HI = 102.450 m.

s = 2.375 − 1.155 = 1.220 m

D = 100 × 1.220 × cos² 8° = 118.4 m

V = 100 × 1.220 × sin 16°2 = 16.81 m

RL = 102.450 + 16.81 − 1.765 = 117.50 m

Finding k and C in the field

Measure staff intercepts at two known distances on level ground and solve D₁ = k s₁ + C and D₂ = k s₂ + C together.

Today

A total station measures distance electronically, but the reduction of slope distance and vertical angle into horizontal distance and height follows the same trigonometry.