This document provides principles and an example calculation for determining horizontal distance, vertical distance, and difference in height using incline sights with the staff normal method in tachymetry. The key principles are that horizontal distance is calculated using trigonometry with the staff intercept, instrument height, and zenith angle, vertical distance also uses trigonometry with the staff intercept and zenith angle, and difference in height considers the instrument height, vertical distance, and the center hair reading. An example calculation demonstrates applying these principles and formulas to find horizontal distance, vertical distance, and difference in height at a staff station given instrument data, staff readings, and angles.

1. TACHYMETRY
LESSON 4
Incline Sights With
the Staff Normal
LOGO

2. Contents
1
PRINCIPLES
2
3
CALCULATION

3. Principles
Incline Sight With The Staff Normal
dH
hi
s = the staff intercept AB
h = the length of the centre hair reading from the staff base
V = the vertical component XY, the height of the centre hair reading above
(or below) the instrument axis
D = the length of the line of sight IX
H = the horizontal distance required.
hi = instrument height

4. Formula staff normal
Horizontal Distance;
H = (Ks + C) cos θ ± h sin θ
Vertical Distance ;
V = (Ks + C) sin θ
First Reduced Level Station ;
RL1 = RLTBM – hi ± V – h cos θ
Difference Height ;
dH = hi . ± V – h cos θ

5. Cycle Diagram
Where
D = Distance
K & C = constant (if not given assume K = 100 & C = 0)
S = staff intercept
H = horizontal distance
V = vertical distance
θ = zenith angle (positive for angles of the elevation,
negative for angles depression)
hi = the height of instrument (always positive)
h = the centre hair reading (always negative)

6. EXAMPLE CALCULATION
Inst.
Stn.
and
Ht.
of
inst.
axis
Staff
stn.
A
B
(1.64m)
Horizontal
angle
Vertical
angle
Zenith
angle, θ
97° 30’ 00” 81° 30’ 00” +8° 30’ 00”
Upper
stadia
Middle
stadia
Lower
stadia
2.025
1.515
Horizontal
distance
H (m)
Vertical
distance
V (m)
Difference
height,
dH (m)
R.L.
at
stn.
1.000
R.L.
at
staff
Remarks
Take the
first RL =
42.353 m
Given K = 100 and C = 0.65
H = (Ks + C) cos θ ± h sin θ
= (((100 . (2.025 – 1.000)) + 0.65) . (Cos +80 30’ )) + (1.515 . Cos +80 30’ )
= (103.150. (Cos +80 30’ )) + 0.224
= 102.240 m
V =
=
=
=
(Ks + C) sin θ
(100 . (2.025 – 1.000) + 0.65) . (Sin +80 30’ )
(103.150. (Sin +80 30’ ))
15.250 m
RLA = RLTBM – hi ± V – h
= RLTBM – dH
dH = hi ± V – h kos θ
= 42.353 + 15.387
= 1.64 + 15.25 - 1.515 kos +80 30’
= 57.740 m
= 15.387 m

7. LOGO