Formula Sheets: Levelling & Contouring | Geomatics Engineering (Surveying) - Civil Engineering (CE) PDF Download

 Page 1


GA TE CE 2026 F orm ula Sheet: Geomatics Engineering - Dis-
tance, Angle, Lev elling, and Con touring
1. Distance Measuremen t
• Distance b y stadia tac heometry:
D =k·s·cos
2
?+c·cos?
where:
– D : Horizon tal distance (m)
– k : Stadia constan t (t ypically 100)
– s : Staff in tercept (m)
– ? : V ertical angle (degrees)
– c : A dditiv e constan t (m, usually 0)
• V ertical comp onen t in stadia:
V =
k·s
2
·sin(2?)+c·sin?
where:
– V : V ertical heigh t difference (m )
• Correction for slop e distance:
D
h
=D
s
·cos?
where:
– D
h
: Horiz on tal distance (m)
– D
s
: Slop e distanc e (m)
– ? : Slop e angle (degrees)
• EDM (Electronic Distance Measuremen t) correction:
D
c
=D
m
·
(
1+
?n
n
0
)
where:
– D
c
: Corrected distance (m)
– D
m
: Measured distance (m)
– ?n : Refractiv e index correction
– n
0
: Standard refractiv e index
2. Angl e Measuremen t
• Horizon tal angle b y theo dolite:
? =
?
( face left+ face righ t)
2n
where:
– ? : Mean horizon tal angle (degrees)
– n : Num b er of observ ations
• V ertical angle correction for curv ature and refraction:
?
c
=?
m
-
d
2R
·(1-k)
where:
1
Page 2


GA TE CE 2026 F orm ula Sheet: Geomatics Engineering - Dis-
tance, Angle, Lev elling, and Con touring
1. Distance Measuremen t
• Distance b y stadia tac heometry:
D =k·s·cos
2
?+c·cos?
where:
– D : Horizon tal distance (m)
– k : Stadia constan t (t ypically 100)
– s : Staff in tercept (m)
– ? : V ertical angle (degrees)
– c : A dditiv e constan t (m, usually 0)
• V ertical comp onen t in stadia:
V =
k·s
2
·sin(2?)+c·sin?
where:
– V : V ertical heigh t difference (m )
• Correction for slop e distance:
D
h
=D
s
·cos?
where:
– D
h
: Horiz on tal distance (m)
– D
s
: Slop e distanc e (m)
– ? : Slop e angle (degrees)
• EDM (Electronic Distance Measuremen t) correction:
D
c
=D
m
·
(
1+
?n
n
0
)
where:
– D
c
: Corrected distance (m)
– D
m
: Measured distance (m)
– ?n : Refractiv e index correction
– n
0
: Standard refractiv e index
2. Angl e Measuremen t
• Horizon tal angle b y theo dolite:
? =
?
( face left+ face righ t)
2n
where:
– ? : Mean horizon tal angle (degrees)
– n : Num b er of observ ations
• V ertical angle correction for curv ature and refraction:
?
c
=?
m
-
d
2R
·(1-k)
where:
1
– ?
c
: Corrected v ertical angle (radians)
– ?
m
: Measured v ertical angle (radians)
– d : Distance to target (m)
– R : Earth’s radius (appro ximately 6,371,000 m)
– k : Refraction co e?icien t (t ypically 0.14)
• Angle closure in a p olygon tra v erse:
?
?
i
= (n-2)·180
?
where:
– ?
i
: In terior angles (degrees)
– n : Num b er of sides in p olygon
3. Lev elling
• Reduced lev el (RL) b y heigh t of instrumen t metho d:
RL
B
= RL
A
+ HI- FS
where:
– RL
A
, RL
B
: Reduced lev els of p oin ts A and B (m)
– HI: Heigh t of instrumen t (m)
– FS: F oresigh t reading (m)
• Rise and fall metho d:
RL
B
= RL
A
+( BS- FS)
where:
– BS: Bac ksigh t reading (m)
• Correction for curv ature:
C
c
=
d
2
2R
where:
– C
c
: Curv ature correction (m)
– d : Distance from instrumen t to staff (m)
– R : Earth’s radius (appro ximately 6,371,000 m )
• Refraction correction:
C
r
=k·
d
2
2R
where:
– C
r
: Refraction correction (m)
– k : Refraction co e?icien t (t ypically 0.14)
• Com bined curv ature and refraction correction:
C
cr
=
d
2
2R
·(1-k)
2
Page 3


GA TE CE 2026 F orm ula Sheet: Geomatics Engineering - Dis-
tance, Angle, Lev elling, and Con touring
1. Distance Measuremen t
• Distance b y stadia tac heometry:
D =k·s·cos
2
?+c·cos?
where:
– D : Horizon tal distance (m)
– k : Stadia constan t (t ypically 100)
– s : Staff in tercept (m)
– ? : V ertical angle (degrees)
– c : A dditiv e constan t (m, usually 0)
• V ertical comp onen t in stadia:
V =
k·s
2
·sin(2?)+c·sin?
where:
– V : V ertical heigh t difference (m )
• Correction for slop e distance:
D
h
=D
s
·cos?
where:
– D
h
: Horiz on tal distance (m)
– D
s
: Slop e distanc e (m)
– ? : Slop e angle (degrees)
• EDM (Electronic Distance Measuremen t) correction:
D
c
=D
m
·
(
1+
?n
n
0
)
where:
– D
c
: Corrected distance (m)
– D
m
: Measured distance (m)
– ?n : Refractiv e index correction
– n
0
: Standard refractiv e index
2. Angl e Measuremen t
• Horizon tal angle b y theo dolite:
? =
?
( face left+ face righ t)
2n
where:
– ? : Mean horizon tal angle (degrees)
– n : Num b er of observ ations
• V ertical angle correction for curv ature and refraction:
?
c
=?
m
-
d
2R
·(1-k)
where:
1
– ?
c
: Corrected v ertical angle (radians)
– ?
m
: Measured v ertical angle (radians)
– d : Distance to target (m)
– R : Earth’s radius (appro ximately 6,371,000 m)
– k : Refraction co e?icien t (t ypically 0.14)
• Angle closure in a p olygon tra v erse:
?
?
i
= (n-2)·180
?
where:
– ?
i
: In terior angles (degrees)
– n : Num b er of sides in p olygon
3. Lev elling
• Reduced lev el (RL) b y heigh t of instrumen t metho d:
RL
B
= RL
A
+ HI- FS
where:
– RL
A
, RL
B
: Reduced lev els of p oin ts A and B (m)
– HI: Heigh t of instrumen t (m)
– FS: F oresigh t reading (m)
• Rise and fall metho d:
RL
B
= RL
A
+( BS- FS)
where:
– BS: Bac ksigh t reading (m)
• Correction for curv ature:
C
c
=
d
2
2R
where:
– C
c
: Curv ature correction (m)
– d : Distance from instrumen t to staff (m)
– R : Earth’s radius (appro ximately 6,371,000 m )
• Refraction correction:
C
r
=k·
d
2
2R
where:
– C
r
: Refraction correction (m)
– k : Refraction co e?icien t (t ypically 0.14)
• Com bined curv ature and refraction correction:
C
cr
=
d
2
2R
·(1-k)
2
4. T rigonometric Lev elling
• Heigh t difference (simple trigonometric lev elling):
h =D·tan?+h
i
-h
s
where:
– h : Heigh t difference b et w een p oin ts (m)
– D : Horizon tal distance (m)
– ? : V ertical angle (degrees)
– h
i
: Heigh t of instrumen t (m)
– h
s
: Heigh t of target/staff (m)
• Heigh t difference with curv ature and refraction:
h =D·tan?+h
i
-h
s
-
d
2
2R
·(1-k)
• Recipro cal lev elling:
h =
(b
1
-f
1
)+(b
2
-f
2
)
2
where:
– b
1
, f
1
: Bac ksigh t and foresigh t from instrumen t at p oin t 1 (m)
– b
2
, f
2
: Bac ksigh t and foresigh t from instrumen t at p oin t 2 (m)
5. Con touring
• Con tour in terv al:
CI =
Range of elev ation
Num b er of con tours
where:
– CI: Con tour in terv al (m)
• Slop e b et w een t w o p oin ts:
S =
h
2
-h
1
D
×100
where:
– S : Slop e (%)
– h
1
, h
2
: Elev ations of t w o p oin ts (m)
– D : Horizon tal distance b et w een p oin ts (m)
• Distance to con tour line (in terp olation):
d =
h
c
-h
1
h
2
-h
1
·D
where:
– d : Distance to con tour line from p oin t 1 ( m)
– h
c
: C on tour elev ation (m)
– h
1
, h
2
: Elev ations at t w o p oin ts (m)
3