Formula Sheets: Theodolite, Compass & Traverse Surveying | Geomatics Engineering (Surveying) - Civil Engineering (CE) PDF Download

 Page 1


GA TE CE 2026 F orm ula Sheet: Geomatics Engineering - T ra v ers-
ing, T riangulation, and T otal Station
1. T ra v ersing
• Co ordinates of a tra v erse p oin t:
E
B
=E
A
+L·sin?, N
B
=N
A
+L·cos?
where:
– E
A
, N
A
: Easting and northing of p oin t A (m)
– E
B
, N
B
: Easting and northing of p oin t B (m)
– L : Horizon tal distance b et w een A and B (m)
– ? : Be aring of line AB (degrees)
• Latitude and departure:
Latitude =L·cos?, Departure =L·sin?
• Closure error in tra v erse:
e
c
=
v
(
?
Departure)
2
+(
?
Latitude)
2
where:
– e
c
: Linear closure error (m)
• Relativ e error of closure:
REC =
e
c
P
where:
– REC: Relativ e error of closure
– P : T otal p erimeter of tra v erse (m)
• Angular closure error:
?
?
i
= (n-2)·180
?
where:
– ?
i
: In terior angles (degrees)
– n : Num b er of sides in tra v erse
• Bo wditc h’s correction for latitude and departure:
C
Li
= -
?
Latitude·L
i
P
, C
Di
= -
?
Departure·L
i
P
where:
– C
Li
, C
Di
: Corrections for latitude and departure of line i (m)
– L
i
: Length of line i (m)
1
Page 2


GA TE CE 2026 F orm ula Sheet: Geomatics Engineering - T ra v ers-
ing, T riangulation, and T otal Station
1. T ra v ersing
• Co ordinates of a tra v erse p oin t:
E
B
=E
A
+L·sin?, N
B
=N
A
+L·cos?
where:
– E
A
, N
A
: Easting and northing of p oin t A (m)
– E
B
, N
B
: Easting and northing of p oin t B (m)
– L : Horizon tal distance b et w een A and B (m)
– ? : Be aring of line AB (degrees)
• Latitude and departure:
Latitude =L·cos?, Departure =L·sin?
• Closure error in tra v erse:
e
c
=
v
(
?
Departure)
2
+(
?
Latitude)
2
where:
– e
c
: Linear closure error (m)
• Relativ e error of closure:
REC =
e
c
P
where:
– REC: Relativ e error of closure
– P : T otal p erimeter of tra v erse (m)
• Angular closure error:
?
?
i
= (n-2)·180
?
where:
– ?
i
: In terior angles (degrees)
– n : Num b er of sides in tra v erse
• Bo wditc h’s correction for latitude and departure:
C
Li
= -
?
Latitude·L
i
P
, C
Di
= -
?
Departure·L
i
P
where:
– C
Li
, C
Di
: Corrections for latitude and departure of line i (m)
– L
i
: Length of line i (m)
1
2. T riangulation Surv ey
• Sine rule for triangle:
a
sinA
=
b
sinB
=
c
sinC
where:
– a , b , c : Sides of triangle (m)
– A , B , C : Opp osite angles (degrees)
• Cosine rule:
a
2
=b
2
+c
2
-2bc·cosA
• Area of triangle:
Area =
1
2
bc·sinA
• Base line correction in triangulation:
D
c
=D
m
·
sin?
c
sin?
m
where:
– D
c
: Corrected base line distance (m)
– D
m
: Measured base line distance (m)
– ?
c
, ?
m
: Corrected and measured angles (degrees)
• Strength of figure (triangulation):
S =
?
(dA)
2
n
where:
– S : Strength of figure
– dA : Angular error (seconds)
– n : Num b er of triangles
3. T o tal Station
• Horizon tal distance measuremen t:
D
h
=S ·cos?
v
where:
– D
h
: Horizon tal distance (m)
– S : Slop e distance measured b y total station (m)
– ?
v
: V ertical angle (degrees)
• V ertical heigh t difference:
h =S ·sin?
v
+h
i
-h
t
where:
– h : Heigh t difference (m)
– h
i
: Instrumen t heigh t (m)
– h
t
: T arget heigh t (m)
• Co ordinates from total station:
E
B
=E
A
+D
h
·sin?
h
, N
B
=N
A
+D
h
·cos?
h
where:
2
Page 3


GA TE CE 2026 F orm ula Sheet: Geomatics Engineering - T ra v ers-
ing, T riangulation, and T otal Station
1. T ra v ersing
• Co ordinates of a tra v erse p oin t:
E
B
=E
A
+L·sin?, N
B
=N
A
+L·cos?
where:
– E
A
, N
A
: Easting and northing of p oin t A (m)
– E
B
, N
B
: Easting and northing of p oin t B (m)
– L : Horizon tal distance b et w een A and B (m)
– ? : Be aring of line AB (degrees)
• Latitude and departure:
Latitude =L·cos?, Departure =L·sin?
• Closure error in tra v erse:
e
c
=
v
(
?
Departure)
2
+(
?
Latitude)
2
where:
– e
c
: Linear closure error (m)
• Relativ e error of closure:
REC =
e
c
P
where:
– REC: Relativ e error of closure
– P : T otal p erimeter of tra v erse (m)
• Angular closure error:
?
?
i
= (n-2)·180
?
where:
– ?
i
: In terior angles (degrees)
– n : Num b er of sides in tra v erse
• Bo wditc h’s correction for latitude and departure:
C
Li
= -
?
Latitude·L
i
P
, C
Di
= -
?
Departure·L
i
P
where:
– C
Li
, C
Di
: Corrections for latitude and departure of line i (m)
– L
i
: Length of line i (m)
1
2. T riangulation Surv ey
• Sine rule for triangle:
a
sinA
=
b
sinB
=
c
sinC
where:
– a , b , c : Sides of triangle (m)
– A , B , C : Opp osite angles (degrees)
• Cosine rule:
a
2
=b
2
+c
2
-2bc·cosA
• Area of triangle:
Area =
1
2
bc·sinA
• Base line correction in triangulation:
D
c
=D
m
·
sin?
c
sin?
m
where:
– D
c
: Corrected base line distance (m)
– D
m
: Measured base line distance (m)
– ?
c
, ?
m
: Corrected and measured angles (degrees)
• Strength of figure (triangulation):
S =
?
(dA)
2
n
where:
– S : Strength of figure
– dA : Angular error (seconds)
– n : Num b er of triangles
3. T o tal Station
• Horizon tal distance measuremen t:
D
h
=S ·cos?
v
where:
– D
h
: Horizon tal distance (m)
– S : Slop e distance measured b y total station (m)
– ?
v
: V ertical angle (degrees)
• V ertical heigh t difference:
h =S ·sin?
v
+h
i
-h
t
where:
– h : Heigh t difference (m)
– h
i
: Instrumen t heigh t (m)
– h
t
: T arget heigh t (m)
• Co ordinates from total station:
E
B
=E
A
+D
h
·sin?
h
, N
B
=N
A
+D
h
·cos?
h
where:
2
– E
A
, N
A
: Co ordinates of instrumen t station (m)
– E
B
, N
B
: Co ordinates of target p oin t (m)
– ?
h
: Horizon tal angle (degrees)
• Resection (three-p oin t metho d) for station co ordinates:
tan?
AB
=
E
B
-E
A
N
B
-N
A
, tan?
AC
=
E
C
-E
A
N
C
-N
A
where:
– E
A
, N
A
: Unkno wn co ordinates of station A (m)
– E
B
, N
B
, E
C
, N
C
: Kno wn co ordinates of p oin ts B and C (m)
– ?
AB
, ?
AC
: Observ ed angles from A to B and C
• Precision of total station measuremen t:
s
D
=
v
s
2
s
+(D·s
?
)
2
where:
– s
D
: Standard error in distance (m)
– s
s
: Instrumen t distance accuracy (m)
– s
?
: Angular accuracy (radians)
– D : Measured distance (m)
3