Formula Sheets: Fundamental Concepts of Surveying | Geomatics Engineering (Surveying) - Civil Engineering (CE) PDF Download

 Page 1


GA TE CE 2026 F orm ula Sheet: Geomatics Engineering - Sur-
v eying and Mapping
1. Principles of Surv eying
• Distance b y pacing:
D =n·s
where:
– D : Distance measured (m)
– n : Num b er of paces
– s : A v erage pace length (m)
• Slop e correction for measured distance:
C
s
=
h
2
2L
or C
s
=L(1-cos?)
where:
– C
s
: Slop e correction (m)
– h : V ertical heigh t difference (m)
– L : Measured slop e distance (m)
– ? : Slop e angle (degrees)
• Corrected horizon tal distance:
D
h
=L-C
s
or D
h
=Lcos?
where:
– D
h
: Horizon tal distance (m)
• Bearing of a line:
? = tan
-1
(
?E
?N
)
where:
– ? : Bearing angle (degrees)
– ?E : Easting difference (m)
– ?N : Northing difference (m)
2. Errors and Their A djustmen t
• Correction for tap e length:
C
t
=
L
m
(l
a
-l
n
)
l
n
where:
– C
t
: Correction for tap e length (m)
– L
m
: Measured length (m)
– l
a
: A ctual tap e length (m)
– l
n
: Nominal tap e length (m)
• T emp erature correction:
C
temp
=aL
m
(T
m
-T
s
)
where:
1
Page 2


GA TE CE 2026 F orm ula Sheet: Geomatics Engineering - Sur-
v eying and Mapping
1. Principles of Surv eying
• Distance b y pacing:
D =n·s
where:
– D : Distance measured (m)
– n : Num b er of paces
– s : A v erage pace length (m)
• Slop e correction for measured distance:
C
s
=
h
2
2L
or C
s
=L(1-cos?)
where:
– C
s
: Slop e correction (m)
– h : V ertical heigh t difference (m)
– L : Measured slop e distance (m)
– ? : Slop e angle (degrees)
• Corrected horizon tal distance:
D
h
=L-C
s
or D
h
=Lcos?
where:
– D
h
: Horizon tal distance (m)
• Bearing of a line:
? = tan
-1
(
?E
?N
)
where:
– ? : Bearing angle (degrees)
– ?E : Easting difference (m)
– ?N : Northing difference (m)
2. Errors and Their A djustmen t
• Correction for tap e length:
C
t
=
L
m
(l
a
-l
n
)
l
n
where:
– C
t
: Correction for tap e length (m)
– L
m
: Measured length (m)
– l
a
: A ctual tap e length (m)
– l
n
: Nominal tap e length (m)
• T emp erature correction:
C
temp
=aL
m
(T
m
-T
s
)
where:
1
– C
temp
: T emp erature correction (m)
– a : Co e?icien t of thermal expansion (t ypically 1.15×10
-5
/°C)
– T
m
: Measured temp erature (°C)
– T
s
: Standard temp erature (°C)
• Least squares adjustmen t (error):
v
i
= ˆ x
i
-x
i
where:
– v
i
: Residual error
– ˆ x
i
: A djusted observ ation
– x
i
: Observ ed v alue
• Standard error of mean:
s
¯ x
=
s
v
n
where:
– s
¯ x
: Standard error of mean
– s : Standard deviation of observ ations
– n : Num b er of observ ations
• Closure error in tra v erse:
e
c
=
v
(
?
?E)
2
+(
?
?N)
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)
3. Maps - Scale
• Map scale (represen tativ e fraction):
S =
D
m
D
g
where:
– S : Scale (dimensionless)
– D
m
: Distance on map (m)
– D
g
: Corresp onding distance on ground (m)
• Area scale:
S
a
=S
2
=
(
D
m
D
g
)
2
where:
– S
a
: Area scale
• Scale factor for distorted maps:
SF =
L
actual
L
nominal
where:
– SF : Scale factor
– L
actual
, L
nominal
: A ctual and nominal lengths (m)
2
Page 3


GA TE CE 2026 F orm ula Sheet: Geomatics Engineering - Sur-
v eying and Mapping
1. Principles of Surv eying
• Distance b y pacing:
D =n·s
where:
– D : Distance measured (m)
– n : Num b er of paces
– s : A v erage pace length (m)
• Slop e correction for measured distance:
C
s
=
h
2
2L
or C
s
=L(1-cos?)
where:
– C
s
: Slop e correction (m)
– h : V ertical heigh t difference (m)
– L : Measured slop e distance (m)
– ? : Slop e angle (degrees)
• Corrected horizon tal distance:
D
h
=L-C
s
or D
h
=Lcos?
where:
– D
h
: Horizon tal distance (m)
• Bearing of a line:
? = tan
-1
(
?E
?N
)
where:
– ? : Bearing angle (degrees)
– ?E : Easting difference (m)
– ?N : Northing difference (m)
2. Errors and Their A djustmen t
• Correction for tap e length:
C
t
=
L
m
(l
a
-l
n
)
l
n
where:
– C
t
: Correction for tap e length (m)
– L
m
: Measured length (m)
– l
a
: A ctual tap e length (m)
– l
n
: Nominal tap e length (m)
• T emp erature correction:
C
temp
=aL
m
(T
m
-T
s
)
where:
1
– C
temp
: T emp erature correction (m)
– a : Co e?icien t of thermal expansion (t ypically 1.15×10
-5
/°C)
– T
m
: Measured temp erature (°C)
– T
s
: Standard temp erature (°C)
• Least squares adjustmen t (error):
v
i
= ˆ x
i
-x
i
where:
– v
i
: Residual error
– ˆ x
i
: A djusted observ ation
– x
i
: Observ ed v alue
• Standard error of mean:
s
¯ x
=
s
v
n
where:
– s
¯ x
: Standard error of mean
– s : Standard deviation of observ ations
– n : Num b er of observ ations
• Closure error in tra v erse:
e
c
=
v
(
?
?E)
2
+(
?
?N)
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)
3. Maps - Scale
• Map scale (represen tativ e fraction):
S =
D
m
D
g
where:
– S : Scale (dimensionless)
– D
m
: Distance on map (m)
– D
g
: Corresp onding distance on ground (m)
• Area scale:
S
a
=S
2
=
(
D
m
D
g
)
2
where:
– S
a
: Area scale
• Scale factor for distorted maps:
SF =
L
actual
L
nominal
where:
– SF : Scale factor
– L
actual
, L
nominal
: A ctual and nominal lengths (m)
2
4. Maps - Co ordinate System
• Con v ersion from geographic to Cartesian co ordinates (simplified):
x =R·?cos?, y =R·?
where:
– x , y : Cartesian co ordinates (m)
– R : Earth’s radius (appro ximately 6,371,000 m )
– ? : Longitude (radian s)
– ? : Latitude (radians)
• Distance b et w een t w o p oin ts (plane co ordinates):
D =
v
(?E)
2
+(?N)
2
where:
– D : Distance (m)
– ?E : Difference in easting (m)
– ?N : Difference in northing (m)
• Azim uth from co ordinates:
a = tan
-1
(
?E
?N
)
where:
– a : Azim uth angle (degrees)
3