Formula Sheets: Tacheometry, Plane Table & Curves | Geomatics Engineering (Surveying) - Civil Engineering (CE) PDF Download

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GA TE CE 2026 F orm ula Sheet: Geomatics Engineering - Hori-
zon tal and V ertical Curv es
1. Horizon tal Curv es
• Radius of circular curv e:
R =
180
p·?
·C
where:
– R : Radius of curv e (m)
– ? : Deflection angle (degrees)
– C : Chord length (m)
• Length of circular curv e:
L =
pR?
180
where:
– L : Length of curv e (m)
• T angen t length:
T =R·tan
(
?
2
)
where:
– T : T angen t length (m)
• Long c hord:
LC = 2R·sin
(
?
2
)
where:
– LC : Long c hord length (m)
• External distance:
E =R·
(
sec
(
?
2
)
-1
)
where:
– E : External d istance (m)
• Mid-ordinate:
M =R·
(
1-cos
(
?
2
))
where:
– M : Mid-ordinate distance (m)
• Deflection angle for a c hord:
d =
180·c
2pR
where:
– d : Deflection angle for c hord (degrees)
– c : Chord length (m)
• T ransition curv e length (spiral):
L
s
=
V
3
C ·R
where:
– L
s
: Length of transition curv e (m)
– V : Design sp eed (m/s )
– C : Rate of c hange of radial acceleration (t ypically 0.3–0.6 m/s³)
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Page 2


GA TE CE 2026 F orm ula Sheet: Geomatics Engineering - Hori-
zon tal and V ertical Curv es
1. Horizon tal Curv es
• Radius of circular curv e:
R =
180
p·?
·C
where:
– R : Radius of curv e (m)
– ? : Deflection angle (degrees)
– C : Chord length (m)
• Length of circular curv e:
L =
pR?
180
where:
– L : Length of curv e (m)
• T angen t length:
T =R·tan
(
?
2
)
where:
– T : T angen t length (m)
• Long c hord:
LC = 2R·sin
(
?
2
)
where:
– LC : Long c hord length (m)
• External distance:
E =R·
(
sec
(
?
2
)
-1
)
where:
– E : External d istance (m)
• Mid-ordinate:
M =R·
(
1-cos
(
?
2
))
where:
– M : Mid-ordinate distance (m)
• Deflection angle for a c hord:
d =
180·c
2pR
where:
– d : Deflection angle for c hord (degrees)
– c : Chord length (m)
• T ransition curv e length (spiral):
L
s
=
V
3
C ·R
where:
– L
s
: Length of transition curv e (m)
– V : Design sp eed (m/s )
– C : Rate of c hange of radial acceleration (t ypically 0.3–0.6 m/s³)
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2. V ertical Curv es
• Length of v ertical curv e:
L =
A·S
2
K
where:
– L : Length of v ertical curv e (m)
– A : Algebraic difference in grades (%)
– S : Sigh t distance (m)
– K : Rate of v ertical curv ature (m/%, dep ends on crest or sag)
• Elev ation of p oin t on v ertical curv e:
y =y
0
+g
1
x+
rx
2
2
where:
– y : Elev ation at distance x (m)
– y
0
: Elev ation at b eginning of curv e (m)
– g
1
: Initial grade (%)
– r : Rate of c hange of grade, r =
g2-g1
L
(%/m)
– g
2
: Final grade (%)
– x : Distance from start of curv e (m)
• Lo cation of highest/lo w est p oin t on curv e:
x
m
= -
g
1
r
where:
– x
m
: Distance to highest/lo w est p oin t (m)
• Elev ation at highest/lo w est p oin t:
y
m
=y
0
+g
1
x
m
+
rx
2
m
2
where:
– y
m
: Elev ation at highest/lo w est p oin t (m)
• Sigh t distance on crest curv e:
L =
{
AS
2
2
v
h1+2
v
h2
2
if S <L
2S -
(2
v
h1+2
v
h2)
2
A
if S >L
where:
– h
1
: Heigh t of driv er’s ey e (t ypically 1.2 m )
– h
2
: Heigh t of ob ject (t ypically 0.15 m )
• Sigh t distance on sag curv e (headligh t illumination):
L =
{
AS
2
2h+2Stanß
if S <L
2S -
2h+2Stanß
A
if S >L
where:
– h : Headligh t heigh t (t ypically 0.6 m)
– ß : Headligh t b eam angle (t ypically 1°)
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