1. January 7, 2021 Lecture Notes 1
COORDINATE SYSTEM
Introduction:
Coordinates: Set of numbers representing the
position of points in space/map with respect to
a certain coordinate system.
Coordinate system: A system whereby the
location/position of points are referred to.
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Characteristics of coordinate system
They have reference axes
The reference axes meet at a point
known as origin
The references axes are in a fixed
known orientation.
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Coordinate system used in
Land surveying
The coordinate system used in surveying
are;
2D-Plane coordinate system
3D-Spatial coordinate system
2D-Plane coordinate is divided into two types
namely;
2D-cartesian coordinate system
2D-polar coordinate system
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2D-cartesian coordinate system
A coordinate system where by a point is
referred to by two offsets from two mutually
perpendicular axes.
In surveying, we have primary axis (X- axis)
that points towards the North and secondary
axis (Y-axis) that points towards the East.
5. January 7, 2021 Lecture Notes 5
2D-cartesian coordinate system cont….
2D-cartesian
Coord. system
Origin E (y)
N (x)
P (NP, EP)
∆NP
∆EP
The intersection of the X- and Y-axis forms the origin.
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2D-cartesian coordinate system cont….
Also known as “Rectangular” coordinates
Locates “XY” values on a flat plane
Consist of four quadrants: I, II, III, IV
Origin is at the center
Distances measured “Horizontally” from the Origin
are “Easting” values
Right of Origin are positive
Left of Origin are negative
Distances measured “Vertically” from the Origin
are “Northing” values
Above Origin are positive
Below Origin are negative
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2D-cartesian coordinate system cont….
Specifying “XY” values or “Coordinates
Both values positive = Quadrant “I”
Both values negative = Quadrant “III”
“X”value negative,“Y” value positive = Quadrant “II”
“X” value positive,“Y”value negative = Quadrant “IV”
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2D-cartesian coordinate system cont….
Origin
Quadrant III Quadrant II
Quadrant IV Quadrant I
N (x)
E (y)
“X” positive, “Y” positive
“X” Negative, “Y” positive
“X” positive, “Y” Negative
“X” Negative, “Y” Negative
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2D-polar coordinate system
A system that involve the distance
SP from the origin to the point
concerned and the angle δ
between a fixed (or zero) direction
and the direction to the point.
A point is referred to by a distance
and the angle from a known fixed
control point.
The angle called azimuth or
bearing is measured in a
clockwise direction (i.e. from the
North direction
N P(SP, δP)
O
δ
δ is called Azimuth
or bearing.
SP
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3D-Spatial coordinate system
These are classified as;
3D-Geographical coordinate system: A point is
referred to on the surface of the earth by its latitude,
longitude and height (Φ,λ, h)
3D-cartesian coordinate system: A point is referred
to by its X, Y, Z coordinates measured with respect
to 3-mutually perpendicular axes with the origin at
the center of the earth.
Astronomical coordinate system: A position of an
object in space is defined by its Right ascension
and declination, or by Hour angle and altitude.
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3D-Geographical coordinate system
The ellipsoidal height (h) of a point is the vertical
distance of the point above the ellipsoid. It is
measured in distance units along the ellipsoidal
normal from the point to the ellipsoid surface
The latitude are measured with respect to the equator
at the center of the earth to observer parallel.
Longitude are measured along the equatorial plane
from the Greenwich meridian to the observer's
meridian.
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3D-Geographical coordinate system cont….
The latitude (Φ) and longitude (λ) angles and
the ellipsoidal height (h) represent the 3D geographic coordinate
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3D-cartesian coordinate system
The system has its origin at the mass-centre of the
Earth with the X- and Y-axes in the plane of the
equator.
The X-axis passes through the meridian of Greenwich,
and the Z-axis coincides with the Earth's axis of
rotation.
The three axes are mutually orthogonal and form a
right-handed system.
All distances are measured from the origin.
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3D-cartesian coordinate system cont….
Geocentric coordinates can be used to define a position
on the surface of the Earth (point P in figure above).
CIO: Conversion International of the Origin
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Relationship between 2D-cartesian
and 2D-Polar coordinate systems
Let consider the
plane below.
P (NP, EP)∆EP
∆NP
E
N
δP SP
∆NP = SP CosδP
∆EP = SP SinδP
SP = {∆NP
2 + ∆EP
2}
TanδP = ∆EP/∆NP
Taking the ratio from the two
equation above, we get
O )(tan 1
N
E
2D-Plane coordinate system
Let consider the
plane below.
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2D-cartesian and 2D-Polar
coordinate systems cont…
The system used to define a direction is called the
whole circle bearing system (WCB). A bearing is
the direction measured clockwise from 00 full circle
to 3600.
From the figure above, if is the forward bearing,
then the back bearing is given as;
Back bearing = forward bearing + 1800 or
= forward bearing – 1800
If FB is > 1800, then
Back bearing = Forward bearing – 1800
If the FB is < 1800, then
Back bearing = Forward bearing + 1800
17. Quadrants cont…
AB
AB
AB
N
E1
tan
AB
AB
AB
E
N1
tan90
AB
AB
AB
N
E1
tan180
1st Quadrant 2nd Quadrant
4th Quadrant3rd Quadrant
AB
AB
AB
E
N1
tan270
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Fundamental Calculations in Surveying
There are two most fundamental calculations
in surveying namely;
Join computations
Polar computations
When computing, the positive sign and
negative sign provided in quadrants should be
put into consideration for increasing the point.
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Join computation method
This involves computing the horizontal
distance (S) and bearing (δ) from the
difference in coordinates (∆E, ∆N) of a
line.
Example
Given the point A(-275342.45,14758.02)
and B(322446.10,-92751.99). Calculate
distance and bearing of AB.
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Join computation method cont…
Solution
First calculate the increments
The bearing is in the fourth quadrant, then
EEE
0
270
E
1
tan
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Join computation method cont…
STN N E
A -275342.45 14758.02
B 322446.10 -92751.99
∆N,∆E 597788.55 -107510.01
25.607379S
''1648349 '0
secSCheck
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Polar computation method
This involves calculating ∆E and ∆N
given the horizontal distance (S) and
Bearing (δ) of the line.
Given :i) coordinate of a control point A &
ii) The bearing(δAP ) and
distance (SAP ) to the unknown point P )
Required: coordinate of the unknown
point , P i.e (NP,EP).
Consider the figure below
23. January 7, 2021 Lecture Notes 23
Polar computation method cont…
E(y)
N(x)
P (NP, EP)
A (NA, EA)
SAP
C ∆EP
∆NP
δAP
2D-Plane coordinate system
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Polar computation method cont…
Then, unknown point P can be calculate as
cosSAC
sinSEPC
sinSEE
cosSNN
