This document discusses different coordinate systems used to define locations on Earth. It describes the Everest spheroid, which defines India's geodetic datum. It then explains geographic and geocentric coordinate systems. The geographic system uses latitude and longitude based on a spherical model of Earth. The geocentric system assigns each point an (x,y,z) coordinate using a Cartesian grid with origins at Earth's center. The document provides formulas to convert between the two systems.

1. Nirmal Lama Polytechnic Institute
Prepared by:- Er. Dipendra Kr.Yadav

2.  In the 1830 Sir George Everest , India’s First surveyor general
mapped out the geodetic reference datum for india.
 Everest spheroid is the reference surface for india and
adjacent countries.
 The origin (initial point) of the ellipsoid is kalyanpur in
central india.
 Its semi-major axis (a) = 6377276.346 m
 Its semi-minor axis (b) = 6356075.413 m
 Its flattening (f) = 1/300.8017
 Its minor axis is parallel to the axis of the Earth’s rotation.

3.  Coordinate system can be thought as a system used to identify
locations on a graph or grid
 There are various coordinate systems available to represent
the location of any point.
 However, all of them fall into two broad categories –
curvilinear and rectangular.
 Curvilinear system uses angular measurements from the origin
to describe one’s position, whereas rectangular coordinate
system uses distance measurements from the origin.
 One example of Curvilinear system is Geographic Coordinate
system (GCS) and one example of rectangular Coordinate
system is Cartesian Coordinate System (CCS).

4.  A GCS uses a 3D spherical surface to define locations on the
earth
 The location of any point on the earths surface can be defined
by a reference using latitude and longitude.
 Latitude and longitude are angles measured from the earth’s
centre to a point on the earth’s surface.
 Thus, the origin of the earth’s surface is the centre of the earth.

5.  Lines of latitude are
called “parallels”
 Lines of longitude are
called “meridians”
 The Prime Meridian passes through Greenwich,
England
 The circle with the greatest circumference is known as
the equator and lies equidistant from the two poles.

6. Latitude:- The angle between
the line joining the point to the cen-
tre of the earth and the equatorial
Plane is called Latitude.
Longitude:-The angular distance
between the prime meridian plane
and the meridian plane of the point
measured in equatorial plane is called
Longitude.

7.  In this system , Horizontal lines or
east-west lines are lines of latitude or
Parallels
 vertical lines or north-south lines are
Lines of longitudes.
 These lines encompass the globe and
form a gridded network called
Graticule.
 Lines of latitude lie at right angles to
lines of longitude and run parallel to
one another.

8.  Each line of latitude represents circle
round the globe.
 Each circle has a different circumfere
nce and area depending on where it
lies relative to the two poles.
 The origin of the graticule (0,0) is defined by where the
equator and prime meridian intersect.
 Latitude and longitude values are traditionally measured either
in decimal degrees or in degrees , minutes and seconds.

9. 2D Rectangular coordinate system
:- (Cartesian coordinate system)
 The CCS is used to uniquely determine each
point in the plane through two numbers called
the x-coordinate and the y-coordinate of the
point.
 The horizontal axis is labelled x and the
vertical axis is labelled y.
 The point of intersection where the axis meet is
called the origin and is normally labelled O.
 The x and y axes define a plane that is referred
to as the xy-plane.

10. 2D Rectangular coordinate system
:- (Cartesian coordinate system)
 To specify a particular point on CCS, we
indicate the x unit first (abscissa) followed by
the y unit (ordinate) in the form (x,y) an ordered
pair.
 The intersection of the two axes creates four
regions, called Quadrants
 The quadrants are labelled anti-clockwise
starting from the north-east quadrant.
 In the 1st quad both coordinates are +ve; in the
2nd quad x-coordinates are –ve and y-
coordinates are +ve; in the 3rd quad both
coordinates are –ve; and in the 4th quad x-
coordinates are +ve and y-coordinates –ve.

11. 3D Rectangular Coordinate
System:- (Geocentric Coordinate
system)
An X,Y,Z cartesian coordinate system may be
associated with the datum such that;
 the positive x-axis lies in the equatorial plane
and passes through 0 degree longitude,
 the positive y-axis lies in the equatorial plane
and passes through 90 degree East longitude,
 the positive z-axis is coincide to the earth’s
rotation axis and passes through 90 degree
north latitude
 Its origin is the center of the earth.

12. →

13. 1. Geographical to Geocentric coordinate system:

14. • Latitude, , and Longitude, , in terms of
Geographic Coordinate System may be expressed in
terms of a geocentric (earth-centered) Cartesian
coordinate system (X, Y
, Z) with
• the Z axis corresponding with the Polar axis positive
northwards,
• the X axis through the intersection of the Greenwich
meridian and equator, and
• the Y axis through the intersection of the equator with
longitude 900E.

15. • If the earth's spheroidal semi major axis is a, semi minor axis b, and
inverse flattening 1/f, then
Where,
• N is the prime vertical radius of curvature at latitude  and is
equal to
•  and  are respectively the latitude and longitude of the point
• h is height above the ellipsoid, and
• e is the eccentricity of the ellipsoid.

16. 2. Geocentric to Geographical Coordinate system:-
 For the reverse conversion, Cartesian coordinates in the geocentric
coordinate system may then be converted to the geographical coordinates in
terms of geographic coordinate system by using the following steps:
 Step 1: Compute D as;
D= 𝑋2 + 𝑌2
 Step 2: Compute 𝑝 as;
𝑝= 2 𝑡𝑎𝑛−1(
𝐷−𝑋
𝑌
)
 Step 3: Calculate approximate latitude 0
as;
0
= 𝑡𝑎𝑛−1[
𝑍
𝐷(1−𝑒2)
]

17.  Step 4: Calculate the approximate radius of the prime
vertical N , using 0
from step 3 and equation;
N =
𝑎
1−𝑒2𝑠𝑖𝑛20
 Step 5: Calculate an improved value for the latitude from
 = 𝑡𝑎𝑛−1(
𝑍+𝑒2𝑁 𝑠𝑖𝑛0
𝐷
)
 Step 6: Repeat the computations of step 4 and 5 until the
change in  between iterations becomes negligible. This final
value 𝑝
is the latitude of the station.

18.  Step 7: Use the following formulas to compute the geodetic
height of the station.
for latitudes less than 45°, use;
h=
𝐷
𝑐𝑜𝑠𝑝
− N
for latitudes greater than 45° ,use;
h= [
𝑍
𝑠𝑖𝑛𝑝
] – N(1-𝑒2)