This document discusses spherical coordinate systems and their applications. It begins by introducing spherical coordinates and their relationship to Cartesian coordinates. It then discusses how spheroids and spheres are used to model the shape of the Earth. It explains how latitude and longitude are used in geographic coordinate systems to specify locations on the Earth's surface. Finally, it discusses the Military Grid Reference System (MGRS) which provides more precise coordinates than typical GPS and how it divides the Earth into grid squares identified by letter-number combinations.

1. Vector Calculus and PDEs
Mathematics
Presentation

2. APPLICATIONS OF
SPHERICAL CO-ORDINATE
SYSTEM

3. Introduction to Spherical Co-ordinate System
• Spherical coordinates are a system of curvilinear coordinates that are
natural for describing positions on a sphere or spheroid.
The coordinate ρ is the distance from P to the origin.
θ is the angle between the positive x-axis and the line segment
from the origin to Q.
ϕ is the angle between the positive z-axis and the line segment
from the origin to P.

4. Relationship b/w Spherical and Cartesian coordinate
System
x = ρ sinϕ cosθ
y = ρ sin ϕsinθ
z = ρ cosϕ.

5. SPHEROIDS AND SPHERES
• The shape and size of a geographic coordinate system’s surface is defined
by a sphere or spheroid
The assumption
that the earth is a sphere
is possible for small-scale
maps (smaller than
1:5,000,000)
To maintain accuracy
for larger-scale maps (scales
of 1:1,000,000 or larger),
a spheroid is necessary to
represent the shape of the
Earth
A sphere is based on a circle, while a spheroid
(or ellipsoid) is based on an ellipse

6. The shape of an ellipse is defined by two radii. longer radius is the semi major
axis & shorter radius is the semi minor axis.

7. Rotating the ellipse around the semi minor axis creates a spheroid. A spheroid
is also known as an oblate ellipsoid of revolution

8. General Properties of a Spheroid
A spheroid is defined by either the semi major axis, a, and the semi minor axis, b, or by a and the
flattening.
The flattening is the difference in length between the two axes expressed as a fraction
f = (a - b) / a
The spheroid parameters for theWorld Geodetic System of 1984
a = 6378137.0 meters
1/f = 298.257223563
The flattening ranges from zero to one.A flattening value of zero means the two axes are equal,
resulting in a sphere.The flattening of the earth is approximately 0.003353
Another quantity that describes the shape of a spheroid, is the square of the
Eccentricity
𝑒2
=
𝑎2
− 𝑏2
𝑎2

9. Location of a Point
• The spherical coordinate system extends polar coordinates into 3D by using an
angle ϕ for the third coordinate.This gives coordinates (r,θ,ϕ) consisting of
Co-ordinate Name Range Definition
r radius 0≤r<∞ distance from the origin
θ azimuth −π<θ≤π
angle from the x-axis in the x–
y plane
ϕ elevation −π/2<ϕ≤π/2 angle up from the x–y plane
The location of any point in spherical is (r,θ,ϕ)

10. Latitude & Longitude
• A Geographic Coordinate System (GCS) uses a 3D spherical surface to define
locations on the Earth
• GCS uses the azimuth and elevation of the spherical coordinate system
• A point is referenced by its longitude and latitude values
• Longitude and latitude are angles measured from the earth’s center to a
point on the Earth’s surface.

11. Latitude
• Horizontal line
• It is the angular distance, in degrees, minutes, and seconds of a point north
or south of the Equator.
• Often referred to as parallels.
• The coordinate ϕ corresponds to latitude
• On the Earth, latitude is measured as angular distance from the equator.
• In spherical coordinates, latitude is measured as the angular distance from
the North Pole

12. At the North Pole,
Φ=o
At the equator,
Φ=
At the South Pole,
Φ=
𝜋 2
𝜋
Latitude

13. Longitude
• Vertical line
• It is the angular distance in degrees, minutes and seconds of a point, East or
West of the Prime (Greenwich) Meridian
• Often referred to as Meridians
• Each longitude line measures 12,429.9 miles
• The coordinate θ corresponds to longitude
• θ is a measurement of angular distance from the horizontal axis.

14. Longitude
At the North pole
Θ=
At the equator
Θ=0 or 𝜋
At the south pole
Θ= -
𝜋 2
𝜋 2

15. Latitude & Longitude
Distance between Lines
If we divide the circumference of the earth (approximately 25,000
miles) by 360 degrees, the distance on the earth's surface for each one
degree of latitude or longitude is just over 69 miles, or 111 km.

16. GPS
Global Positioning System
• Space-based satellite navigation system
• Developed in 1973 to overcome the limitations of previous navigation systems
• Provides location and time information in all weather conditions, anywhere on or
near the Earth

17. GPS
• Any desired location can be found by entering its coordinates in ourGPS
device.
• We only need to know the latitude and longitude of that location to know
exactly where it is.
• Today GPS is a network on 30 satellites

18. WarTarget Definition
• Wars in the modern world rely more on the precision of technology and
weapons than on manpower
• The Military Grid Reference System (MGRS) is the geo-coordinate
standard used by NATO militaries for locating points on the earth.
• It locates a point more accurately than a commonGPS device.

19. Format of MGRS Grid Squares
• The MGRS divides the surface of the earth into bands of squares of longitude wide and
of latitude tall
• It extends from 80 degrees south latitude to 84 degrees north latitude
• Each square is formatted as a letter-number combination, with numbers increasing from
West to East from prime meridian near the International Date Line
• The letter code runs from C through X, omitting "I" and "O" to avoid confusion with the
digits 1 and 0, from South to North
• The next two letters identify the 10 km grid square number within that grid zone, where the
first letter is the column and the second is the row
• "16TDM" is the grid square for Chicago
o
6 𝑜 8 𝑜

20. Format of MGRS Coordinates
MGRS coordinates within the grid identified are handled as a series of numerical data,
presented as one number of 2, 4, 6, 8 or 10 digits
These are called "n+n" coordinates.A 10-digit MGRS coordinate is a 5+5 coordinate and so on
The most common coordinate standard in MGRS is 4+4, which gives a resolution of 10 meters
The first half of the coordinate counts the number of resolution increments (10 meters for a
4+4 coordinate) east from the southwest corner of the grid square.
The second half gives the number of resolution increments north.

21. MGRS of USA

22. Advantages of MGRS Coordinates
• While the most common use of GPS devices is turn-by-turn navigation
systems, MGRS is meant to be used in conjunction with a separate map.
• MGRS maps have square grids that use the same units of measurement for
east/west as they do for north/south.
• The coordinates translate directly to distances on the ground, making path
and travel time estimates much easier.