Spherical Co-ordinate system (Applications)

Applications and Introduction of Spherical Coordinate system

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 sin ϕ cos θ y = ρ sin ϕ sin θ z = ρ cos ϕ .

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 the World 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  

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 3 D 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. O ften 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 =     Latitude

13. Longitude Vertical line It is the angular distance in degrees, minutes and seconds of a point, E ast 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 At the south pole Θ = -      

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 D eveloped 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 our GPS 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. War Target 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 common GPS 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 S outh 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 " 16 TDM" is the grid square for Chicago    

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 T hese 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.