Vector Calculus and PDEs

Mathematics Presentation

APPLICATIONS OF SPHERICAL CO-ORDINATE

SYSTEM

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.

Relationship b/w Spherical and Cartesian coordinate System

x = ρ sinϕ cosθy = ρ sin ϕsinθz = ρ cosϕ.

SPHEROIDS AND SPHERES

• The shape and size of a geographic coordinate system’s surface is defined by a sphere or spheroid

The assumptionthat the earth is a sphere is possible for small-scalemaps (smaller than 1:5,000,000)

To maintain accuracyfor larger-scale maps (scales of 1:1,000,000 or larger),a spheroid is necessary to represent the shape of theEarth

A sphere is based on a circle, while a spheroid (or ellipsoid) is based on an ellipse

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

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

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 theflattening.

The flattening is the difference in length between the two axes expressed as a fractionf = (a - b) / a

The spheroid parameters for the World Geodetic System of 1984a = 6378137.0 meters1/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 theEccentricity

𝑒2 =𝑎2 − 𝑏2

𝑎2

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≤πangle 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)

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.

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

At the North Pole,Φ=o

At the equator,Φ=

At the South Pole,Φ=

𝜋 2

𝜋

Latitude

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.

Longitude

At the North poleΘ=

At the equatorΘ=0 or 𝜋

At the south poleΘ= -

𝜋 2

𝜋 2

Latitude & Longitude

Distance between LinesIf 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.

GPSGlobal 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

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

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.

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

6𝑜 8𝑜

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.

MGRS of USA

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.