Monday, January 3, 2011

Spherical Co-ordinate System


1. Curvilinear co-ordinates

2. Used for describing positions on a sphere or spheroid

3. Denoted as set (r,  theta  (azimuthal angle), phi (polar angle) ) 

4.  Zenith
In spherical coordinates, the polar angle is the angle measured from the z-axis, denoted phi in this work, and also variously known as the zenith angle and colatitude. ,

 phi=90 degrees-delta ( where delta is the latitude) from the positive z-axis with 0<=phi<=pi

5. It is a logical extension of the usual polar coordinates notation, with theta remaining the angle in the xy-plane and phi becoming the angle out of that plane.

6. The symbol rho is sometimes also used in place of r, and phi and psi instead of phi. 

7. The spherical coordinates (r,theta,phi) are related to the Cartesian coordinates (x,y,z) by
where r in [0,infty)theta in [0,2pi), and phi in [0,pi], and the inverse tangent must be suitably defined to take the correct quadrant of (x,y) into account.

8. In terms of Cartesian coordinates,

9. The line element is
the area element
and the volume element

10.  In geography and astronomy, the elevation and azimuth (or quantities very close to them) are called the latitude and longitude, respectively; and the radial distance is usually replaced by an altitude (measured from a central point or from a sea level surface).
11. Spherical coordinates are useful in analyzing systems that have some degree of symmetry about a point, such as volume integrals inside a sphere, the potential energy field surrounding a concentrated mass or charge, or global weather simulation in a planet's atmosphere. A sphere that has the Cartesian equation x2 + y2 + z2 = c2 has the simple equation r = c in spherical coordinates.

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