Spherical Coordinates

The spherical coordinates of a point P are defined to be ( r,f,q) , where r is the distance from P to the origin, f is the angle formed by the z-axis and the ray from the origin to P, and q is the polar angle from polar coordinates.

Specifically, the cartesian coordinates ( x,y,z) of a point P are related to the spherical coordinates ( r,f,q) of P through two right triangles. Relationships among x, y, q, and the polar distance r are contained in the familiar polar coordinate triangle. Relationships among r, z, r, and f are conveyed by a second right triangle. These 2 triangles are at the heart of spherical coordinates. For example, the triangles imply the relationships

x = rcos( q) z = rcos( f) y = rsin( q) r = rsin( f)

(1)

so that if we eliminate r using the fact that r = r sin( f) , we obtain

x = rsin( f) cos( q) ,    y = rsin( f) sin( q) ,    z = rcos( f)

(2)

This is the coordinate transformation that maps spherical coordinates into Cartesian coordinates.      

EXAMPLE 3    Transform the point ( 4,p/3,p/2) from spherical into Cartesian coordinates.       

Solution: The transformation (2) implies that

x = 4sin æ è  p 3 ö ø cos æ è  p 2 ö ø = 4·  Ö3 2 ·0 = 0 y = 4sin æ è  p 3 ö ø sin æ è  p 2 ö ø = 4·  Ö3 2 ·1 = 2Ö3 z = 4cos æ è  p 3 ö ø = 2

Thus, ( 4,p/3,p/2) in spherical coordinates is the same point as ( 0, 2Ö3, 2) in Cartesian coordinates.

       

In spherical coordinates, r = rsin( f) and z = rcos( f) , so that the polar x²+y² = r² becomes

x2+y2 = r2sin2( f)

Moreover, r²+z² = r², so that we have the identity

x2+y2+z2 = r2

(3)

Thus, if R is constant, then r = R is a sphere of radius R centered at the origin. In addition, we usually restrict q to [ 0,2p] and f to [ 0,p] so that the sphere is covered only once. Restricting f and q to smaller intervals yields smaller sections of a sphere.      

EXAMPLE 4    What section of the sphere r = 1 is given by f in [ 0,p/2] , q in [ 0,2p] ?

Solution: Since f = p/2 is the xy-plane, the set of points r = 1, f in [ 0,p/2] , q in [ 0,2p] is the part of the unit sphere above the xy-plane-i.e., the upper hemisphere.

Similarly, f = k for k constant is a cone with sides at angle k to the vertical, and q = c for c constant is a vertical plane of the form y = tan( c) x

LiveGraphics3d Applet                 LiveGraphics3d Applet f = k for k constant q = c for c constant