Unit Sphere

Date: 01/21/2002 at 16:43:23
From: Jordan Kratzer
Subject: The "Unit sphere"

I know that there is a unit circle and that it has to do with sine 
and cosine and the placement of certain points on that circle. Now my 
question is this, is there such thing as a "unit sphere" that has to 
do with other such trigonometric functions and the placement of points 
on said sphere?

Date: 01/23/2002 at 11:20:15
From: Doctor Douglas
Subject: Re: The "Unit sphere"

Hi, Jordan,

Thanks for submitting your question to the Math Forum.

Yes, the unit sphere in three-dimensional space consists of all points
(x,y,z) such that the distance from the origin r satisfies the 
equation 

   r^2 = x^2 + y^2 + z^2 = 1

Because the unit sphere is a surface, and not a circle, two angles are
required to specify a given point on the sphere. For example, a common
set of such angles are the "azimuthal angle" (you can think of lines 
of longitude on a globe) and the "polar angle," the angle between a 
vertical line drawn through the line x = y = 0 and the line that 
connects the point of interest to the origin. In the globe context, 
the polar angle is similar to latitude, although the polar angle is 
zero at the north pole and 90 degrees or pi/2 radians at the equator.

If the azimuthal angle is q and the polar angle is p, then these 
angles are related to the coordinates (x,y,z) by

   z = cos(p)                      p = arctan[sqrt(x^2+y^2)/z]
   y = sin(p)sin(q)                q = arctan[y/x]
   x = sin(p)cos(q)

These formulas assume that p and q are measured in radians.  

I hope this helps.

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/