Pythagorean Triples

Three positive integers a, b, and c are called a Pythagorean triple if
they are the lengths of three sides of a right triangle.  Pythagoras'
theorem says that a^2 + b^2 = c^2 (assuming c is the length of the
hypoteneuse).  

Can you describe all Pythagorean triples?  That is, can you give a
method or algorithm for generating all positive integer solutions to the
equation x^2 + y^2 = z^2 ?

You will quickly see that this question is the same as the following:
find all points on the unit circle (the set of points in the x,y plane
which are solutions to the equation x^2 + y^2 = 1 ) with both the x and
y coordinate rational (a rational number is a number that can be
expressed as a fraction m/n of integers, n not zero).

Can you describe all the rational points on the 2-sphere?  The n-sphere?