
Off the top of my head, I might try:

Given: p_1, p_2, p_3, p_4

Find:  p_c (center of sphere determined by p_1, ..., p_4), 
       dist(p_c, p_i) (radius)

p_c is the same distance from our four points, so
dist(p_c,p_1) = dist(p_c,p_2) = dist(p_c,p_3) = dist(p_c,p_4) 

Of course, we can square the whole thing to get rid of square roots:
distsq(p_c,p_1) = distsq(p_c,p_2) = distsq(p_c,p_3) = distsq(p_c,p_4) 

Plug in the variables into the distance formula, simplify, and the 
x^2_c, y^2_c, and z^2_c terms cancel out, leaving you with three 
linearly independent equations and three unknowns (x_c, y_c, z_c).  
Solve using your favorite method.  :)


All the best,

- John :)

-------------------------------------------------------------------------------
Dr. John S. Eickemeyer                 ::     "The Lord God is subtle,
Information Technology Institute      ::::     but malicious He is not."
National Computer Board, Singapore     ::
Email: johne@iti.gov.sg                ::                     - Albert Einstein