MA.JOE@forsythe.stanford.edu (Joe Bayus) (12/30/89)
Three points define a plane...so the smallest sphere to enclose
three points is the same problem as the smallest circle to enclose
three points...
The triangle defined by the three points has three sides (if co-
linear, the problem is the same... the triangle is just very flat!),
which I will label as:
ab = the longest side, length = L
ac = the next-longest side, length = N
bc = the shortest side, length = S
If the triangle is not acute (i.e., one of the angles is = or
greater than 90 degrees), then the radius of the smallest enclo-
sing sphere (circle) is L/2, and the focus (center) is the mid-
point of the longest side.
If the triangle is acute, the smallest enclosing sphere is the
sphere whose midplane circle includes the three points (note that
the colinear situation is covered as a non-acute triangle); the
radius of this sphere is:
R = L / ( 2 * arcsin(S/2L) )
The focus (center) of the sphere lies on the bisector-line of the
longest side, within the triangle, at this distance from the longest
side:
D [distance from longest side] = R * tan ( arcsin(S/2L) )
Note that the maximum radius is L / SQR [square root of] 3 , which
occurs when the triangle is equilateral.
Hope this fuller explanation is helpful... I did not see the
posting which asked the original question.