MA.JOE@forsythe.stanford.edu (Joe Bayus) (01/03/90)
Major error on the formulas for smallest sphere enclosing
three points which form an acute triangle....sorry about that,
but brain death comes in many forms...real formulae are as follows:
----
Definitions:
Acute triangle abc
ab = longest side
L = Length of ab
ac = next-longest side
N = Length of ac
bc = shortest side
S = Length of bc
f = focus (center) of sphere
l = center-point of longest side
D = Distance to focus from longest side (length of line lf)
R = Radius of sphere (length of lines fa, fb, and fc)
E = Distance from vertex acb (opposite longest side) of inter-
section with shortest side of perpendicular drawn from
vertex cab
G = Length of perpendicular from vertex cab to intersection with
shortest side
Formulae:
E = ( S**2 - L**2 + N**2 ) / ( 2*S )
G = SQR ( N**2 - E**2 )
D = ( L*E ) / ( 2*G )
R = D * N / E
I realize that this is only fundamental geometry, and
that the expressions require further work for (x,y) coordinate
usage in a graphics program, but it's a start... I would be inter-
ested in hearing about any refinements to these formulae, trans-
forms or reductions, etc.