









Good I had a bad feeling about this problem because of a special case
with no solution that worried me.

Four coplanar points in the shape of a square have no unique sphere 
that they are on the surface of.
Similarly 4 colinear point have no finite sized sphere that they are on the
surface of.

These algorithms being geometrical designed rather than algebraically design
meet these problems neatly.

When determining which plane the 3 points are on if they are colinear
the algorithm should afil or return infinite R.
When intersecting the two lines there are 2 possibilities
they are the same line  (the 4 points were on a planar circle)
they are different lines but parallel. There is a sphere of in radius.

This last case can be achieved with 3 colinier points and any 4th point
by taking the 4th point and pairs of the first 3 parallel lines will be produced

it can also be achieved by

If all 4 points are coplanar but are not on one circle. 

It seems to me that the algorithm only fails when the 4 points are coplanar.
The algorithm always fails when the points are coplanar.
(4 points being colinear => coplanar)

Testing if the 4th point is coplanar when the plane of the first 3 points
has been found is trivial.

