Then you will have cases where you *cannot* find a closest point on the
circumference.  How do you define that case where the circles' planes may be
perpendicular?  There is no solution then for one of the circles, because *all*
points on one will be equidistant to the most extended point on the other.  And
what about coplanar circles with perpendicular centers, such as occurring on a
cone or cylinder?  Then *neither* will have a closest point!
    You can do some simple geometric tests to throw those cases out, then
proceed to construct a truncated conic section that contains the two circles as
limits.  That is, as long as the circles to not intersect!  Now you open a new
can of worms.
    If you have the circles on a single plane, the solution is obvious- draw a
line from center to center as long as they are no concentric.  If they are
non-coplanar and non-intersecting, construct a conic section using the two
circles as caps, and the shortest path on the surface will yield your answer.
    For more than two dimensions, the ways this process can go wrong are
numerous.  For 4 dimensions and up, I can visualize that there could be more
than one solution.  In 3 dimensions, you can always treat any two non-coplanar
planes as you might any two non-colinear lines in a plane.  They will always
have an angle of intersection, which can be seen as being between 0 and 90
degrees.  In some such cases, you might have one closest point on one circle and
*two* closest points on the second- think of one circle passing through the line
of intersection of the two planes.
    This is a mathematically non-trivial problem.

Cheers!

Chip Shults
My robotics, space and CGI web page - http://home.cfl.rr.com/aichip

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