On 10/21/09, rcs@xmission.com <rcs@xmission.com> wrote:
Two nits ...
(a) There's an edge case in the intersecting-plane test: The two circles could lie in the same plane. In this case, they might intersect (or not), or be tangent, but they can't be linked. This isn't a theory problem, but it might be a programming problem: Computing the intersection of two nearly-parallel planes is a numerical challenge. We could have two circles that have no chance of linkage, because they are moderately far apart, but still cause a "can't tell" in the linked-p subroutine.
Sure --- I've implicitly been considering things "in general". The planes might be coincident, or distinct but parallel. There are also limiting cases where "circles" and "spheres" include lines and planes resp, or degenerate to points or (inversive) infinity. The circles might have one or two common points, be tangent, coincide ...
(b) Fred's separability test seems to work if we allow spheres as separators (instead of just allowing planes).
Sketch proof of (b) --- which was also suggested privately by Dan --- If neither circle meets the plane of the other, any other plane containing the line L where their planes meet will separate them. Otherwise, consider the coaxal system of circles through the pair of points (in general) where one circle A meets L. Choose a circle C of this system which does not meet the other circle B --- this is possible provided A and B do not both meet L in pairs of points which overlap. Construct the unique sphere containing A and C, then offset its radius down or up, depending upon whether B lies within or outside C. The new sphere S separates A and B. Conversely, if A and B do both meet L in pairs of points which overlap, any sphere separating A and B would have to meet the plane of B in circle C, with C disjoint from B, and C separating B from both points of A --- which I'll take to be evidently impossible --- though if put up against a wall, I'd have to admit to a certain vagueness about exactly what my axioms might be here ... Fred Lunnon