Abstract:Two spheres with centers p and q and signed radii r and s are said to be in contact if |p − q| 2 = (r − s) 2 . Using Lie's line-sphere correspondence, we show that if F is a field in which −1 is not a square, then there is an isomorphism between the set of spheres in F 3 and the set of lines in a suitably constructed Heisenberg group that is embedded in (F [i]) 3 ; under this isomorphism, contact between spheres translates to incidences between lines.In the past decade there has been significant progress in un… Show more
“…The crucial step is to establish the following incidence result for points and curves in R 3 . The following proof is closely modeled on the arguments in [29] by the second author, which are in turn based on arguments of Sharir and Zlydenko [19].…”
We prove that if P is a set of n points in C 2 , then either the points in P determine Ω(n 1−ε ) complex distances, or P is contained in a line with slope ±i. If the latter occurs then each pair of points in P have complex distance 0.
“…The crucial step is to establish the following incidence result for points and curves in R 3 . The following proof is closely modeled on the arguments in [29] by the second author, which are in turn based on arguments of Sharir and Zlydenko [19].…”
We prove that if P is a set of n points in C 2 , then either the points in P determine Ω(n 1−ε ) complex distances, or P is contained in a line with slope ±i. If the latter occurs then each pair of points in P have complex distance 0.
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