Computing circular separability

Abstract

Two sets of planar points S1 and S2 are circularly separable i f there is a circle that encloses S1 but excludes S2. We show that deciding whether two sets are circularly separable can be accomplished in O(n) time via Megiddo's linear programming algorithm. We also show that a smallest separating circle can be found in O(n) time, and largest separating circles can be found in O(n1ogn) time. Finally we establish that all these results are optimal.

Robust and accurate vectorization of line drawings

“…From their result, it immediately follows that the preimage of a discrete segment can be represented using at most two segments and four half-lines. For discrete circular arcs, we proved in [17] that the preimage could only have three possible shapes, using a lemma from O'Rourke et al [31]. These shapes are presented in Fig.…”

Computing Largest Circles Separating Two Sets of Segments

“…[20] We proceed by reduction to the maximum gap problem for which Ω(n log n) is a lower bound in the linear decision-tree model of computation. [16] Let X = {x 1 , x 2 , .…”

“…Bhattacharya [2] computed in O(n log n) time the set of centers of all circles that separate two given point sets. O'Rourke, Kosaraju and Megiddo [20] presented optimal algorithms for the circular separability of point sets. They determine the circular separability of two given point sets and find the smallest separating circle in linear time and all the largest separating circles in O(n log n) time.…”

Optimal Consensus Set for nD Fixed Width Annulus Fitting

“…Those are however not adapted for annulus fitting. The circle fitting method proposed by O'Rourke et al [13,16] that transforms a circle separation problem into a plane separability problem, is also not well suited because the fixed width of the digital circles translates into non fixed vertical widths for the planes. In this case, the problem is complicated (See [14] for some ideas on how to handle this difficulty).…”