TY - GEN

T1 - An approximate arrangement algorithm for semi-algebraic curves

AU - Milenkovic, Victor

AU - Sacks, Elisha

PY - 2006

Y1 - 2006

N2 - We present an arrangement algorithm for plane curves. The inputs are (1) continuous, compact, x-monotone curves and (2) a module that computes approximate crossing points of these curves. There are no general position requirements. We assume that the crossing module output is e accurate, but allow it to be inconsistent, meaning that three curves are in cyclic y order over an x interval. The curves are swept with a vertical line using the crossing module to compute and process sweep events. When the sweep detects an inconsistency, the algorithm breaks the cycle to obtain a linear order. We prove correctness in a realistic computational model of the crossing module. The number of vertices in the output is V = 2n + N + min(3kn, n2/2) and the running time is O(Vlogn) for n curves with N crossings and k inconsistencies. The output arrangement is realizable by curves that are 0(ε + knε) close to the input curves, except in kn neighborhoods of the curve tails. The accuracy can be guaranteed everywhere by adding tiny horizontal extensions to the segment tails, but without the running time bound. An implementation is described for semi-algebraic curves based on a numerical equation solver. Experiments show that the extensions only slightly increase the running time and have little effect on the error. On challenging data sets, the number of inconsistencies is at most 3N, the output accuracy is close to ε, and the running time is close to that of the standard, non-robust floating point sweep.

AB - We present an arrangement algorithm for plane curves. The inputs are (1) continuous, compact, x-monotone curves and (2) a module that computes approximate crossing points of these curves. There are no general position requirements. We assume that the crossing module output is e accurate, but allow it to be inconsistent, meaning that three curves are in cyclic y order over an x interval. The curves are swept with a vertical line using the crossing module to compute and process sweep events. When the sweep detects an inconsistency, the algorithm breaks the cycle to obtain a linear order. We prove correctness in a realistic computational model of the crossing module. The number of vertices in the output is V = 2n + N + min(3kn, n2/2) and the running time is O(Vlogn) for n curves with N crossings and k inconsistencies. The output arrangement is realizable by curves that are 0(ε + knε) close to the input curves, except in kn neighborhoods of the curve tails. The accuracy can be guaranteed everywhere by adding tiny horizontal extensions to the segment tails, but without the running time bound. An implementation is described for semi-algebraic curves based on a numerical equation solver. Experiments show that the extensions only slightly increase the running time and have little effect on the error. On challenging data sets, the number of inconsistencies is at most 3N, the output accuracy is close to ε, and the running time is close to that of the standard, non-robust floating point sweep.

KW - Arrangements

KW - Robust computational geometry

UR - http://www.scopus.com/inward/record.url?scp=33748080785&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33748080785&partnerID=8YFLogxK

U2 - 10.1145/1137856.1137892

DO - 10.1145/1137856.1137892

M3 - Conference contribution

AN - SCOPUS:33748080785

SN - 1595933409

SN - 9781595933409

T3 - Proceedings of the Annual Symposium on Computational Geometry

SP - 237

EP - 245

BT - Proceedings of the Twenty-Second Annual Symposium on Computational Geometry 2006, SCG'06

PB - Association for Computing Machinery (ACM)

T2 - 22nd Annual Symposium on Computational Geometry 2006, SCG'06

Y2 - 5 June 2006 through 7 June 2006

ER -