An Inconsistency Sensitive Arrangement Algorithm for Curve Segments

We present a robust arrangement algorithm for algebraic curves based on floating point arithmetic. The algorithm performs a line sweep, tests the consistency of each sweep update, and modifies the input to prevent inconsistent updates. The output arrangement is realizable by semi-algebraic curves that are close to the input curves. We present a new performance model for robust computational geometry in which running times and error bounds are expressed in terms of the number of input inconsistencies. An inconsistency is a combinatorial property that is derivable with a given set of numerical algorithms, but that is not realizable. The running time of the arrangement algorithm is O((n + N) log n + k(n + N) log n) for n curves with N intersection points and with k = O(n) inconsistencies. The distance between the realization curves and the input is O( + kn ) where is the curve intersection accuracy. The output size is always the standard O(n+N). We show experimentally that k is zero for generic inputs and is tiny even for highly degenerate inputs. Hence, the algorithm running time on real-world inputs equals that of a standard sweep and the realization error equals the curve intersection error.