Patent Document ID: 7671860
Application ID: 11213553
Patent Flag: 1

Claim One:
1. A computer implemented method for determining at least some coordinates of at least some points on an implicit curve resulting from an intersection of a plurality of graphical objects, the method comprising: determining, by a computer, one or more parameterization regions of the implicit curve resulting from the intersection; determining, by the computer, one or more convergence regions of the one or more parameterization regions, wherein the determining one or more convergence regions comprises (a) and (b): (a) testing for convergence for a given parameterization region via a Kantorovich condition while adjusting the given parameterization region by Newton steps; (b) responsive to determining that the given parameterization region does not converge after adjusting the given parameterization region via Newton steps, subdividing the given parameterization region into a plurality of subdivisions and separately testing for convergence of the plurality of subdivisions; receiving, by the computer, a sparse parameterized representation of at least a portion of the implicit curve, the representation comprising sparse convergence region descriptor data for describing the one or more convergence regions and sparse starting point descriptor data for deriving starting points associated with the one or more convergence regions; at runtime, based at least on the sparse starting point descriptor data of the starting points associated with the one or more convergence regions, deriving, by the computer, the coordinates of at least some of the starting points associated with the one or more convergence regions; and applying, by the computer, Newton iteration starting from the derived coordinates of the at least some of the starting points to determine at least some coordinates of one or more points on the implicit curve; wherein applying Newton iteration starting from the derived coordinates of the at least some of the starting points is guaranteed to converge to a solution.