Sub-quadratic algorithm for finding the intersections between two sets of spherical arcs

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Is there an algorithm that can find the intersections between two sets of spherical great-circle arcs (on the same sphere) with an worst-case runtime complexity better than the $O(n × m)$ brute-force algorithm? Intersections between arcs in the same set are irrelevant, I only need to find the intersections between two arcs in different sets.



The arcs are defined by two points on the sphere, taking the shorter great-circle path (the one with length < π, assume none of the pairs of points are antipodal). The arcs are not necessarily the same length.







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    Is there an algorithm that can find the intersections between two sets of spherical great-circle arcs (on the same sphere) with an worst-case runtime complexity better than the $O(n × m)$ brute-force algorithm? Intersections between arcs in the same set are irrelevant, I only need to find the intersections between two arcs in different sets.



    The arcs are defined by two points on the sphere, taking the shorter great-circle path (the one with length < π, assume none of the pairs of points are antipodal). The arcs are not necessarily the same length.







    share|cite|improve this question






















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      up vote
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      down vote

      favorite











      Is there an algorithm that can find the intersections between two sets of spherical great-circle arcs (on the same sphere) with an worst-case runtime complexity better than the $O(n × m)$ brute-force algorithm? Intersections between arcs in the same set are irrelevant, I only need to find the intersections between two arcs in different sets.



      The arcs are defined by two points on the sphere, taking the shorter great-circle path (the one with length < π, assume none of the pairs of points are antipodal). The arcs are not necessarily the same length.







      share|cite|improve this question












      Is there an algorithm that can find the intersections between two sets of spherical great-circle arcs (on the same sphere) with an worst-case runtime complexity better than the $O(n × m)$ brute-force algorithm? Intersections between arcs in the same set are irrelevant, I only need to find the intersections between two arcs in different sets.



      The arcs are defined by two points on the sphere, taking the shorter great-circle path (the one with length < π, assume none of the pairs of points are antipodal). The arcs are not necessarily the same length.









      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Aug 17 at 7:31









      taylor swift

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