Time-optimal computation of the rectilinear convex hull with arbitrary orientation of sets of segments and circles
We explore an extension to rectilinear convexity of the classic problem of computing the convex hull of a set of geometric objects. Namely, we solve the problem of computing the rectilinear convex hull witharbitraryorientation for asetofsegmentsandcircles.Wedescribe efficient algorithms to compute a...
| Authors: | , , , |
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| Format: | article |
| Publication Date: | 2025 |
| Country: | España |
| Institution: | Universitat Politècnica de Catalunya (UPC) |
| Repository: | UPCommons. Portal del coneixement obert de la UPC |
| Language: | English |
| OAI Identifier: | oai:upcommons.upc.edu:2117/427276 |
| Online Access: | https://hdl.handle.net/2117/427276 https://dx.doi.org/10.1007/s10898-025-01482-9 |
| Access Level: | Open access |
| Keyword: | Rectilinear convex hull Segments Simple polygonal line Circles Theory of computation Computational geometry Àrees temàtiques de la UPC::Matemàtiques i estadística::Geometria::Geometria computacional |
| Summary: | We explore an extension to rectilinear convexity of the classic problem of computing the convex hull of a set of geometric objects. Namely, we solve the problem of computing the rectilinear convex hull witharbitraryorientation for asetofsegmentsandcircles.Wedescribe efficient algorithms to compute and maintain the objects appearing on the boundary of the rectilinear convex hull of such sets, while we rotate the coordinate axes by an angle that goes from 0 to 2p. We first consider a set of n segments. If the segments are not necessarily disjoint, we describe an algorithm that runs in optimal (n logn) time and O(na(n)) space, where a(n) is the extremely slowly growing inverse of Ackermann’s function. If instead the segments form a simple polygonal chain, we describe an algorithm that improves the previous space complexity to (n). We then extend the techniques used in these algorithms to a set of n circles. The resulting algorithm runs in optimal (n logn) time and (n) space. |
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