English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Open

Presents the pdf analysis as premium feature

PDF Analysis

Insights

Presents the summarisation as premium feature

Summarisation

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

In this paper we consider the problem of monitoring an art gallery modeled as a polygon, the edges of which are arcs of curves. We consider two types of guards: edge guards (these are edges of the polygon) and mobile guards (these are either edges or straight-line diagonals of the polygon). Our focus is on piecewise-convex polygons, i.e., polygons that are locally convex, except possibly at the vertices, and their edges are convex arcs. We reduce the problem of monitoring a piecewise-convex polygon to the problem of 2-dominating a constrained triangulation graph with edges or diagonals, where 2-dominance requires that every triangle in the triangulation graph has at least two of its vertices in the 2-dominating set. We show that, given a triangulation graph Tp of a polygon P with n ≥ 3 vertices: (1) ⌊n+1/3⌋ diagonal guards are always sufficient and sometimes necessary, and (2) ⌊2n+1/5⌋ edge guards are always sufficient and ⌊2n/5⌋ edge guards are sometimes necessary, in order to 2-dominate Tp. We also show that a diagonal (resp., edge) 2-dominating set of size ⌊2n+1/5⌋ can be computed in O(n2) time and O(n) space. Based on these results we prove that, in order to monitor a piecewise-convex polygon P witn n ≥ 2 vertices: (1) ⌊n+1/3⌋ mobile guards or ⌊2n+1/5⌋ edge guards are always suffcient, and (2) ⌊n/3⌋ mobile edge guards are sometimes necessary. A mobile (resp., edge) guard set for P of size ⌊n+1/3⌋ (resp., ⌊2n+1/5⌋ or ⌊3n/7⌋ can be computed in O(n log n + T(n)) time and O(n) space, where T(n) denotes the time for computing a diagonal (resp., edge) 2-dominating set of size ⌊n+1/3⌋ (resp., ⌊2n+1/5⌋ or ⌊3n/7⌋ for a trinagulation graph with n vertices.

Guarding curvilinear art galleries with edge or mobile guards