Optimizing generalized kernels of polygons

Alejandra Martinez-Moraian (), David Orden (), Leonidas Palios (), Carlos Seara () and Paweł Żyliński ()
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Alejandra Martinez-Moraian: Universidad de Alcalá
David Orden: Universidad de Alcalá
Leonidas Palios: University of Ioannina
Carlos Seara: Universitat Politècnica de Catalunya
Paweł Żyliński: University of Gdańsk

Journal of Global Optimization, 2021, vol. 80, issue 4, No 7, 887-920

Abstract: Abstract Let $$\mathcal {O}$$ O be a set of k orientations in the plane, and let P be a simple polygon in the plane. Given two points p, q inside P, we say that p $$\mathcal {O}$$ O -sees q if there is an $$\mathcal {O}$$ O -staircase contained in P that connects p and q. The $$\mathcal {O}$$ O -Kernel of the polygon P, denoted by $$\mathcal {O}$$ O - $$\mathrm{Kernel }(P)$$ Kernel ( P ) , is the subset of points of P which $$\mathcal {O}$$ O -see all the other points in P. This work initiates the study of the computation and maintenance of $$\mathcal {O}$$ O - $$\mathrm{Kernel }(P)$$ Kernel ( P ) as we rotate the set $$\mathcal {O}$$ O by an angle $$\theta $$ θ , denoted by $$\mathcal {O}$$ O - $$\mathrm{Kernel }_{\theta }(P)$$ Kernel θ ( P ) . In particular, we consider the case when the set $$\mathcal {O}$$ O is formed by either one or two orthogonal orientations, $$\mathcal {O}=\{0^\circ \}$$ O = { 0 ∘ } or $$\mathcal {O}=\{0^\circ ,90^\circ \}$$ O = { 0 ∘ , 90 ∘ } . For these cases and P being a simple polygon, we design efficient algorithms for computing the $$\mathcal {O}$$ O - $$\mathrm{Kernel }_{\theta }(P)$$ Kernel θ ( P ) while $$\theta $$ θ varies in $$[-\frac{\pi }{2},\frac{\pi }{2})$$ [ - π 2 , π 2 ) , obtaining: (i) the intervals of angle $$\theta $$ θ where $$\mathcal {O}$$ O - $$\mathrm{Kernel }_{\theta }(P)$$ Kernel θ ( P ) is not empty, (ii) a value of angle $$\theta $$ θ where $$\mathcal {O}$$ O - $$\mathrm{Kernel }_{\theta }(P)$$ Kernel θ ( P ) optimizes area or perimeter. Further, we show how the algorithms can be improved when P is a simple orthogonal polygon. In addition, our results are extended to the case of a set $$\mathcal {O}=\{\alpha _1,\dots ,\alpha _k\}$$ O = { α 1 , ⋯ , α k } .

Date: 2021
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DOI: 10.1007/s10898-021-01020-3

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