On the visibility locations for continuous curves

Abstract The problem of determining visibility locations (VLs) on/inside a domain bounded by a planar C 1 -continuous curve (without vertices), such that entire domain is covered, is discussed in this paper. The curved boundary has been used without being approximated into lines or polygons. Initially, a few observations regarding the VLs for a curved boundary have been made. It is proposed that the set of VLs required to cover the domain be placed in a manner that the VLs and the lines connecting them form a spanning tree. Along with other observations, an algorithm has been provided which gives a near optimal number of VLs. The obtained number of VLs is then compared with a visibility disjoint set, called as witness points, to obtain a measure of the ‘nearness’ of the number of VLs to the optimum. The experiments on different curved shapes illustrate that the algorithm captures the optimal solution for many shapes and near-optimal for most others.

[1]  Jirí Matousek,et al.  Separating an object from its cast , 1997, SCG '97.

[2]  Min Liu,et al.  Computing global visibility maps for regions on the boundaries of polyhedra using Minkowski sums , 2009, Comput. Aided Des..

[3]  John K. Johnstone,et al.  A parametric solution to common tangents , 2001, Proceedings International Conference on Shape Modeling and Applications.

[4]  Min Liu,et al.  On minimal orthographic view covers for polyhedra , 2009, 2009 IEEE International Conference on Shape Modeling and Applications.

[5]  Gershon Elber,et al.  Mold accessibility via Gauss map analysis , 2004, Proceedings Shape Modeling Applications, 2004..

[6]  K. C. Hui,et al.  Mould design with sweep operations - a heuristic search approach , 1992, Comput. Aided Des..

[7]  G. Roth,et al.  View planning for automated three-dimensional object reconstruction and inspection , 2003, CSUR.

[8]  Menelaos I. Karavelas.  Guarding curvilinear art galleries with edge or mobile guards via 2-dominance of triangulation graphs , 2011, Comput. Geom..

[9]  Gershon Elber,et al.  The Kernel of a Freeform Surface and its duality with the Convex Hull of its Tangential Surface , 2006, Int. J. Shape Model..

[10]  Joseph S. B. Mitchell,et al.  Locating Guards for Visibility Coverage of Polygons , 2010, Int. J. Comput. Geom. Appl..

[11]  Jirí Matousek,et al.  Separating an object from its cast , 2002, Comput. Aided Des..

[12]  Csaba D. Tóth,et al.  Guarding curvilinear art galleries with vertex or point guards , 2009, Comput. Geom..

[13]  J. O'Rourke.  Art gallery theorems and algorithms , 1987 .

[14]  Gershon Elber,et al.  Geometric covering , 2013, Comput. Aided Des..

[15]  Gershon Elber,et al.  Geometric multi-covering , 2014, Comput. Graph..

[16]  Lawrence R. Rabiner,et al.  A tutorial on hidden Markov models and selected applications in speech recognition , 1989, Proc. IEEE.

[17]  Ralph R. Martin,et al.  Two-dimensional visibility charts for continuous curves , 2005, International Conference on Shape Modeling and Applications 2005 (SMI' 05).

[18]  M. Ramanathan,et al.  Shortest path in a multiply-connected domain having curved boundaries , 2013, Comput. Aided Des..

[19]  Andrea Bottino,et al.  A new lower bound for evaluating the performances of sensor location algorithms , 2009, Pattern Recognit. Lett..