Visibility between two edges of a simple polygon

One of the most recurring themes in many computer applications such as graphics automated cartography, image processing and robotics is the notion of visibility. We are concerned with the visibility between two edges of a simplen-vertex polygon. Four natural definitions of edge-to-edge visibility are proposed. There existO(nlogn) algorithms and complicatedO(nlog logn) algorithms to solve this problem partially and indirectly. A linear running time, and thus optimal algorithm is presented to determine edge-to-edge visibility under any of the four definitions. This simple, efficient, and direct algorithm without computing the triangulation of the simple polygon also identifies the visibility region if it exists.

[1]  David Avis,et al.  A combinational approach to polygon similarity , 1983, IEEE Trans. Inf. Theory.

[2]  Tomás Lozano-Pérez,et al.  An algorithm for planning collision-free paths among polyhedral obstacles , 1979, CACM.

[3]  Nils J. Nilsson,et al.  A mobius automation: an application of artificial intelligence techniques , 1969, IJCAI 1969.

[4]  David Rappaport A linear algorithm for eliminating hidden-lines from a polygonal cylinder , 2005, The Visual Computer.

[5]  Godfried T. Toussaint A linear-time algorithm for solving the strong hidden-line problem in a simple polygon , 1986, Pattern Recognit. Lett..

[6]  Hossam Ahmed Elgindy,et al.  Hierarchical decomposition of polygons with applications , 1985 .

[7]  Bruce Randall Donald,et al.  Hypothesizing Channels through Free-Space in Solving the Findpath Problem , 1983 .

[8]  Leonidas J. Guibas,et al.  Visibility and intersection problems in plane geometry , 1989, Discret. Comput. Geom..

[9]  David Maier,et al.  Hysterical B-trees , 1981, Inf. Process. Lett..

[10]  T. Asano An Efficient Algorithm for Finding the Visibility Polygon for a Polygonal Region with Holes , 1985 .

[11]  Godfried T. Toussaint,et al.  Applications of a two-dimensional hidden-line algorithm to other geometric problems , 2005, Computing.

[12]  Kurt Mehlhorn,et al.  Data Structures and Algorithms 1: Sorting and Searching , 2011, EATCS Monographs on Theoretical Computer Science.

[13]  Godfried T. Toussaint Shortest Path Solves Translation Separability of Polygons , 1986, IAS.

[14]  Joseph O'Rourke,et al.  Worst-case optimal algorithms for constructing visibility polygons with holes , 1986, SCG '86.

[15]  Kurt Mehlhorn,et al.  Intersecting a line and a simple polygon , 1984 .

[16]  Kurt Mehlhorn,et al.  A new data structure for representing sorted lists , 1980, Acta Informatica.

[17]  Michael Ian Shamos,et al.  Computational geometry: an introduction , 1985 .

[18]  Kurt Mehlhorn,et al.  Sorting Jordan sequences in linear time , 1985, SCG '85.

[19]  Donald Ervin Knuth,et al.  The Art of Computer Programming , 1968 .

[20]  Godfried T. Toussaint,et al.  PATTERN RECOGNITION AND GEOMETRICAL COMPLEXITY. , 1980 .

[21]  F. Frances Yao On the priority approach to hidden-surface algorithms , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

[22]  Larry S. Davis,et al.  Computational Models of Space: Isovists and Isovist Fields , 1979 .

[23]  Robert E. Tarjan,et al.  Design and Analysis of a Data Structure for Representing Sorted Lists , 1978, SIAM J. Comput..

[24]  Godfried T. Toussaint,et al.  Separation of two monotone polygons in linear time , 1984, Robotica.

[25]  Leonidas J. Guibas,et al.  Visibility and intersectin problems in plane geometry , 1985, SCG '85.

[26]  Herbert Freeman,et al.  An Algorithm for the Solution of the Two-Dimensional "Hidden-Line" Problem , 1967, IEEE Trans. Electron. Comput..

[27]  Robert M. Haralick,et al.  Decomposition of Two-Dimensional Shapes by Graph-Theoretic Clustering , 1979, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[28]  Godfried T. Toussaint,et al.  Translating Polygons in the Plane , 1985, STACS.

[29]  G. Toussaint Solving geometric problems with the rotating calipers , 1983 .

[30]  Godfried T. Toussaint,et al.  A simple linear hidden-line algorithm for star-shaped polygons , 1985, Pattern Recognit. Lett..

[31]  J. Stoer,et al.  Convexity and Optimization in Finite Dimensions I , 1970 .

[32]  Godfried T. Toussaint,et al.  Shortest path solves edge-to-edge visibility in a polygon , 1986, Pattern Recognit. Lett..

[33]  Dave Hightower A solution to line-routing problems on the continuous plane , 1969, DAC '69.

[34]  V. Chvátal A combinatorial theorem in plane geometry , 1975 .

[35]  D. T. Lee,et al.  Computing the visibility polygon from an edge , 1986, Comput. Vis. Graph. Image Process..

[36]  Robert E. Tarjan,et al.  A linear-time algorithm for triangulating simple polygons , 1986, STOC '86.

[37]  Bernard Chazelle,et al.  A theorem on polygon cutting with applications , 1982, 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982).

[38]  Hans Rohnert,et al.  Shortest Paths in the Plane with Convex Polygonal Obstacles , 1986, Inf. Process. Lett..

[39]  D. T. Lee,et al.  An Optimal Algorithm for Finding the Kernel of a Polygon , 1979, JACM.

[40]  M. Shamos Problems in computational geometry , 1975 .

[41]  Godfried T. Toussaint,et al.  SOME NEW RESULTS ON MOVING POLYGONS IN THE PLANE. , 1983 .

[42]  Leonidas J. Guibas,et al.  Linear time algorithms for visibility and shortest path problems inside simple polygons , 2011, SCG '86.

[43]  David Avis,et al.  A Linear Algorithm for Computing the Visibility Polygon from a Point , 1981, J. Algorithms.

[44]  J. Griffiths Bibliography of hidden - line and hidden - surface algorithms , 1978 .

[45]  Godfried T. Toussaint,et al.  An Optimal Algorithm for Determining the Visibility of a Polygon from an Edge , 1981, IEEE Transactions on Computers.

[46]  D. T. Lee,et al.  Euclidean shortest paths in the presence of rectilinear barriers , 1984, Networks.

[47]  D. T. Lee,et al.  Visibility of a simple polygon , 1983, Comput. Vis. Graph. Image Process..

[48]  Kurt Mehlhorn,et al.  A new data structure for representing sorted lists , 2004, Acta Informatica.

[49]  Nils J. Nilsson,et al.  A Mobile Automaton: An Application of Artificial Intelligence Techniques , 1969, IJCAI.

[50]  Robert E. Tarjan,et al.  Triangulating a Simple Polygon , 1978, Inf. Process. Lett..