Corrections to Lee's visibility polygon algorithm

Abstract

Abstract.We present a modification and extension of the (linear time) visibility polygon algorithm of Lee. The algorithm computes the visibility polygon of a simple polygon from a viewpoint that is either interior to the polygon, or in its blocked exterior (the cases of viewpoints on the boundary or in the free exterior being simple extensions of the interior case). We show by example that the original algorithm by Lee, and a more complex algorithm by El Gindy and Avis, can fail for polygons that wind sumciently. We present a second version of the algorithm, which does not extend to the blocked exterior case. CR Cate#ories: F.2.2.Keywords: computational geometry, visibility. Introduction.We consider the following problem: Given a simple polygon P and a viewpoint z in the plane, find all points on the boundary of P that are 'visible' from z. The position of the viewpoint z determines three cases : z in the interior or exterior of P, or on the boundary of P. The exterior case can be further categorized as free exterior or blocked exterior depending on whether there exists a ray from z which does not intersect P or there is no such ray. The boundary and free exterior cases can be handled as simple extensions of the interior case. So from an algorithmic viewpoint, there are basically two cases, the interior case and the blocked exterior case.Two linear time algorithms for computing the visibility polygon of P from an interior viewpoint have been published, Lee [7] and E1 Gindy and Avis [1], and Lee also presented a modification for the blocked exterior case in his paper. Lee's algorithm is simpler in structure; in particular it requires only one stack which eventually yields the visibility polygon, as opposed to three in the E1

Terrain Visibility With Multiple Viewpoints

“…Each set is recursively divided into two until each subset has only one viewpoint. The Voronoi diagram of one viewpoint is its visibility polygon, which can be computed in O(n) time [20]. Each of these diagrams can be transformed into a list of intervals such that each interval defines a portion of the terrain that is assigned to a particular viewpoint (or none).…”

“…Therefore well-known linear-time algorithms to construct visibility polygons can be applied (e.g. [20]). In 2.5D the viewshed is more complex.…”

Draining a polygon—or—rolling a ball out of a polygon

“…This takes O(n log n) preprocessing time and supports O(log n) time per query ray [CEG + 94]. Next we construct the visibility polygon from each vertex of in overall O(n 2 ) time [JS87]. From these visibility polygons, for each v i we construct a gravity diagram D i .…”

Mesh Generation and Optimal Triangulation

“…The visibility graph can be computed in time On 2 b y traversing the boundary of the polygon once per vertex, performing a simple stack algorithm in each traversal 79,116,120,130 . A more complicated algorithm reduces the time to On log n + E 106 .…”