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Unformatted text preview: AMS 345/CSE 355 (Fall, 2010) Joe Mitchell COMPUTATIONAL GEOMETRY Homework Set # 1 – Solution Notes (1). [20 points] Construct a simple polygon P , having each of its edges parallel to the x- or y-axis, and a placement of some number of vertex guards in P such that the guards see every vertex of P , but there is at least one point on the boundary , ∂P , of P that is not seen by any guard. Try to make your example as “small” as possible (having the fewest number of vertices in P ). Optional: Can you argue that your example is the smallest possible? In the example below (left), with n = 8 vertices, we see that rectilinear simple polygon P has all of its vertices guarded by the two guards, g 1 and g 2, however, all points interior to the edge from v 0 to v 1 are not seen by the guards. (There are also points interior to P that are not seen.) We claim that there are no examples smaller than this (with smaller n ). First, note that for a rectilinear polygon P , it must be that n is even (since edges alternate between vertical and horizontal as one goes around the boundary of P ). Thus, the possible values of n are n = 4 , 6 , 8 , . . . . For n = 4 ( P is a rectangle), any one vertex guard sees all of P . For n = 6, P is a rectangle with a rectangular chunk removed from one corner, as in the figure below, right. It is easy to check that any placement of one vertex guard (at v 0 or v 3) that sees all of the vertices of P also sees all of the edges and interior of P ; similarly, any placement of a pair of vertex guards that sees all of the vertices of P (e.g., v 1 and v 4) also sees all of P . Thus, our example with n = 8 is smallest possible. g2 v0 v1 g1 v5 v4 v3 v2 v1 v0 (2). [20 points] (a). O’Rourke, problem 2, section 1.1.4, page 9. What is the answer to Klee’s question for clear visibility (Section 1.1.2)? More specifically, let G ′ ( n ) be the smallest number of point guards that suffice to clearly see every point in any polygon of n vertices. Point guards are guards who may stand at any point of P ; these are distinguished from vertex guards who may be stationed only at vertices. Are clearly seeing guards stronger or weaker than usual guards? What relationship between G ′ ( n ) and G ( n ) follows from their relative strength? ( G ( n ) is defined in Section 1.1.2) Does Fisk’s proof establish ⌊ n/ 3 ⌋ sufficiency for clear visibility? Try to determine G ′ ( n ) exactly. (Basically, the whole question boils down to asking you to find G ′ ( n ) ; he just asks it repeatedly.) Clear visibility guards are “weaker” than ordinary visibility guards, since the clear visibility polygon with respect to a guard is a subset of the (ordinary) visibility polygon with respect to the same guard. This implies, in particular, that g ′ ( P ) ≥ g ( P ) for any polygon P (since any set of points that clearly guards...
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- Spring '08
- Polygons, vertices, Point guard, vertex guards