By Godfried T Toussaint

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A. 18-21; 1973. , Seidel,R. Report 83-577; 1983. " Cornell Univ. v. " Proc. of the 12th Annual ACM Symposium on Theory of Computing; pp. 135-145; 1980. P. 402-405; 1979. J. 87-93; 1977. [Sei] Seidel,R. " Univ. of British Columbia, Dept. of Computer Science, Report 81-14; 1981. I. Thesis; 1978. C. 780-787; 1981. Simple On-Line Algorithms for Convex Polygons 37 Appendix A: In this appendix we give detailed description of the function CONTAINS_LTP for testing whether a point p lies in the region Iq' , corresponding to a non-leaf node q which is not the root, defined in section III.

P . ) φ DIAM(P) then P Φ D I S K ( p . , p . ) . ) and consider any other pair of points p 6 P , r ^ s ^ j . ) . f1»i'j// x , respectively. Now d(p ,p ) < d(x ,x ) . ) . )) it follows that which is after all the diameter of DIAM(P) . )) . D. p. in FPT(P). ) . ) u ui share the edge · Since c,T lies above 0 and c,K. p. 1 J must intersect c,c, K. , . p. ) , c, must lie below 0 . Similarly, for all 0 . It follows that p, r . ) , c, . ) . D. ,ρ, l j k. ρ l j K. ,p, ) . J k 1 The necessity part of the theorem follows straightforwardly from the definition.

Seidel 28 So far we have shown how the left and right tangent points for some point p can be determined quickly. Now it remains to show bow the tree T representing P has to be modified to a tree T' that represents P1 . Let us assume that the point p lies outside the polygon P and let Up and rip denote the left and right tangent point of P for p , respectively. The chain C of vertices representing P' is obtained from the chain C of vertices representing P by deleting the chain D of vertices between Up and rip , when P is traversed in counterclockwise order, and replacing it by the point p .