R. Klein. Algorithmische Geometrie. Springer, Heidelberg, 2nd edition, 2005.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... of the polygon (e.g. x2) 0 in Figure 8(b) and a negative value of the turn at a vertex xi indicates that xi is a concave vertex of the polygon (e.g. x3) 0 in Figure 8(b) It is well known that if polyarc(xo, Xn)is a simple polygon, then I (polyarc(xo, Xn) l = 360 (see, e.g. Klein [5], Lemma 4.16, p. 182) Note that the sign of the turn depends on the direction in which we traverse the polygonal arc, i.e. polyarc(xo, x) polyarc(xn, Xo) where polyarc(xn,Xo) Xk Xk)i=n , O. We thus have Proposition 5 The turn of a simple polygon C is (C) 4 360 . 6.2 ....

R. Klein. Algorithmische Geometrie, Addison-Wesley, Bonn, 1997.

....computes the minimum of the distance functions associated with each re nement interval i. This means to compute the lower contour of all function graphs (see Figure 15) i t . v Figure 15: The lower contour of an arrangement of function graphs in time interval i. Computational Geometry [Kle97] gives us a solution to this problem. By using a combination of divide and conquer and sweep technique, we can compute the lower contour of k di erent t monotonic function graphs, which are de ned over the same time interval and where any two function graphs intersect each other in at most two ....

R. Klein. Algorithmische Geometrie. Addison-Wesley, 1997.

....the x axis by Equation 10. Since by multiplying two complex numbers their polar angles are added, the sequence 2a k turns by an angle of towards the second quadrant with each iteration. Once 2a k is in the second quadrant, 2Re(a k ) is negative. This is illustrated in Figure 1 (see also [Hip94, IKL97, Kle97]) We show that D can be chosen large enough such that there is an index n 0 with y n 0 0 and y n0 2 c 2 which proves Lemma 2.5. Of course, we are interested in the smallest D for which the above inequalities holds. Let n 0 be the rst index such 10 that y n 0 0, that is, cos (n 0 ....

R. Klein. Algorithmische Geometrie. Addison-Wesley, 1997.

....If we multiply two complex numbers, then the radii are multiplied and the angles are added. Hence, the sequence 2a k turns by an angle of towards the second quadrant with each iteration. Once 2a k is in the second quadrant, 2Re(a k ) is negative. This is illustrated in Figure 2 (see also [Hip94, IKL97, Kle97]) Hence, y k becomes negative as soon as there is an integer l 0 with k arctan q (4 Gamma c) c Gamma arctan q c= 4 Gamma c) 2 ( 2 l; 3=2 l) 9 Note that since arctan(x) 2, we can choose l = 0 in the above expression and there is a k with k arctan q (4 Gamma c) c Gamma ....

R. Klein. Algorithmische Geometrie. Addison-Wesley, 1997.

....a simple optimal algorithm with a competitive ratio of 1 2m m = m Gamma 1) m Gamma1 . If the rays are allowed to contain branching vertices, then we obtain a geometric tree. Icking presents an algorithm to search in a geometric tree with m leaves that has a competitive ratio of 8m Gamma 3 [Ick94, Kle97]. This algorithm can be applied to search in a simple polygon and achieves a competitive ratio of 8n Gamma 3 if the polygon has n vertices. In this paper we show how to adapt the optimal deterministic strategy to search on m concurrent rays to geometric trees, i.e. trees embedded in ....

....path from p to q is denoted by shp(p; q) The union of all shortest paths from v 3 T v 7 v 8 v 5 v 4 v 6 v 1 v 2 Figure 1: A geometric tree T embedded in IE 2 with 11 leaf rays and 8 vertices. a fixed point p to the vertices of P forms a geometric tree called the shortest path tree of p [Kle97]. It is denoted by T p . If P is a path in P , then the visibility polygon vis(P) of P is defined as set of all points in P that are seen by at least one point of P. A maximal line segment of the boundary of vis(P) that does not belong to the boundary of P is called a window of vis(P) A window ....

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R. Klein. Algorithmische Geometrie. Addison-Wesley, 1997.

....lies entirely in P but shares only the two vertices with the boundary. That means that a diagonal cuts a simple polygon in exactly two simple sub polygons (see Figure 1 on the left) This is always possible for a simple polygon as stated by the next theorem, a proof is given for example by Klein [5] or by Toussaint [12] where also an interesting discussion about previous proofs can be found. Theorem 1.1 Every simple polygon P with more than three vertices can be partitioned into two subpolygons in O(n) time by inserting some diagonal of P . Figure 1: On the left, only d is a diagonal, ....

R. Klein. Algorithmische Geometrie. Addison Wesley, 1997.

....[ R 0 is a bumpy set. Proof: Choose R as base and R 0 as bump. ffl R is convex and therefore a bumpy set, ffl R 0 is convex and therefore conv(R 0 ) R [ R 0 and ffl R R 0 is a one dimensional, connected, proper line segment as R R 0 consists of only two points (see, e.g. [Kle97]) q.e.d. Lemma 5 Let R 1 ; R 2 be convex sets with int(R 0 R 1 ) 6= Then R = R 0 [ R 1 is a bumpy set. Proof: If R is convex it trivially is a bumpy set. If not we define the convex set R 0 as the base of the bump. Let B 1 ; B 2 ; BL be the connected components of R 1 n R 0 . As R ....

....a bumpy set. Proof: Induction over the number of vertices n of the polygon. For n = 3 we have a convex triangle which trivially is a bumpy set. Now take any polygon R with more than 3 vertices. Consider a triangulation of R into n Gamma 2 triangles (such a triangulation always exists, see e.g. [Kle97], O R93] and take any triangle R 1 of that triangulation such that R 1 has two edges on the boundary of R. Define R 1 as bump and R 0 : R n R 1 as base. Then the triangulation of R 0 consists of n Gamma 3 triangles, such that the number of vertices of R 0 is n Gamma 1. Therefore R 0 is a ....

Rolf Klein. Algorithmische Geometrie. Addison-Wesley, 1997.

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Klein, R. 1997. Algorithmische Geometrie. Bonn: Addison-Wesley.

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R. Klein. Algorithmische Geometrie. Addison-Wesley Longman, 1997.

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R. Klein. Algorithmische Geometrie. Addison-Wesley, Bonn, 1997.

....in S such that TVR(p,S) L is not empty. Even though the Euclidean Voronoi Diagram for line segments is also valid for this model, there is another well known algorithm that can be adapted to our problem and fits like a glove. Take Fortune s sweepline algorithm as described, for example in [5]. Let a horizontal sweepline move downwards and construct the Euclidean Voronoi Diagram of S step by step until the sweepline reaches S r (p) with p being the point in S closer to L.Atthispoint, the parabolic wavefront coincides with the boundary of the Time Voronoi Region of p in L ....

R. Klein, Algorithmische Geometrie, Addison-Wesley-Longman, 1997.

....is one solution of the quadratic equation t (H 1)t H 1, and z denotes the other solution. That these terms do in fact solve the above recursion can be easily verified by induction, using the identities zz = H 1=z z and z 3) How to obtain these solutions can be found in Klein [6]. Now let us assume that C 9, hence H 3. Then z cannot be a real number, so z is the complex conjugate of z. The coe#cients a n can also be expressed by a n = vz =2Re(vz ) 5) where Re(w) denotes the real part, c,ofacomplexnumberw = c di.Ifwe represent complex numbers by points in ....

R. Klein. Algorithmische Geometrie. Addison-Wesley, Bonn, 1997.

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R. Klein. Algorithmische Geometrie. Springer, Heidelberg, 2nd edition, 2005.

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Rolf Klein. Algorithmische Geometrie. Addison-Wesley, 1997.

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R. Klein. Algorithmische Geometrie. Addison-Wesley, 1997.

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R. KLEIN, Algorithmische Geometrie, Addison-Wesley (1997).

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