Fournier, A. and Montuno, D. Y., "Triangulating Simple Polygons and Equivalent Problems," ACM Transactions on Graphics, 1984, 153-74.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the special cases of vertical horizontal segments) The usual vector based representation of a polygon is also the list of points P 1 ; P 2 ; P n describing its boundary as a closed polyline. Many algorithms have been designed for polygon partition, mostly for polygon triangulation [10, 9, 44, 4, 41]. However, triangulation results in a large, possibly prohibitive, number of convex components in the partition. Given a polygon with n vertices, the number of triangles in the partition is n Gamma 2. Thus, we would like to minimize the number of convex elements in the partition. There exists an ....

A. Fournier and D.Y. Montuno. Triangulating Simple Polygons and Equivalent Problems. ACM Transactions on Graphics, 3:153--174, 1984.

....the loop and holes present. Loop Hole (a) b) Figure 1: a) A valid polygon with the loop and holes indicated. b) An invalid polygon. The dashed circle highlights the overlapping edges. The earliest trapezoidation algorithms, such as C hazelle and Incerpi [1] and Fournier and Montuno [2], deal well with the trapezoidation of general polygons. The Fournier and Montuno [2] method also copes with holes, in that it is able to trapezoid only the inside of the loop (ignoring the holes all together) However the Zalik and Clapworthy [4] algorithm is the first trapezoidation technique ....

....with the loop and holes indicated. b) An invalid polygon. The dashed circle highlights the overlapping edges. The earliest trapezoidation algorithms, such as C hazelle and Incerpi [1] and Fournier and Montuno [2] deal well with the trapezoidation of general polygons. The Fournier and Montuno [2] method also copes with holes, in that it is able to trapezoid only the inside of the loop (ignoring the holes all together) However the Zalik and Clapworthy [4] algorithm is the first trapezoidation technique presented that is able to trapezoid both the loop and holes. The algorithm decomposes a ....

[Article contains additional citation context not shown here]

A. Fournier and D. Y. Montuno, "Triangulating simple polygons and equivalent problems", ACM Trans. Graphics, 3(2):153-174, 1984.

....to O(nk log c) assuming the c orientations are given in a form that permits binary search. Two remarks close this subsection. First, the analysis of the running time can be sharpened in a simple polygon if there are two opposite basis vectors. ff we use the algorithm of Fournier and Montuno [15] to change the triangulation to a trapezoidation with sides parallel to these basis vectors, then c oriented paths can follow the edges of the trapezoids. Now, application of lemma 5.4 computes two minimum link paths from r to r . If we travel from p to q and arbitrarily choose which path to take ....

Alain Fournier and Delphin Y. Montuno. Triangulating simple polygons and equivalent problems. A CM Transactions on Graphics, 3(2):153-174, 1984.

....the largest empty lune determined by two points, or the on line computation of a circular visibility region in a simple polygon. 2 The hierarchical vertical decomposition The hierarchical vertical decomposition of a simple polygon P is based on its vertical decomposition (or trapezoidal map) [6, 8]. Recall that the trapezoidal map is obtained by drawing a vertical line segment through every vertex of P, cutting the interior of P into two parts. The result is a partition of P into trapezoids (which can degenerate into triangles) separated by vertical sides. We call these vertical sides ....

....The number of regions in 74 is O(n) The total size of all regions in 74 is O(nlogn) 74 can be computed in time O(nlogn) Proof: We give an algorithm to construct 7 in time O(n log n) The properties will follow from the construction. We start out with T, constructed in time O(nlogn) from P [6, 8]. We create the leaves of 7 from 7 . The construction now proceeds in phases. In every phase, we merge adjacent regions in pairs. Note that such a merge is admissible unless both regions have three doors (because that would result in a region with four doors) Assume we have m regions at the ....

A. Fournier and D. Y. Montuno. Triangulating simple polygons and equivalent problems. ACM Trans. Graph., 3:153-174, 1984.

....the following: Lemma 6. The multi point planar point location problem can be solved using O( n k) log m n) I O operations. After computing the trapezoid decomposition of a simple polygon, the polygon can be triangulated in O(n) I Os using a slightly modified version of an algorithm from [15]: Lemma 7. A simple polygon with N vertices can be triangulated in O(n log m n) I O operations. 4 Line Segment Intersection In order to construct the optimal algorithm for the red blue line segment intersection problem, we will first consider the problem of sorting a set of nonintersecting ....

A. Fournier and D. Y. Montuno. Triangulating simple polygons and equivalent problems. ACM Trans. on Graphics, 3(2):153--174, 1984.

....of polygons that has also been studied before and this concerns the problem of decomposing a polygon into trapezoids. In fact, many polygon triangulation algorithms start out by first obtaining a trapezoidization of the polygon and subsequently converting this trapezoidization into a triangulation [12]. Here the trapezoidization is merely a step towards another goal. Furthermore, we should point out that in the work on trapezoids, triangles are allowed and considered as degenerate trapezoids. On the other hand, trapezoidizations are used in the manufacturing industry as the main goal in ....

Fournier, A., and D.Y. Montuno, Triangulating simple polygons and equivalent problems, in ACM Transactions on Graphics, 3, 2, pp. 153--174, 1984.

....of polygons that has also been studied before and this concerns the problem of decomposing a polygon into trapezoids. In fact, many polygon triangulation algorithms start out by first obtaining a trapezoidization of the polygon and subsequently converting this trapezoidization into a triangulation [14]. Here the trapezoidization is merely a step towards another goal. Furthermore, we should point out that in the work on trapezoids, triangles are allowed and considered as degenerate trapezoids. On the other hand, trapezoidizations are used in the manufacturing industry as the main goal in ....

Fournier, A., and D.Y. Montuno, Triangulating simple polygons and equivalent problems, in ACM Transactions on Graphics, 3, 2, pp. 153--174, 1984.

....the largest empty lune determined by two points, or the on line computation of a circular visibility region in a simple polygon. 2 The hierarchical vertical decomposition The hierarchical vertical decomposition of a simple polygon P is based on its vertical decomposition (or trapezoidal map) [6, 8]. Recall that the trapezoidal map is obtained by drawing a vertical line segment through every vertex of P , cutting the interior of P into two parts. The result is a partition of P into trapezoids (which can degenerate into triangles) separated by vertical sides. We call these vertical sides ....

....of regions in H is O(n) ffl The total size of all regions in H is O(n log n) ffl H can be computed in time O(n log n) Proof: We give an algorithm to construct H in time O(n log n) The properties will follow from the construction. We start out with T , constructed in time O(n log n) from P [6, 8]. We create the leaves of H from T . The construction now proceeds in phases. In every phase, we merge adjacent regions in pairs. Note that such a merge is admissible unless both regions have three doors (because that would result in a region with four doors) Assume we have m regions at the ....

A. Fournier and D. Y. Montuno. Triangulating simple polygons and equivalent problems. ACM Trans. Graph., 3:153--174, 1984.

....monotone polygon (OSMP) is a monotone polygon whose upper (or lower) chain is a straight line, which is called the distinguished edge. Monotone polygon triangulation consists of two phases: decomposition into onesided monotone polygons [3, 46] and triangulating the one sided monotone polygons [40, 36]. Without loss of generality, all vertices are assumed to have distinct x coordinates. Monotone Polygon Triangulation : 1. Decomposition into OSMPs: CHAPTER 3. APPLICATIONS OF ANSVP 63 (a) Merge the vertices of the lower and upper chains according to their x coordinates. b) For each edge (v i ....

FOURNIER, A., AND MONTUNO, D. Y. Triangulating Simple Polygons and Equivalent Problems. ACM Transactions on Graphics 3 (1984), 153--174. BIBLIOGRAPHY 128

....n) In order to construct 0 it is necessary to construct a horizontal trapezoidization of P and a vertical trapezoidization of P . Either trapezoidization of P can be constructed in linear time by a complex algorithm [2] or in O (n log n) time by a number of algorithms; see for example [3] [5], and [6] If an O (n log n) time trapezoidization algorithm is used then the overall time complexity of our algorithm remains at O (n log n) All polygons used in this paper are assumed to be simple and without adjacent collinear edges. We need some basic terminology. Let P = fv 0 ; v 1 ; ....

A. Fournier and D. Y. Montuno. Triangulating simple polygons and equivalent problems. ACM Trans. Graph., 3(2):153174, 1984.

....triangle strips, which are more efficient to render on current graphics systems than are triangles. Algorithms to generate long strips from triangles exist [EAV99] but they are intended for static models and are rather slow. Several asymptotically efficient polygon triangulation algorithms [FM84,Cha91] are known but most are difficult to implement and they produce generally skinny triangles. It is possible to ensure triangle fatness using additional (Steiner) points [BDE92,Mit93,KU99] For example, in two dimensions, a polygon may be tiled using triangles with angles at most 7 8p using O(n ....

....The diagonals (shown in thin solid lines) of the trapezoids that connect two vertices of the PSLG partition the PSLG into a set of uni monotone polygons. Uni monotone polygons consist of a single v monotone chain and another line segment. For a discussion and proofs, we refer the reader to [Sei91,FM84]. It can be shown (we omit the proof here) that the line segments mentioned above, call them monotone segments, are all PSLG segments, and thus small in length for our application. We will exploit this fact while triangulating these monotone polygons. MONOTONE TRIANGULATION Simple O(n) ....

[Article contains additional citation context not shown here]

A. Fournier and D. Montuno. Triangulating simple polygons and equivalent problems. ACM Transactions on Graphics , 3:153---174, 1984.

....n) In order to construct 0 it is necessary to construct a horizontal trapezoidization of P and a vertical trapezoidization of P . Either trapezoidization of P can be constructed in linear time by a complex algorithm [2] or in O (n log n) time by a number of algorithms; see for example [3] [5], and [6] If an O (n log n) time trapezoidization algorithm is used then the overall time complexity of our algorithm remains at O (n log n) All polygons used in this paper are assumed to be simple and without adjacent collinear edges. We need some basic terminology. Let P = fv 0 ; v 1 ; ....

A. Fournier and D. Y. Montuno. Triangulating simple polygons and equivalent problems. ACM Trans. Graph., 3(2):153174, 1984.

....whether the two vertices of the original polygon lie on the same side. This is a linear time operation. ffl Triangulation of monotone polygons: A monotone polygon can be triangulated in linear time by using a simple greedy algorithm which repeatedly cuts off the convex corners of the polygon [FM84]. Hence, all the monotone polygons can be triangulated in O(n) time. Partitioning a Simple Polygon using Non Intersecting Chains Given a simple polygon P and a number of nonintersecting polygonal chains C, the output of the algorithm is a list of disjoint regions partitions of the polygon due to ....

A. Fournier and D. Y. Montuno. Triangulating simple polygons and equivalent problems. ACM Trans. Graph., 3:153--174, 1984.

....holes. For simple polygons without holes, the lower bound of Asano et al. does not hold, however. This fact, and the importance of the polygon triangulation problem, in turn prompted several researchers to work on methods for beating the (n log n) time bound for this problem. Fournier and Montuno [11] and Chazelle and Incerpi [5] showed, even prior to the Asano et al. lower bound result, that to triangulate a simple polygon in linear time it is sucient to produce a trapezoidal decomposition (trapezoidation) of a simple polygon. A trapezoidation is formed by shooting a vertical (visibility) ....

....a randomized algorithm for computing the trapezoidation of a simple polygon. The expected running time of our algorithm is linear in the size of the polygon. As already mentioned, from the trapezoidation, a triangulation of the polygon can be obtained in linear time using well known methods [5, 11]. Thus, our algorithm provides a randomized algorithm for polygon triangulation that runs in linear expected time. In addition, our algorithm is considerably simpler than Chazelle s optimal deterministic algorithm; hence, it e ectively responds to the open problem posed by Chazelle as to the ....

A. Fournier and D. Y. Montuno. Triangulating simple polygons and equivalent problems. ACM Trans. Graph., 3(2):153-174, 1984.

....core of this problem precisely is to find for each segment endpoint the segment immediately above it. The ability to compute a trapezoid decomposition of a simple polygon also leads to an O(sort(N) polygon triangulation algorithm using a slightly modified version of an internal memory algorithm [61]. Furthermore, if one takes a closer look at the algorithm for EPD one realizes that it works in general with K query points, that are not necessarily endpoints of the segments. Therefore the result also leads to an O( n k) log m n) I O solution to the batched planar point location problem. But ....

....solved in O(n log m n) I Os, then a simple polygon with N vertices can be triangulated in O(n log m n) I O operations. 77 Proof : After computing the trapezoid decomposition of a simple polygon, the polygon can be triangulated in O(n) I Os using a slightly modified version of an algorithm from [61]. 2 ( b a a above b b a b a a b b a ya y b ya y b Figure 2: Using EPD to compute the trapezoid decomposition of a simple polygon. Figure 3: Comparing segments. Two segments can be related in four different ways. We define a segment AB in the plane to be above another ....

A. Fournier and D. Y. Montuno. Triangulating simple polygons and equivalent problems. ACM Trans. on Graphics, 3(2):153--174, 1984. 164

....stack) else (q i is adjacent to both x 1 and x j ) Fig.3(c) Add diagonals (q i ; x 2 ) q i ; x 3 ) q i ; x j Gamma1 ) and stop. end For general simple polygons, the time complexity was not improved till quite recently when a number of O(n log n) algorithms [GJPT78, B. C82, HM83, FM84] were proposed in the late 70 s and 80 s. Still the gap between the known lower bound, Omega Gamma n) and upper bounds kept challenging computational geometers for long. Finding a linear time algorithm remained open 8 problem; TV86] s algorithm was mistakenly thought to be linear for some ....

....sequences of spiraling and antispiraling chains. The number of such chains is the sinuosity of the polygon. In particular start shaped polygons have a sinuosity of 1. Its complexity is n log s. The attempts to start with the montonization or diagonal splitting path failed till [CI84] and [FM84] showed the equivalence of trapezoidal decomposition with triangulation for simple 9 (a) b) c) s1 s4 s2 s3 s5 s7 s8 s9 s6 d0 d1 d2 Figure 3: Generating monotone polygons from the trapezoid formation polygons, and many recent efforts have been concentrated on polygon ....

[Article contains additional citation context not shown here]

A. Fournier and D. Y. Montuno. Triangulating simple polygons and equivalent problems. ACM Trans. Graph., 3(2):153--174, 1984.

....of the original polygon lie on the same side of the horizontal line. This is a linear time operation. ffl Triangulation of monotone polygons: A monotone polygon can be triangulated in linear time by using a simple greedy algorithm which repeatedly cuts off the convex corners of the polygon [FM84] Hence, all the monotone polygons can be triangulated in O(n) time. In our algorithm for boundary evaluation, we represent the trimming boundary of a surface patch as a simple polygon. During ray shooting and trimmed surface intersection operations, we have to perform point location queries in ....

A. Fournier and D. Y. Montuno. Triangulating simple polygons and equivalent problems. ACM Trans. Graph., 3:153--174, 1984.

....contain holes. For simple polygons without holes, the lower bound of Asano et al. does not hold, however. This fact, and the importance of the polygon triangulation problem, in turn prompted several researchers to work on methods for beating O(n log n) time for this problem. Fournier and Montuno [14] and Chazelle and Incerpi [6] showed, even prior to the Asano et al. lower bound result, that to triangulate a simple polygon in linear time it is sucient to produce a trapezoidal decomposition (trapezoidation) of a simple polygon. In addition, Yap [31] showed that a similar result holds in a ....

....a randomized algorithm for computing the trapezoidation of a simple polygon. The expected running time of our algorithm is linear in the size of the polygon. As already mentioned, from the trapezoidation, a triangulation of the polygon can be obtained in linear time using well known methods [6, 14]. Thus, our algorithm provides a randomized algorithm for polygon triangulation that runs in linear expected time. In addition, our algorithm is considerably simpler than Chazelle s celebrated optimal deterministic algorithm; hence, it addresses the open problem posed by Chazelle and others as to ....

A. Fournier and D. Y. Montuno. Triangulating simple polygons and equivalent problems. ACM Trans. Graph., 3(2):153-174, 1984.

....of polygons that has also been studied before and this concerns the problem of decomposing a polygon into trapezoids. In fact, many polygon triangulation algorithms start out by first obtaining a trapezoidization of the polygon and subsequently converting this trapezoidization into a triangulation [12]. Here the trapezoidization is merely a step towards another goal. Furthermore, we should point out that in the work on trapezoids, triangles are allowed and considered as degenerate trapezoids. On the other hand, trapezoidizations are used in the manufacturing industry as the main goal in ....

Fournier, A., and D.Y. Montuno, Triangulating simple polygons and equivalent problems, in ACM Transactions on Graphics, 3, 2, pp. 153--174, 1984.

....core of this problem precisely is to find for each segment endpoint the segment immediately above it. The ability to compute a trapezoid decomposition of a simple polygon also leads to an O(sort(N) polygon triangulation algorithm using a slightly modified version of an internal memory algorithm [61]. Furthermore, if one takes a closer look at the algorithm for EPD one realizes that it works in general with K query points, that are not necessarily endpoints of the segments. Therefore the result also leads to an O( n k) log m n) I O solution to the batched planar point location problem. But ....

....solved in O(n log m n) I Os, then a simple polygon with N vertices can be triangulated in O(n log m n) I O operations. 77 Proof : After computing the trapezoid decomposition of a simple polygon, the polygon can be triangulated in O(n) I Os using a slightly modified version of an algorithm from [61]. # ## b a a above b b a b a a b b a ya y b ya y b Figure 2: Using EPD to compute the trapezoid decomposition of a simple polygon. Figure 3: Comparing segments. Two segments can be related in four di#erent ways. We define a segment AB in the plane to be above another segment ....

A. Fournier and D. Y. Montuno. Triangulating simple polygons and equivalent problems. ACM Trans. on Graphics, 3(2):153--174, 1984. 164

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Fournier, A. and Montuno, D. Y., "Triangulating Simple Polygons and Equivalent Problems," ACM Transactions on Graphics, 1984, 153-74.

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A. Fournier and D.Y. Montuno. Triangulating simple polygons and equivalent problems. In ACM Transactions on Graphics, Vol.3, No.2, pp.153-174, 1984

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A. Fournier and D. Montuno. Triangulating simple polygons and equivalent problems. ACM Transactions on Graphics , 3:153---174, 1984.

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A. Fournier and D. Y. Moutuno, "Triangulating Simple Polygons and Equivalent Polygons ", ACM Transactions on Graphics, Vol. 3, No. 2, pp. 153-174, 1984. 27

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A. Fournier and D.Y. Montuno. Triangulating simple polygons and equivalent problems. ACM Transaction on Graphics, 3:153--174, 1984.

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