J. D. Hobby. Practical segment intersection with finite precision output. Comput. Geom. Theory Appl., 13(4):199--214, 1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....handle many special cases. The number of special cases induced by degeneracies can be in the dozens already for simple algorithms. The geometric objects retain their geometric structure. That is, the circles are not transformed into pseudo circles (in contrast with, for example, snap rounding [15]) Thus, all the geometric rules and axioms regarding the geometric objects (circles, in our case) will still be valid. Implementations using exact arithmetic with floating point filtering, can be sped up, since the perturbation will cause the predicates to be evaluated using the floatingpoint ....

J. D. Hobby. Practical segment intersection with finite precision output. Comput. Geom. Theory Appl., 13(4):199--214, 1999.

....to pass through the hooks of # (i.e. grid points nearest to the intersections of # with other line segments) But this may generate new intersections (derived hooks) and the cascaded e#ects must be carefully controlled. The grid model of Greene Yao has been taken up by several other authors [Hob99, GM95, GGHT97] Extension to higher dimensions is harder: there is a solution of Fortune [For98] in 3 dimension. Further developments include the numerically stable algorithms in [FM91] The interesting twist here is the use of pseudolines rather than polylines. Ho#mann, Hopcroft, and Karasick ....

John D. Hobby. Practical segment intersection with finite precision output. Computational Geometry: Theory and Applications, 13:199--214, 1999.

....may see 100 digits though many of these may be wrong without the user realizing this. 8 in T # ) In the robustness literature, rounding problems are often a composition of two problems: a construction problem followed by a bona fide rounding problem. For instance, the problem of snap rounding [36] for intersecting line segments is such a composite problem. In EGC, we prefer to reduce such composite problems to two distinct steps: first compute some geometric object T , and then rounding T to a lower precision version T # . The first step is considered solved using EGC. The second step is ....

J. D. Hobby. Practical segment intersection with finite precision output. Comput. Geom. Theory Appl., 13(4):199--214, Oct. 1999.

....This ersatz geometry can only preserve a few of the properties found in Euclidean geometry. A very natural and popular finite precision geometry is the grid (usually regular, but this is not essential) Greene and Yao [32] investigated line arrangement computation in this geometry. See also [35, 33]. Here, line segment may become polygonal lines so that their intersections preserve properties such as non braiding and connected intersection. But we give up the properties such as the intersection of two lines is a single point. Approximate Predicates and Fat Geometry. Another approach to ....

J. Hobby. Practical segment intersection with finite precision output. Technical report, Bell Labs, 1993. Tech. Report.

....Therefore it is important to quantify the error that the output produced by such a method will have, and formulate methods to deal with this problem. The reader might notice the connection between our work on raster displays and the vast body of work on finite resolution computational geometry [6, 9, 5, 8, 16]. One of the main directions (especially in the context of geometric rounding is to modify the output of an exact algorithm (for example for computing line intersections in the plane) so that when snapped to a grid, the topology of the arrangement is preserved (or at least is consistent with the ....

HOBBY, J. Practical segment intersection with finite precision output. Technical Report 93/2-27, Bell Laboratories, 1993.

....Therefore it is important to quantify the error that the output produced by such a method will have, and formulate methods to deal with this problem. The reader might notice the connection between our work on raster displays and the vast body of work on finite resolution computational geometry [6, 9, 5, 8, 16]. One of the main directions (especially in the context of geometric rounding is to modify the output of an exact algorithm (for example for computing line intersections in the plane) so that when snapped to a grid, the topology of the arrangement is preserved (or at least is consistent with the ....

HOBBY, J. Practical segment intersection with finite precision output. Technical Report 93/2-27, Bell Laboratories, 1993.

....algorithm requires more complicated global information. Devising a simple, practical, and efficient three dimensional rounding algorithm is a significant open problem. Other work. Greene and Yao were the first to suggest a rounding scheme for polygonal subdivisions in two dimensions [8] Hobby [11] and Greene [9] give algorithms to compute the snap rounding of the arrangement formed by a set of intersecting edges. Guibas and Marimount [10] show how to maintain the snap rounded arrangement of edges under insertion and deletion of edges; they also give elementary proofs of basic topological ....

J. Hobby, Practical segment intersection with finite precision output, Computational geometry: theory and applications, to appear.

....doubled, as did the number of intersections between segments. In some applications it is desirable to round the intersection points to have the same bit length as the original data; our implementation does not do this. Such rounding can be accomplished by modifying the Bentley Ottmann algorithm[17, 19, 26]. 13 ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl ffl 10 Gamma 8 10 Gamma 6 10 Gamma 4 0.01 1 0 20 40 60 80 100 Exact orientation tests, percent Figure 10: Exact orientation tests required for polyhedral intersection, as a function of rotation ....

J. Hobby, Practical segment intersection with finite precision output, 1993, submitted.

.... se[13] 9999999827968.000000 se[14] 100000000376832.000000 se[15] 999999986991104.000000 se[16] 10000000272564224.000000 se[17] 99999998430674944.000000 se[18] 999999984306749440.000000 se[19] 9999999980506447872.000000 se[20] 100000002004087734272.000000 se[21] 1000000020040877342720.000000 se[22]=9999999778196308361216.000000 se[23] 99999997781963083612160.000000 se[24] 1000000013848427855085568.000000 se[25] 10000000714945030854279168.000000 se[26] 100000002537764290115403776.000000 se[27] 1000000062271131048573140992.000000 se[28] 10000000622711310485731409920.000000 ....

....se[13] 0.0000000000000149011612 se[14] 0.0000000000000372529020 se[15] 0.0000000000000931322551 se[16] 0.0000000000002328306276 se[17] 0.0000000000005820765961 se[18] 0.0000000000014551914361 se[19] 0.0000000000036379785903 se[20] 0.0000000000090949461504 se[21] 0. 0000000000227373658096 se[22]= 0.0000000000568434153914 se[23] 0.0000000001421085332742 se[24] 0.0000000003552713401245 se[25] 0.0000000008881783641890 se[26] 0.0000000022204458272057 se[27] 0.0000000055511146790366 se[28] 0.0000000138777869196360 se[29] 0.0000000346944659668225 se[30] 0.0000000867361649170562 se[31] ....

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J. Hobby. Practical segment intersection with finite precision arithmetic. Technical report, AT&T Bell Labs., October 1993.

....with f has higher z coordinate) then oe(f) is above (or overlaps) oe(f 0 ) In the worst case Q has O(n 4 ) vertices and can be computed in time O(n 4 ) Other work. Greene and Yao were the first to suggest a rounding scheme for polygonal subdivisions in two dimensions [8] Hobby [11] and Greene [9] give algorithms to compute the snap rounding of the arrangement formed by a set of intersecting edges. Guibas and Marimount [10] show how to maintain the snap rounded arrangement of edges under insertion and deletion of edges; they also give elementary proofs of basic topological ....

J. Hobby, Practical segment intersection with finite precision output, Computational geometry: theory and applications, to appear.

....be invalidated by small perturbations of its faces or vertices. However, many applications are insensitive to changes in the combinatorial structure. If the structure is permitted to change, there are methods to round polygons or other planar objects made up of line segments to the integer grid [70, 66] or any nonuniform grid [102, 104] In general, rounding algorithms, particularly for curved or higher dimensional structures, are as yet inadequately developed. There are important applications, such as operations on algebraic curves and surfaces, where bit length estimates appear to rule out the ....

Hobby, J. Practical segment intersection with finite precision arithmetic, Manuscript, AT&T Bell Labs, October 1993.

....is applied to geometric objects after they have been generated by a geometric algorithm, and therefore it solves the problem of exponential cost (albeit at the price of reducing accuracy) without any modification of existing geometric algorithms. Unlike Green and Yao s algorithm and snap rounding [15, 19] (see also [18] shortest path geometric rounding 1) introduces the minimum possible geometric error, 2) introduces the minimum combinatorial change, and 3) can round vertices to any rounding lattice with connected rounding cells. The other methods can only round to the integer lattice. We argue ....

....lattice point. Advantages: bounded error, good for graphics applications, might be generalizable to other lattices. Disadvantages: introduces Omega Gamma n log jabj) excess lattice points onto the segment, where n is the number of vertices to which the segment it pulled. Snap Rounding [15, 19]: Various researchers have discovered this technique for rounding line segments to the integer grid. 2 Each vertex rounds to the nearest lattice point. To round ab, determine rounding cells of rounded vertices that intersect ab. Replace ab by the polygonal curve that visits the lattice points of ....

J. D. Hobby. Practical segment intersection with finite precision output. submitted for publication.

.... which also appears in our work on rounded arithmetic algorithms [9, 14] We also show how use this rounding algorithm in combination with a technique to reduce the amount of precision needed to construct an arrangement of lines or line segments (and round them to the integer grid) 12] Hobby [6] presents a simple technique for rounding an arrangement of line segments to the integer grid. The basic idea is to hook each edge to every vertex whose rounding cell intersects the edge. A rounding cell of a grid point is the set of points nearer to that grid point than any other. This ....

J. Hobby. "Practical Segment Intersection with Finite Precision Arithmetic". Manuscript, AT&T Bell Labs, October 1993.

....increases its complexity by a logarithmic factor, which is tight. We also look briefly at the algorithmic issues involved. These are closely related to rounding techniques for robust handling of arrangements of line segments, although we ask slightly different questions. See papers by Hobby [11, 12, 13], Milenkovic [17, 18, 19, 20] and Guibas et al. 7, 8] for more on arrangements of segments. 2 Exact minimum link paths Under the real RAM model of computation, a minimum link path from s to t in P can be computed by a simple greedy algorithm. We let (P; s; t) denote the number of links in ....

J. Hobby. Practical segment intersection with finite precision output. Manuscript, 1994.

....as little as possible, and which can be computed efficiently. 1. 1 Some Approaches to Rounding The problem of dealing with finite precision and robustness in geometric algorithms is fundamental, and there has been considerable work done on developing good approaches to this problem (e.g. see [2, 8, 9, 10, 11, 13, 15, 16, 17, 19, 22, 23]) Of particular relevance to this paper is the previous work done on producing rounded versions of arrangements of line segments. At a high level, of course, the goal of such a method is to round the given set of line segments so that each rounded version of a segment is close to the original ....

....possibly at a reduced bit complexity than a naive method might use. One approach to the segment rounding problem that has been shown to be very promising, from the standpoint of the combinatorial complexity of the rounded representation, is the snap rounding paradigm introduced by Greene and Hobby [15] and studied in more detail by Guibas and Marimont [11] Given a set S of n line segments in the plane and a regular pixel grid G, this approach involves defining pixels in G as being hot if they contain segment endpoints or segment intersection points (the point features of the arrangement) ....

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J. Hobby. Practical segment intersection with finite precision output. Technical Report 93/2-27, Bell Laboratories (Lucent Technologies), 1993.

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J. D. Hobby. Practical segment intersection with finite precision output. Comput. Geom. Theory Appl., 13(4):199--214, 1999.

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John Hobby. Practical segment intersection with finite precision output. Computational Geometry: Theory and Applications, 13(4), 1999. 31

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John Hobby. Practical segment intersection with finite precision output. Computational Geometry: Theory and Applications, 13(4), 1999.

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HOBBY, J. Practical segment intersection with finite precision output. Technical Report 93/2-27, Bell Laboratories, 1993.

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John D. Hobby, "Practical Segment Intersection with Finite Precision Output," submitted for publication.

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