J. Canny, B.R. Donald, and E.K. Ressler. A rational rotation method for robust geometric algorithms. In Proc. of the 8th ACM Symp. on Computational Geometry, pages 251--260, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....non essential (e.g. in motion planning) For instance, rigid transformations are important in solid modeling, but they involve trigonometric functions. We can get arbitrarily good approximations by using rational rigid transformations . Solutions in 2 and 3 dimensions are given by Canny et al. CDR92] and Milenkovic and Milenkovic [MM93] respectively. APPLICATIONS We now consider issues in implementing specific algorithms under the EGC paradigm. The rapid growth in the number of such algorithms means the following list is quite partial. We attempt to illustrate the range of activities in ....

J. F. Canny, B. Donald, and E.K. Ressler. A rational rotation method for robust geometric algorithms. Proc. 8th ACM Symp. on Computational Geometry, pages 251-- 160, 1992. Berlin.

....as are needed to determine its sign. This will slow down the computation, but techniques have been developed to keep the performance penalty relatively small [73, 143] Besides these general approaches, there have been a number papers dealing with robust computation in specific problems [10, 13, 34, 60, 71, 72, 84, 102]. Dealing with degeneracies. Most algorithms described in the computational geometry literature make the assumption that the input is in general position. For example, for computing the intersections in a set of line segments it is often assumed that no three segments meet in a common point, that ....

J. Canny, B. R. Donald, and E. K. Ressler. A rational rotation method for robust geometric algorithms. In Proc. 8th Annu. ACM Sympos. Cornput. Geom., pages 251-260, 1992.

....ISR. Unfortunately, the available exact arithmetic data types do not support the calculations of sines and cosines which are necessary for calculating rotations. Instead we use only angles for which the sines and cosines can be expressed as rational numbers with small enumerator and denominator [3]. We keep an array Z of approximations to the sines of integer degree angles between 0 89. We emphasize that once we x an angle we have the exact sine and cosine of . What we cannot do is obtain the exact values of the trigonometric functions of a prescribed arbitrary angle. Since our choice ....

....trigonometric functions of a prescribed arbitrary angle. Since our choice of rotation angles is heuristic to begin with, the precise angle is immaterial, and the angle we use is never more than one degree o the prescribed angle. Moreover, there are techniques 11 to achieve better approximations [3], but we prefer not to use them because of performance reasons. How big should c be There are advantages and drawbacks in using few kd tress, say even one kd tree compared to using many. When using one kd tree, we are prone to get many false points in the range queries, resulting in more time ....

J. Canny, B. R. Donald, and E. K. Ressler. A rational rotation method for robust geometric algorithms. In Proc. 8th Annu. ACM Sympos. Comput. Geom., pages 251-260, 1992.

....sin and cos by using a rational parameterization of the unit circle, t) 1 t 2 1 t 2 ; 2t 1 t 2 : For 1 t 1, t) nearly uniformly (j 0 (t)j 2) covers the unit circle to the right of the y axis. The left half is parameterized by re ection. Canny, Donald, and Ressler [3] describe how to generate a dense set of rational values for t to uniformly cover the unit circle to any desired degree of precision. Di erent numerical representations can be used with the geometric representations. The coordinates of (x; y) can be integers, rationals (ordered pairs of ....

....the nearest oating point approximations to the output coordinates. Applying shortest path rounding yields a combinatorial structure consistent with these approximations. There are also methods for nding good integer approximations, if that is desired. As previously mentioned, Canny et al. [3] give a method for nding accurate, low precision rational coordinates on the unit circle. Part of their method involves using continued fractions to nd good rational approximations to real numbers. Hence, their techniques can be used to round to exact rational polar coordinates or exact rational ....

J. Canny, B. R. Donald, and E. K. Ressler. A rational rotation method for robust geometric algorithms. In Proc. 8th Annu. ACM Sympos. Comput. Geom., pages 251-260, 1992.

....See Figure 6 for an illustration. This is in contrast with most existing motion planning software for which tight or narrow passages constitute a signi cant hurdle. We remark that our tool can only handle translational motion; steps towards robust handling of polygon rotation are discussed in [23]. The input to our algorithms consists of two polygonal sets P and Q (each being an arbitrary collection of simple polygons) with a total of m and n vertices respectively. Our algorithms consist of the following three steps: 1) Decompose the polygons of P into convex subpolygons P 1 ; P 2 ; ....

J. Canny, B. R. Donald, and E. K. Ressler. A rational rotation method for robust geometric algorithms. In Proc. 8th Annu. ACM Sympos. Comput. Geom., pages 251-260, 1992.

....optimal Mopt (M; ffl) with accuracy ffl and bit size proven to be no more than 1:5b. In practice, the second algorithm generates an approximation with 1:5 and is much faster than the third algorithm. The best bit sizes which one could obtain using previously known results in two dimensions [1] are more than 3b bits for numerator and denominator. Applications are described for the approximation functions in the area of solid modeling. 1 Introduction Certain numerical issues must be resolved in order to implement an algorithm of computational geometry as a computer program. The ....

.... ffl) which takes an arbitrary orthogonal matrix M and accuracy ffl 0 as input and returns an orthogonal matrix that has rational entries and which approximates M to within ffl: max jvj=1 j(M Gamma M approx )vj ffl: 2) At the ACM geometry conference last year, Canny, Donald, and Ressler [1] presented a technique for generating rational two dimensional rotation matrices. For any two dimensional rotation matrix M , their technique can generate a rational matrix with accuracy 2 Gammab and rational entries with b bit numerators and a common b bit denominator (bit size b) This is ....

J. Canny, B. Donald, E.K. Ressler. A Rational Rotation Method for Robust Geometric Algorithms. Proceedings of the Eighth Symposium on Computational Geometry, ACM, pages 251--160, June 1992.

....using a rational parameterization of the unit circle, t) 1 Gamma t 2 1 t 2 ; 2t 1 t 2 : For Gamma1 t 1, t) nearly uniformly (j 0 (t)j 2) covers the unit circle to the right of the y axis. The left half is parameterized by reflection. Canny, Donald, and Ressler [3] describe how to generate a dense set of rational values for t to uniformly cover the unit circle to any desired degree of precision. Different numerical representations can be used with the geometric representations. The coordinates of (x; y) can be integers, rationals (ordered pairs of ....

....the nearest floating point approximations to the output coordinates. Applying shortest path rounding yields a combinatorial structure consistent with these approximations. There are also methods for finding good integer approximations, if that is desired. As previously mentioned, Canny et al. [3] give a method for finding accurate, low precision rational coordinates on the unit circle. Part of their method involves using continued fractions to find find good rational approximations to real numbers. Hence, their techniques can be used to round to exact rational polar coordinates or exact ....

J. Canny, B. R. Donald, and E. K. Ressler. A rational rotation method for robust geometric algorithms. In Proc. 8th Annu. ACM Sympos. Comput. Geom., pages 251--260, 1992.

....the resolution of its arithmetic. In computational geometry, the real RAM model of computation is generally assumed for the purposes of proof, although it is notoriously difficult to implement [16] For computing with angular quantities, one approach is to use rational, or exact , arithmetic [4, 23]. 3 Exact Algorithms Suppose an algorithm satisfies only the first the condition for completeness: it is guaranteed to find a solution when one exists. Such algorithms are thorough in that they will not overlook potential solutions; yet they are not necessarily complete. As suggested by John ....

John Canny, Bruce Donald, and Gene Ressler. A rational rotation method for robust geometric algorithms. December 1991.

....then the convex hull might contain new edges joining derived vertices. These new edges are not part of some original edge. Triangulating a polygon (and extracting the triangles as new polygons) also has the same effect. Many applications use either convex hulls or triangulations. Canny et al. [2] describe a scheme for carrying out exact rotations of points in two dimensions. Only a dense set of rotation matrices have rational coefficients, and therefore it is necessary to approximate the rotation angle. Under their scheme, an approximate rotation by with m bits of accuracy ( 1 Sigma ....

....uses lists of floating point numbers whose (implicit) sum is the number being represented. Each number in the list has a different exponent, and it covers a different range of the bits of the represented number. Canny et al. in their work on rational rotation matrices in two dimensions [2], discuss the use of continued fractions for generating good rational approximations to real numbers. This work can also be applied to the problem of finding a lower precision representation for a fraction p=q. It is possible to find p 0 =q 0 which approximates p=q to m bits of accuracy such ....

J. Canny, B. Donald, E.K. Ressler. A Rational Rotation Method for Robust Geometric Algorithms. Proceedings of the Eighth Symposium on Computational Geometry, ACM, pages 251--160, June 1992.

No context found.

J. Canny, B.R. Donald, and E.K. Ressler. A rational rotation method for robust geometric algorithms. In Proc. of the 8th ACM Symp. on Computational Geometry, pages 251--260, 1992.

No context found.

J. Canny, B. R. Donald, and E. K. Ressler. A rational rotation method for robust geometric algorithms. In Proc. 8th Annu. ACM Sympos. Comput. Geom., pages 251--260, 1992.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC