J. Keyser, T. Culver, D. Manocha, and Shankar Krishnan. MAPC: A library for efficient and exact manipulation of algebraic points and curves. In Proc. 15th Annu. Sympos. Comput. Geom., pages 360--369, 1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....cost of extended precision arithmetic [2, 7] Curved objects are often represented implicitly, as the zeroes of a multivariate polynomial. Geometric predicates on curved objects are much more complicated and expensive than on linear objects, and require much more sophisticated algorithms [10]. Using connections with symbolic algebra, the inertia test may eventually lead to a reasonably efficient analog of the orientation test on curved objects. The following illustrates the situation in R 3 ; for more background see [3] Suppose p 1 ; p 2 ; p 3 are real polynomials in x; y; z. Then ....

J. Keyser, T. Culver, D. Manocha, S. Krishnan, MAPC: a library for efficient and exact manipulation of algebraic points and curves, Proc. 15th Annual Symp. Comp. Geom, pp. 360--369, 1999.

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John Keyser, Tim Culver, Dinesh Manocha, and Shankar Krishnan. MAPC: A library for efficient and exact manipulation of algebraic points and curves. In Proc. 15th Annual ACM Symposium on Computational Geometry, pages 360--369, 1999.

....The intersection points correspond to the roots of r(u) f(a bu; c du) In this case, the substitution operation reduces a two dimensional problem to a univariate problem. This approach for curve line intersection is used in the curve curve intersection algorithm presented in Keyser et al. [16]. Each of two curves f(s; t) 0, g(s; t) 0 is intersected with each of the four walls of an axis aligned box. The algorithm examines the configuration of the curve line intersections and infers the presence or absence of a root. 3.2 Curve curve root projection The resultant of a pair of ....

....and g. The roots are effectively projected onto the s axis. This operation may also be viewed as eliminating the variable t from the system. Elimination here carries the same meaning as it does in Gaussian elimination. Projection is used in the curve curve intersection algorithm (Keyser et al. [16]) The x resultant and the y resultant are used together to compute possible locations for Fig. 1. Curve line intersection. The curve f(s; t) 0 is a torus ellipsoid intersection curve, pulled back to the domain of one of the surfaces. The line is t = const. 5 Fig. 2. Curve curve root ....

John Keyser, Tim Culver, Dinesh Manocha, and Shankar Krishnan. MAPC: A library for efficient and exact manipulation of algebraic points and curves. In Proc. 15th Annual ACM Symposium on Computational Geometry, pages 360--369, 1999.

....surfaces, boundary evaluation of algebraic primitives and Minkowski sums of polyhedra. In such cases, the basic primitives like points, curves and surfaces are represented using algebraic numbers and polynomial equations. Techniques for exact representation of these primitives have been proposed [12, 30, 16]. The number of bits required for an exact representation can grow significantly with the algebraic degree 1 . Given an accurate representation of the primitives, the underlying computations that have to be performed in these applications are: Isolating and evaluating the roots of a polynomial ....

....have to be performed in these applications are: Isolating and evaluating the roots of a polynomial system. Accurately evaluating the sign of an algebraic expression, such as a matrix determinant. One of the commonly used approaches for root isolation is based on multivariate Sturm sequences [36, 29, 30, 16]. However, these techniques tend to be slow because of coefficient growth when computations involving moderate degree algebraic primitives are performed. Another approach is to approximate the roots of the system using numerical iterative techniques. These approaches can suffer from convergence ....

[Article contains additional citation context not shown here]

John Keyser, Tim Culver, Dinesh Manocha, and Shankar Krishnan. MAPC: A library for efficient and exact manipulation of algebraic points and curves. In Proc. 15th Annual ACM Symposium on Computational Geometry, pages 360--369, 1999.

....curve curve intersection implementation. of the computational problems faced. These timings do not contain optimizations such as those described in [10] Such optimizations have improved the speeds for all computations shown here by more than an order of magnitude. Alternative approaches [7] have yielded more than two orders of magnitude speedup. We give here some timing results for a basic implementation of a curve curve intersection routine. The results are presented in Fig. 5, and the program was run on an an SGI R10000 200 MHz processor. The machine was loaded at the times these ....

J. Keyser, T. Culver, D. Manocha, and S. Krishnan. Mapc: A library for efficient and exact manipulation of algebraic points and curves. Technical Report TR98038, University of North Carolina, Chapel Hill, 1998.

....algorithm uses multivariate Sturm sequences as proposed by Milne [26] and is described further in the appendix. The overall approach we use will work with any method which allows algebraic numbers to be represented as described above, and we are exploring other methods which appear more promising [17]. We use multivariate Sturm sequences here because they offer a direct and relatively simple way of achieving the representation. 7 f 1 f 2 f 3 f 4 f 5 f 6 f 1 f 2 f 3 f 4 f 5 f 6 e 1 e 2 e 3 e 4 e 5 e 6 e 7 e 8 e 9 e 10 e 11 e 12 Fig. 3. A ....

J. Keyser, T. Culver, D. Manocha, and S. Krishnan. Mapc: A library for efficient and exact manipulation of algebraic points and curves. Technical Report TR98038, University of North Carolina, Chapel Hill, 1998.

....of algebraic primitives, Minkowski sums of polyhedra etc. In such cases, some of the basic primitives such a s points, curves and surfaces are represented using algebraic numbers. Techniques for exact representation of these primitives have been proposed for different applications [Can87, KCMK99, CKM99] The number of bits used for exact representation can grow significantly with the algebraic degrees. Given an accurate representation of the primitives, some of the underlying computations in these applications are: # Isolating the roots of a multivariate polynomial system. # Isolating ....

....expression, such as a matrix determinant. The roots of algebraic systems cannot in general be expressed as closed form expressions, and algorithms for computing them are based on iterative techniques. One of the commonly used approaches is based on multi variate Sturm sequences [Mil92, Ped91, KCMK99] However, for computations involving moderate degree algebraic primitives, these algorithms can be slow. 1.1 Main Results In this paper, we present a new approach to represent algebraic numbers with arbitrary precision and use it to develop novel algorithms for computing roots of polynomial ....

[Article contains additional citation context not shown here]

John Keyser, Tim Culver, Dinesh Manocha, and Shankar Krishnan. MAPC: A library for efficient and exact manipulation of algebraic points and curves. In Proc. 15th Annual ACM Symposium on Computational Geometry, pages 360--369, 1999.

....E SOLID, which performs Boolean operations based primarily on exact rational arithmetic. Performance of this system is improved by using a varietyof techniques like precision driven computation and floating points filters whenever possible. Results from this system have been published in Ref. [39,40,37]. 8.2. Bounding Errors in Cascaded Computations Another important issue in the context of robustness is the error accumulation in cascaded geometric computations. Our experience with the Bradley fighting vehicle and submarine model shows that extremely large CAD models are designed using Boolean ....

J. Keyser, T. Culver,D.Manocha, and S. Krishnan. Mapc: A library for efficient and exact manipulation of algebraic points and curves. In Proceedings of the 15th Annual Symposium on Computational Geometry, pages 360--369, 1999.

....of algebraic primitives, Minkowski sums of polyhedra etc. In such cases, some of the basic primitives such a s points, curves and surfaces are represented using algebraic numbers. Techniques for exact representation of these primitives have been proposed for different applications [Can87, KCMK99, CKM99, MN99, KLPY99] The number of bits used for exact representation can grow significantly with the algebraic degrees 1 . Given an accurate representation of the primitives, some of the underlying computations in these applications are: ffl Isolating the roots of a multivariate polynomial ....

....such as a matrix determinant. The roots of algebraic systems cannot in general be expressed as closed form expressions, and algorithms for computing them are based on iterative techniques. One of the commonly used approaches is based on multi variate Sturm sequences [Mil92, Ped91, KKM99, KCMK99, CKM99] However, for computations involving moderate degree algebraic primitives, these algorithms can be slow. 1.1 Main Results In this paper, we present representations and novel algorithms for computing roots of polynomial systems and evaluating signs of algebraic predicates. No ....

[Article contains additional citation context not shown here]

John Keyser, Tim Culver, Dinesh Manocha, and Shankar Krishnan. MAPC: A library for efficient and exact manipulation of algebraic points and curves. In Proc. 15th Annu. ACM Sympos. Comput. Geom., pages 360--369, 1999.

No context found.

J. Keyser, T. Culver, D. Manocha, and Shankar Krishnan. MAPC: A library for efficient and exact manipulation of algebraic points and curves. In Proc. 15th Annu. Sympos. Comput. Geom., pages 360--369, 1999.

No context found.

J. Keyser, T. Culver, D. Manocha, and S. Krishnan. MAPC: A library for efficient and exact manipulation of algebraic points and curves. In Proc. 15th Annu. ACM Sympos. Comput. Geom., pages 360--369, 1999.

No context found.

J. Keyser, T. Culver, D. Manocha, and S. Krishnan. MAPC: A library for efficient and exact manipulation of algebraic points and curves. Technical Report TR98-038, University of N. Carolina, Chapel Hill, 1998.

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