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Resolution independent curve rendering using programmable graphics hardware
 Transactions on Graphics
"... Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of ..."
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Cited by 31 (2 self)
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Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from Permissions
Interval Arithmetic, Affine Arithmetic, Taylor Series Methods: Why, What Next?
, 2003
"... In interval computations, the range of each intermediate result r is described by an interval r. To decrease excess interval width, we can keep some information on how r depends on the input x = (x 1 ; : : : ; xn ). There are several successful methods of approximating this dependence; in these m ..."
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Cited by 4 (0 self)
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In interval computations, the range of each intermediate result r is described by an interval r. To decrease excess interval width, we can keep some information on how r depends on the input x = (x 1 ; : : : ; xn ). There are several successful methods of approximating this dependence; in these methods, the dependence is approximated by linear functions (affine arithmetic) or by general polynomials (Taylor series methods). Why linear functions and polynomials? What other classes can we try? These questions are answered in this paper.
Useful Computations Need Useful Numbers
"... Most of us have taken the exact rational and approximate numbers in our computer algebra systems for granted for a long time, not thinking to ask if they could be significantly better. With exact rational arithmetic and adjustableprecision floatingpoint arithmetic to precision limited only by the ..."
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Cited by 2 (2 self)
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Most of us have taken the exact rational and approximate numbers in our computer algebra systems for granted for a long time, not thinking to ask if they could be significantly better. With exact rational arithmetic and adjustableprecision floatingpoint arithmetic to precision limited only by the total computer memory or our patience, what more could we want for such numbers? It turns out that there is much more that can be done that permits us to obtain exact results more often, more intelligible results, approximate results guaranteed to have requested error bounds, and recovery of exact results from approximate ones. 1
Robust Adaptive Polygonal Approximation of Implicit Curves
"... We present an algorithm for computing a robust adaptive polygonal approximation of an implicit curve in the plane. The approximation is adapted to the geometry of the curve because the length of the edges varies with the curvature of the curve. Robustness is achieved by combining interval arithmet ..."
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Cited by 1 (0 self)
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We present an algorithm for computing a robust adaptive polygonal approximation of an implicit curve in the plane. The approximation is adapted to the geometry of the curve because the length of the edges varies with the curvature of the curve. Robustness is achieved by combining interval arithmetic and automatic differentiation.
A recursive taylor method for algebraic curves and surfaces
 Proc. Comp. Methods for Algebraic Spline Surfaces
, 2003
"... Abstract. This paper examines recursive Taylor methods for multivariate polynomial evaluation over an interval, in the context of algebraic curve and surface plotting as a particular application representative of similar problems in CAGD. The modified affine arithmetic method (MAA), previously shown ..."
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Abstract. This paper examines recursive Taylor methods for multivariate polynomial evaluation over an interval, in the context of algebraic curve and surface plotting as a particular application representative of similar problems in CAGD. The modified affine arithmetic method (MAA), previously shown to be one of the best methods for polynomial evaluation over an interval, is used as a benchmark; experimental results show that a second order recursive Taylor method (i) achieves the same or better graphical quality compared to MAA when used for plotting, and (ii) needs fewer arithmetic operations in many cases. Furthermore, this method is simple and very easy to implement. We also consider which order of Taylor method is best to use, and propose that second order Taylor expansion is generally best. Finally, we briefly examine theoretically the relation between the Taylor method and the MAA method. 1
Scan Converting Spirals
 Proc. of WSCG 2002
"... Scanconversion of Archimedes ' spiral (a straight line in polar coordinates) is investigated. It is shown that an exact algorithm requires transcendental functions and, thus, cannot have a fast and exact integer implementation. Piecewise polynomial approximations are discussed and a simple alg ..."
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Cited by 1 (1 self)
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Scanconversion of Archimedes ' spiral (a straight line in polar coordinates) is investigated. It is shown that an exact algorithm requires transcendental functions and, thus, cannot have a fast and exact integer implementation. Piecewise polynomial approximations are discussed and a simple algorithm based on piecewise circular approximation is derived. Variations of the algorithms allow to scan convert other types of spirals.
Complex Numerical Values of the Wright! function
"... This paper details an e±cient general method and a Maple implementation for the direct numerical evaluation of the Wright! function to arbitrary precision over C. Because! is discontinuous along two rays in C this is nontrivial, and the implementation demonstrates the utility of computer algebra sup ..."
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This paper details an e±cient general method and a Maple implementation for the direct numerical evaluation of the Wright! function to arbitrary precision over C. Because! is discontinuous along two rays in C this is nontrivial, and the implementation demonstrates the utility of computer algebra support for signed zero and control of rounding modes.
Constructive Solid Geometry with Projection: An Approach to Piano Movers ’ Problem
"... Abstract. Configuration space obstacles are regions in a configuration space which represent forbidden configuration of the object due to the presence of other objects. They can be regarded as geometric objects which are representable using Boolean combinations of equations and inequalities. Moreove ..."
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Abstract. Configuration space obstacles are regions in a configuration space which represent forbidden configuration of the object due to the presence of other objects. They can be regarded as geometric objects which are representable using Boolean combinations of equations and inequalities. Moreover, they can also be represented using existential quantifiers which correspond to geometric projections. If the projected variables only occur algebraically then it is possible to eliminate quantifiers and represent configuration space obstacles in the semialgebraic form. However, no matter how it is done, the quantifier elimination is computationally hard and the output in semialgebraic representation is often large and cumbersome. Therefore, we are looking for a possibility to work directly with the quantified representation of the configuration space obstacles. We investigate combination of tools which can be used in conjunction with this quantified representation. The main idea is the use of interval evaluation on equations and inequalities involving some transcendental functions. We suggested an extended version of Constructive Solid Geometry system which has trigonometric functions as well as the usual Boolean operators. The combination of this system together with interval arithmetic allows simplification and spatial subdivision and pruning to be done in a natural way. We also raise the possibility of an extended Constructive Solid Geometry system which would have projection and boundary formation as operators. This would allow compact representation of the configuration space, but presents computational problems which are, as yet, unsolved. Supported by the scholarship Technolgiestipendien Südostasien 2005/2006 (Postdoc)