Comparison of interval methods for plotting algebraic curves

Martin, Ralph R.; Shou, Huahao; Voiculescu, Irina; Bowyer, Adrian; Wang, Guojin

doi:10.1016/s0167-8396(02)00146-2

Cited by 67 publications

(63 citation statements)

References 20 publications

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“…We consider only advanced interval analysis (IA) techniques here; please refer to [12] for a concise overview. The First Affine Form (AF1) [14] is defined as:…”

Section: Preliminariesmentioning

confidence: 99%

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Visualizing Arcs of Implicit Algebraic Curves, Exactly and Fast

Emeliyanenko

Berberich

Sagraloff

2009

Advances in Visual Computing

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Abstract. Given a Cylindrical Algebraic Decomposition of an implicit algebraic curve, visualizing distinct curve arcs is not as easy as it stands because, despite the absence of singularities in the interior, the arcs can pass arbitrary close to each other. We present an algorithm to visualize distinct connected arcs of an algebraic curve efficiently and precise (at a given resolution), irrespective of how close to each other they actually pass. Our hybrid method inherits the ideas of subdivision and curve-tracking methods. With an adaptive mixed-precision model we can render the majority of algebraic curves using floating-point arithmetic without sacrificing the exactness of the final result. The correctness and applicability of our algorithm is borne out by the success of our webdemo 1 presented in [10].

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“…We consider only advanced interval analysis (IA) techniques here; please refer to [12] for a concise overview. The First Affine Form (AF1) [14] is defined as:…”

Section: Preliminariesmentioning

confidence: 99%

“…The efficient technique to further shrink the interval bounds is by exploiting the derivative information [12]. In order to evaluate a polynomial f (x) on X = [x, x], we first evaluate its derivative f on X using the same interval method.…”

Section: Preliminariesmentioning

confidence: 99%

Visualizing Arcs of Implicit Algebraic Curves, Exactly and Fast

Emeliyanenko

Berberich

Sagraloff

2009

Advances in Visual Computing

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“…This approach has two ingredients: the method to compute f (B) and the termination condition. Many interesting methods are proposed to compute f (B) and an excellent survey is given in [16]. For the termination condition, the simplest one is to give a threshold, say, the size of a pixel.…”

Section: (D)mentioning

confidence: 99%

Topology determination and isolation for implicit plane curves

Cheng

Gao

2009

Proceedings of the 2009 ACM Symposium on Applied Computing

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Abstract. A method is proposed to generate an isolation for a plane curve, which is a set of boxes covering the curve, having the same topology as the curve, and approximating the curve to any given precision. The method uses symbolic computation to guarantee correctness and uses interval analysis whenever possible to enhance efficiency. This leads to a quite effective hybrid method for plane curve isolation.

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“…Thus its operation interval result can be narrower and more accurate, especially in long chain calculation. So the affine algorithm has been widely used in artificial intelligence systems analysis [6], system stability analysis [7], circuit response boundary analysis [8] and computer graphics [9]. Yet errors will still exist when using affine algorithm to calculate  .…”

Section: Introductionmentioning

confidence: 99%