Tight bounding ball for affine IFS attractor

Martyn, Tomasz

doi:10.1016/s0097-8493(03)00089-x

Cited by 11 publications

(3 citation statements)

References 9 publications

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“…Sharp et al [3,12] determine bounding circles with a given fixed center for the purpose of fitting the attractor on the screen. Martyn [8] gives an algorithm that seeks the tightest bounding sphere of an IFS fractal, via some potentially expensive subroutines.…”

Section: Other Boundsmentioning

confidence: 99%

Apollonian Circumcircles of IFS Fractals

Vass

2013

Preprint

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Euclidean triangles and IFS fractals seem to be disparate geometrical concepts, unless we consider the Sierpiński gasket, which is a self-similar collection of triangles. The "circumcircle" hints at a direct link, as it can be derived for three-map IFS fractals in general, defined in an Apollonian manner. Following this path, one may discover a broader relationship between polygons and IFS fractals. * * Previously titled "Explicit Bounding Circles for IFS Fractals". Some of these results appeared in the author's doctoral dissertation [14].

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Section: Other Boundsmentioning

confidence: 99%

Apollonian Circumcircles of IFS Fractals

Vass

2013

Preprint

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“…Martyn [10] showed that the solution of this system is the centroid of the attractor with particular weights. To obtain a better approximation, he presented a heuristic iterative method called "balancing the attractor".…”

Section: Related Workmentioning

confidence: 99%

Approximate convex hull of affine iterated function system attractors

Mishkinis

Gentil

Lanquetin

et al. 2012

Chaos, Solitons & Fractals

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In this paper, we present an algorithm to construct an approximate convex hull of the attractors of an affine iterated function system (IFS). We construct a sequence of convex hull approximations for any required precision using the self-similarity property of the attractor in order to optimize calculations. Due to the affine properties of IFS transformations, the number of points considered in the construction is reduced. The time complexity of our algorithm is a linear function of the number of iterations and the number of points in the output convex hull. The number of iterations and the execution time increases logarithmically with increasing accuracy. In addition, we introduce a method to simplify the approximation of the convex hull without loss of accuracy.

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“…Many authors have taken this approach, including the simple bounding sphere of Hart and DeFanti [16], the tighter sphere of Rice [17], and the anisotropic sphere of Martyn [18]. Convex bounds include the linear programming method of Lawlor and Hart [19], the bound refinement technique of Chu and Chen [20], and a heapbased convex bound refinement method [21,Appendix C].…”

Section: Bounding Iterated Function Systemsmentioning

confidence: 99%

GPU-Accelerated Rendering of Unbounded Nonlinear Iterated Function System Fixed Points

Lawlor

2012

ISRN Computer Graphics

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Nonlinear functions, including nonlinear iterated function systems, have interesting fixed points. We present a non-Lipschitz theoretical approach to nonlinear function system fixed points which generalizes to noncontractive functions, compare several methods for evaluating such fixed points on modern graphics hardware, and present a nonlinear generalization of Barnsley's Deterministic Iteration Algorithm. Unlike the many existing randomized rendering algorithms, this deterministic method avoids noncoherent branching and memory access and takes advantage of programmable texture mapping hardware. Together with the performance potential of modern graphics hardware, this allows us to animate high-quality and high-definition fixed points in real time.

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Tight bounding ball for affine IFS attractor

Cited by 11 publications

References 9 publications

Apollonian Circumcircles of IFS Fractals

Apollonian Circumcircles of IFS Fractals

Approximate convex hull of affine iterated function system attractors

GPU-Accelerated Rendering of Unbounded Nonlinear Iterated Function System Fixed Points

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