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A constructive theory of shape

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  • García-Morales, Vladimir

Abstract

We formulate a theory of shape valid for objects of arbitrary dimension whose contours are path connected. We apply this theory to the design and modeling of viable trajectories of complex dynamical systems. Infinite families of qualitatively similar shapes are constructed giving as input a finite ordered set of characteristic points (landmarks) and the value of a continuous parameter κ∈(0,∞). We prove that all shapes belonging to the same family are located within the convex hull of the landmarks. The theory is constructive in the sense that it provides a systematic means to build a mathematical model for any shape taken from the physical world. We illustrate this with a variety of examples: (chaotic) time series, plane curves, space filling curves, knots and strange attractors.

Suggested Citation

  • García-Morales, Vladimir, 2021. "A constructive theory of shape," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    • Handle: RePEc:eee:chsofr:v:152:y:2021:i:c:s0960077921007803
    DOI: 10.1016/j.chaos.2021.111426
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    References listed on IDEAS

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    1. Kwang‐Rae Kim & Ian L. Dryden & Huiling Le & Katie E. Severn, 2021. "Smoothing splines on Riemannian manifolds, with applications to 3D shape space," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 83(1), pages 108-132, February.
    2. Freeman Dyson, 2004. "A meeting with Enrico Fermi," Nature, Nature, vol. 427(6972), pages 297-297, January.
    3. Peter E. Jupp & John T. Kent, 1987. "Fitting Smooth Paths to Spherical Data," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 36(1), pages 34-46, March.
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