6533b822fe1ef96bd127d666
RESEARCH PRODUCT
A constructive theory of shape
Vladimir García-moralessubject
Convex hullConnected spacePure mathematicsSeries (mathematics)Dynamical systems theoryPlane curveGeneral MathematicsApplied MathematicsGeneral Physics and AstronomyStatistical and Nonlinear PhysicsNumerical Analysis (math.NA)ConstructiveAttractorFOS: MathematicsMathematics - Numerical AnalysisParametric equationMathematicsdescription
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 $\kappa \in (0,\infty)$. 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.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2021-11-01 |