A constructive theory of shape

Abstract

We formulate a theory of shape valid for objects of arbitrary dimension whose contours are path connected. 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, ∞). 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 and knots. In particular, we construct an infinite family of fractal curves whose shape approaches the one of Hilbert's curve in the limit κ → 0, yielding, for κ finite, fractal space-filling curves with nonlinear contours.