A constructive theory of shape

TL;DR

This paper develops a comprehensive mathematical framework for modeling and analyzing shapes of arbitrary dimension, enabling systematic construction of complex shapes from landmarks and parameters, with applications across dynamical systems and physical objects.

Contribution

It introduces a constructive, parameterized shape theory applicable to diverse objects, linking shape construction to landmarks and providing proofs of shape properties within convex hulls.

Findings

Shapes of the same family lie within the convex hull of landmarks

The theory applies to chaotic time series, knots, and strange attractors

Provides a systematic method for shape modeling from physical data

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 $\kappa \in (0,\infty)$. 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.