# A constructive theory of shape

Vladimir García-Morales  Departament de Física de la Terra i Termodinàmica,
Universitat de València, E-46100 Burjassot, Spain<EMAIL_ADDRESS>
###### 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.

_Keywords:_ viable evolutions; parametric equations; space-filling curves;
knots; time series

##  I Introduction

A fact that has captivated the human mind since ancient times is the diversity
of shapes found in nature Ball . It is, however, only in the last half-century
that mathematical theories have been developed to study shape more rigorously
and systematically, from topological Borsuk ; Mardesic  and statistical
Kendall1 ; Kendall2 ; Dryden  points of view. The great scientific interest of
shape is that it is a unifying concept Leyton . Besides being a property of
all living systems Thompson , shape is of crucial importance in architecture
Alexander  and art Leyton ; Leypain ; Leyarch . Physical explanations for the
origin of shape in non-equilibrium systems have been proposed Bejan  that are
of great interest in engineering and optimization problems. Quantitative
analysis of the shapes of biological organs is often necessary in agronomy,
medicine, genetics, ecology, and taxonomy Bookstein ; Iwata2 ; Costa .

Shapes and images are essentially sets Aubin3 . Therefore, the analysis,
processing, evolution, regulation and control of shape require the
consideration of set-valued maps. Viability theory Aubin1 ; Aubin2  and the
study and reconstruction of invariant sets Blanchini1 ; Blanchini2 ; Nagumo ;
Takens  provide pathways to address this problem. Here, we present an
alternative approach linking it to a nonlinear dynamical systems perspective.
Although our theory is general in scope, it is able to provide specific
mathematical models of shapes and their viable trajectories. We say that a
trajectory is viable in a set $S$ if, for an initial condition in $S$, the
trajectory remains in $S$ in any future time.

Spline approximation, Fourier analysis Kuhl  and wavelet transforms Barache ;
Osowski  are popular methods for analyzing shape. The two general approaches
to shape are: region-based (the region in the image corresponding to the
analyzed object is considered); and boundary-based (shape is characterized in
terms of its silhouette) Barache ; Pavlidis . Although we shall focus in this
article in the second approach, representing shape as a parametric contour and
endowing this representation with ‘nice’ properties, the first approach is
also possible with help of the concept of $\kappa$-families of scenes
introduced in this article. Besides generality of the method, a most desirable
property that is sought is _ease_ : once the contour of the shape is traced
and a _finite_ sequence of points in this contour (the landmarks) is selected,
a mathematical model for the specific shape and for an infinite family of
qualitatively related shapes can be immediately written down. There is no need
of calculating any coefficients. This requirement is, clearly, not satisfied
by any of the above methods. It is this property of the theory here presented
what makes it _constructive_. The theory is intended to be used as a
mathematical tool for scientists to provide quick and handy mathematical
models of complex shapes. The theory is also parsimonious because a minimal
amount of information suffices to generate an infinite family of qualitatively
related shapes. Most remarkably, all shapes belonging to the family are
located within the set constituted by the convex hull of the landmarks. Thus,
if the landmarks are meant to be points on the trajectory of a complex
dynamical system (possibly subject to perturbations) an infinite number of
viable trajectories for the dynamical system can be obtained, all contained in
the convex hull of the landmarks.

Our theory of shape is based in nonlinear $\mathcal{B}_{\kappa}$-embeddings, a
mathematical structure that has been recently introduced by the author
homotopon ; JPHYSCOMPLEX  and applied to the problem of finding all the roots
of a polynomial in the complex plane homotopon . The theory is able to
represent any path-connected shape and to resolve details of the shape at
different scales as a continuous smoothing parameter $\kappa$ is varied. This
work is motivated by the following general problem, to which we give a general
solution: _For a given finite sequence of data (points) in $\mathbb{R}^{n}$
find a mathematical model for the shape of the data that: a) can fit the given
data to any degree of accuracy; b) the interpolation between the data is a
non-oscillating, non-piecewise, infinitely differentiable function of its
variable (‘time’) argument $t$; c) relates each individual shape to closely
related shapes (deformations) obtained by varying the continuous parameter
$\kappa$; d) places any $\kappa$-deformed shape within the convex hull of the
original data_.

The outline of this article is as follows. In Section II we present our
definitions of shape and $\kappa$-family of shapes preceded by all the
concepts necessary to understand the definitions. Nonlinear embeddings are
introduced and the properties of interest here are worked out to make the
theory self-contained. In Section III we prove that the $\kappa$-family of
shapes is contained within the convex hull of the landmarks. Then, in Section
IV this general result is applied to different classes of dynamical systems
and it is established how viable trajectories can be constructed for them. In
Section V we give examples of application of the theory to curves in 1D,
periodic shapes (waveforms), curves in 2D (including fractal space-filling
curves Hilbert ), curves in 3D (e.g. knots Kaufman  and strange attractors).
The impatient reader can directly jump to this section going back to Section
II when needed. In Section VI we present some conclusions and discussion of
the theory and sketch some directions for possible future work.

##  II Nonlinear embeddings and the definition of shape

We consider $N$ points in $D$-dimensional Euclidean space extracted from a
contour. We shall call these points _characteristic points_ or _landmarks_ of
the shape. We choose a labeling for these landmarks so that they are ordered
one after the other in succession starting with point $0$ up to point $N-1$.
These landmarks can be obtained directly from the boundary by judicious,
intuitive or automated choice. Neither a chain code Freeman ; Bribiesca  nor
tracing the whole contour is needed (as it is the case with Fourier methods as
Kuhl ). The labeling of the landmarks is important because, as we shall see,
it is related to time. In the case of closed contours any cyclic permutation
of the landmarks is equivalent. The points are thus, specified by vectors
$\mathbf{r}_{0},\mathbf{r}_{1},\ldots,\mathbf{r}_{N-1}$. By introducing a
Kronecker delta convolution, we can rewrite the $n$-th point ($0\leq n\leq
N-1$) as

$\mathbf{r}_{n}=\sum_{j=0}^{N-1}\mathbf{r}_{j}\delta_{jn}$ (1)

where $\delta_{nj}=1$ if $n=j$ and $\delta_{nj}=0$ otherwise, is the Kronecker
delta. We note that, indeed, we can write, equivalently,

$\mathbf{r}_{n}=\frac{\sum_{j=0}^{N-1}\mathbf{r}_{j}\delta_{jn}}{\sum_{j=0}^{N-1}\delta_{jn}},$
(2)

since $\sum_{j=0}^{N-1}\delta_{jn}=1$ because $0\leq n\leq N-1$. The
construction of a nonlinear embedding begins by noting that the Kronecker
delta admits the following simple representation JPHYSCOMPLEX

$\delta_{nj}=\mathcal{B}\left(n-j,\frac{1}{2}\right)$ (3)

where

$\displaystyle\mathcal{B}(x,y)$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\left(\frac{x+y}{|x+y|}-\frac{x-y}{|x-y|}\right)$ (4)
$\displaystyle=$ $\displaystyle{\begin{cases}\text{sign}(y)&{\text{if
}}|x|<|y|\\\ \text{sign}(y)/2&{\text{if }}|x|=|y|,y\neq 0\\\
0&{\text{otherwise}}\end{cases}}$

is the $\mathcal{B}$-function VGM1 ; VGM2 ; JPHYSCOMPLEX , with
$x,y\in\mathbb{R}$. We can thus use Eq. (3) to replace the Kronecker deltas
entering in Eq. (2). We obtain

$\mathbf{r}_{n}=\frac{\sum_{j=0}^{N-1}\mathbf{r}_{j}\mathcal{B}\left(n-j,\frac{1}{2}\right)}{\sum_{j=0}^{N-1}\mathcal{B}\left(n-j,\frac{1}{2}\right)},$
(5)

We can now ‘fuzzify’ this expression by means of the approach sketched in
fuzzypap ; JPHYSCOMPLEX . First, we replace the discrete variable
$n\in\mathbb{Z}$ by a continuous time variable $t\in\mathbb{R}$. We then
define

$\mathbf{r}(t):=\frac{\sum_{j=0}^{N-1}\mathbf{r}_{j}\mathcal{B}\left(t-j,\frac{1}{2}\right)}{\sum_{j=0}^{N-1}\mathcal{B}\left(t-j,\frac{1}{2}\right)},$
(6)

Obviously, at times $t=n+\delta$, where $-1/2<\delta<1/2$

$\mathbf{r}(n+\delta)=\mathbf{r}_{n}$ (7)

The fuzzification approach is completed by replacing all
$\mathcal{B}$-functions in Eq. (6) by the $\mathcal{B}_{\kappa}$-function
fuzzypap  given by

$\mathcal{B}_{\kappa}(x,y):=\frac{1}{2}\left[\tanh\left(\frac{x+y}{\kappa}\right)-\tanh\left(\frac{x-y}{\kappa}\right)\right]$
(8)

This is the so-called _replacement mode I_ in JPHYSCOMPLEX . We then define
the nonlinear embedding

$\mathbf{r}_{\kappa}(t):=\frac{\sum_{j=0}^{N-1}\mathbf{r}_{j}\mathcal{B}_{\kappa}\left(t-j,\frac{1}{2}\right)}{\sum_{j=0}^{N-1}\mathcal{B}_{\kappa}\left(t-j,\frac{1}{2}\right)}.$
(9)

We observe, that in the limit $\kappa\to 0$, Eq. (9) becomes Eq. (6)

$\lim_{\kappa\to 0}\mathbf{r}_{\kappa}(t)=\mathbf{r}(t)$ (10)

Because of Eqs. (7) and (10), Eq. (9) is, in turn, equal to Eq. (2) in the
limit $\kappa\to 0$ when $t\in[0,N-1]$ is an integer. When $\kappa\to\infty$
we have, since $\mathcal{B}_{\kappa}(x,y)\to y/\kappa$ JPHYSA ; homotopon

$\lim_{\kappa\to\infty}\mathbf{r}_{\kappa}(t)=\frac{\sum_{j=0}^{N-1}\mathbf{r}_{j}\frac{1}{2\kappa}}{\sum_{j=0}^{N-1}\frac{1}{2\kappa}}=\frac{1}{N}\sum_{j=0}^{N-1}\mathbf{r}_{j}=\mathbf{r}^{*}$
(11)

where we have introduced the _centroid_ (‘center of masses’) $\mathbf{r}^{*}$
of the distribution of landmarks.

We note that, since JPHYSA ; JPHYSCOMPLEX

$\sum_{j=0}^{N-1}\mathcal{B}_{\kappa}\left(t-j,\frac{1}{2}\right)=\mathcal{B}_{\kappa}\left(t-\frac{N-1}{2},\frac{N}{2}\right),$
(12)

we can write Eq. (9) as

$\mathbf{r}_{\kappa}(t)=\sum_{j=0}^{N-1}\mathbf{r}_{j}\frac{\mathcal{B}_{\kappa}\left(t-j,\frac{1}{2}\right)}{\mathcal{B}_{\kappa}\left(t-\frac{N-1}{2},\frac{N}{2}\right)}\quad
t\in[0,N-1],t\in\mathbb{R}.$ (13)

This equation defines _open_ shapes. We shall also consider _closed_ shapes.
In the letter, although they are still specified by a finite set of landmarks,
these return periodically as $t$ is varied from $-\infty$ to $\infty$. We have

$\mathbf{r}_{n+kN}=\mathbf{r}_{n},\qquad\forall k\in\mathbb{Z},$ (14)

so that landmark $\mathbf{r}_{0}$ is the successor of landmark
$\mathbf{r}_{N-1}$. In this case, instead of Eq. (9), we shall consider the
following expression

$\mathbf{r}_{\kappa}(t):=\frac{\sum_{j=0}^{N-1}\mathbf{r}_{j}\Pi_{\kappa}\left(t-j,\frac{1}{2};N\right)}{\sum_{j=0}^{N-1}\Pi_{\kappa}\left(t-j,\frac{1}{2};N\right)},\quad
t\in[0,\infty),t\in\mathbb{R},$ (15)

where

$\Pi_{\kappa}\left(x,y;T\right)=\mathcal{B}_{\kappa}\left(\sin\frac{\pi
x}{T},\sin\frac{\pi y}{T}\right)$ (16)

is the periodic $\mathcal{B}_{\kappa}$-function introduced in fuzzypap . From
this equation, we note that

$\mathbf{r}_{\kappa}(t+kN)=\mathbf{r}_{\kappa}(t),\qquad\forall
k\in\mathbb{Z}.$ (17)

When $\kappa\to\infty$, Eq. (15) reduces to

$\lim_{\kappa\to\infty}\mathbf{r}_{\kappa}(t)=\frac{1}{N}\sum_{j=0}^{N-1}\mathbf{r}_{j}=\mathbf{r}^{*},$
(18)

i.e. the shape collapses to centroid. Eqs. (13) and (15) are the main
equations of our approach.

We now formally define shape. Let $\mathbf{r}_{0}$, $\mathbf{r}_{1}$,
$\ldots$, $\mathbf{r}_{N-1}$ be a sequence of $N$ points in $D$-dimensional
Euclidean space $\mathbb{R}^{D}$. These points may be repeated in the
collection. We call these points _characteristic points of the shape_ or,
equivalently, _landmarks_. Over these landmarks, we define a $\kappa$_-family
of open shapes_ by Eq. (13) and a $\kappa$_-family of closed shapes_ by Eq.
(15). For each specific value of $\kappa\in(0,\infty)$ we say that the latter
equations define an open (resp. closed) shape. We say that a $\kappa$-family
of shapes is equal to another if both have exactly the same landmarks in the
same order.

We define the $\epsilon$-shape of a given $\kappa$-family as the resulting
shape $\mathbf{r}_{\epsilon}(t)$ when $\kappa=\epsilon$ is nonzero but
vanishingly small. In all examples in Section V the $\epsilon$-shape is
visually indistinguishable of the shapes obtained for $\kappa=0.01$. We say
that the $\epsilon$-shape is a faithful representation of the shape to be
modelled if, to sufficient degree of accuracy, fits the original shape (note
that from the original shape we are only taking a finite set of $N$
landmarks). All $\epsilon$-shapes in the examples are faithful. A faithful
representation can always be obtained by taking a sufficient number $N$ of
landmarks that are representative of its contour.

Eqs. (13) and (15) constitute _parametric equations_ of the shapes and it is
useful to see parameter $t$ as time. Although in this work we shall focus on
landmarks in $\mathbb{R}^{D}$, other vector spaces are possible as well. Any
landmark $n$, $n=0,1,\ldots,N-1$, can be recovered from Eqs. (13) and (15) by
putting $t=n$ in these equations and taking the limit $\kappa\to 0$

$\mathbf{r}_{n}=\lim_{\kappa\to 0}\mathbf{r}_{\kappa}(n)$ (19)

The process of obtaining a particular shape begins by specifying a finite
suitable set of landmarks. We discuss a variety of examples on how to
construct mathematical models for $\kappa$-families of shapes. In general the
landmarks can be specified in a wide variety of methods. For example, they can
be obtained: (1) from experimentally measured data or model; (2) randomly
(according to a probability distribution) or at will; (3) by reading them
directly from a pre-existing figure; (4) recursively.

The $\kappa$-family constitutes a mathematical model in which shapes are
parametrized by means of a ‘time’ real parameter $t$ (a larger number of
parameters is possible) and a smoothing real parameter $\kappa$. The shapes
within the family are controlled by means of $\kappa$ and are located between
the interpolated landmarks (the $\epsilon$-shape) and the centroid of the
landmarks.

It is possible to hierarchically embed families of shapes in more complex
structures that we shall call _scenes_. A scene is any disconnected set of
shapes and can be expressed as a direct product of them. Let
$\mathbf{r}_{\kappa_{0}}(t,0)$, $\mathbf{r}_{\kappa_{1}}(t,2)$, $\ldots$,
$\mathbf{r}_{\kappa_{M}-1}(t,M-1)$ be a different $\kappa$-family of shapes
given either by Eq. (13) or by Eq. (15) each. We can construct a type-I
$\eta$-family of scenes $\mathbf{S}_{\eta}(t)$ as

$\displaystyle\mathbf{S}_{\eta}(t,q)$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{M-1}\mathbf{r}_{\kappa_{j}}(t,j)\frac{\mathcal{B}_{\eta}\left(q-j,\frac{1}{2}\right)}{\mathcal{B}_{\eta}\left(q-\frac{M-1}{2},\frac{M}{2}\right)}$
$\displaystyle t\in[0,N-1],q\in[0,M-1],t,q\in\mathbb{R}.$

and a type-II $\eta$-family of scenes as

$\displaystyle\mathbf{S}_{\eta}(t,q)$ $\displaystyle=$
$\displaystyle\frac{\sum_{j=0}^{M-1}\mathbf{r}_{\kappa_{j}}(t,j)\Pi_{\eta}\left(q-j,\frac{1}{2};M\right)}{\sum_{j=0}^{M-1}\Pi_{\eta}\left(t-j,\frac{1}{2};M\right)}$
$\displaystyle t,q\in[0,\infty),t,q\in\mathbb{R}.$

We see that shapes play within a scene an analogous role to landmarks within a
shape. In the limit $\eta\to 0$ all shapes within the scene are disconnected.
In the limit $\eta\to\infty$ the shapes within the scene mix with each other
and collapse to the average shape

$\lim_{\eta\to\infty}\mathbf{S}_{\eta}(t,q)=\frac{1}{M}\sum_{j=0}^{M-1}\mathbf{r}_{\kappa_{j}}(t,j)$
(22)

Each shape $m$ within a $\kappa$-family of scenes can be recovered from Eq.
(II) and (II) by putting $q=m\in[0,M-1]$ and taking the limit $\eta\to 0$.

$\lim_{\eta\to 0}\mathbf{S}_{\eta}(t,n)=\mathbf{r}_{\kappa_{n}}(t,n)$ (23)

Scenes can be hierarchically embedded in increasingly complex structures in a
similar way as landmarks are embedded in shapes and the latter are embedded in
scenes. Every element in the complex structure, landmark, shape, or scene, can
thus be recovered as sketched above. We find that our theory thus satisfies
Leyton’s criteria for a successful theory of shape: 1) _maximization of
transfer_ and 2) _maximization of recoverability_ Leyton . The first criterion
is satisfied because complex structures in our theory can always be seen as
the result of the transfer of simpler structures since these are embedded in
the former. The second criterion is also satisfied because all elements that
are being transferred can be recovered out of simple operations (taking
limits) on the superior structures. Leyton’s generative theory of shape is
abstract and makes use of group theoretical notions (the wreath product being
at its core) Leyton . We believe than our theory of shape is simpler because,
being general, it is also specific in the following sense: a mathematical
model can always be fully worked out explicitly and systematically for any
shape taken as example (and this is why we call it a _constructive_ theory).

##  III Convexity and transformations

The $\kappa$-families of shapes obtained from Eqs. (13) and (15) have several
important mathematical properties that we now discuss. We have the following
result.

######  Theorem III.1.

Let _conv_ r denote the convex hull of the points $\mathbf{r}_{0}$,
$\mathbf{r}_{1}$, $\ldots$, $\mathbf{r}_{N-1}$ in $D$-dimensional Euclidean
space $\mathbb{R}^{D}$. Then: i) the points $\mathbf{r}_{\kappa}(t)$ in any
open shape of a type-1 $\kappa$-family of shapes given by Eq. (13) satisfy
$\mathbf{r}_{\kappa}(t)\in$ _conv_ r, $\forall\kappa\in(0,\infty)$ and
$\forall t\in[0,N-1],t\in\mathbb{R}$; ii) the points $\mathbf{r}_{\kappa}(t)$
in any closed shape of a type-2 $\kappa$-family of shapes given by Eq. (15)
satisfy $\mathbf{r}_{\kappa}(t)\in$ _conv_ r, $\forall\kappa\in(0,\infty)$ and
$\forall t\in[0,\infty),t\in\mathbb{R}$.

###### Proof.

Since $\text{conv}_{\mathbf{r}}$ is a convex set and
$\mathbf{r}_{j}\in\text{conv}_{\mathbf{r}}$, for $j=0,\ldots,N-1$ we have that
any affine combination

$\mathbf{r}^{\prime}=\sum_{j=0}^{N-1}\lambda_{j}\mathbf{r}_{j}$ (24)

satisfies $\mathbf{r}^{\prime}\in\text{conv}_{\mathbf{r}}$ for any non-
negative real numbers $\lambda_{0}$, $\ldots$, $\lambda_{N-1}$ such that
$\lambda_{0}+\lambda_{1}+\ldots\lambda_{N-1}=1$ Rockafellar . If we,
therefore, take

$\lambda_{j}=\frac{\mathcal{B}_{\kappa}\left(t-j,\frac{1}{2}\right)}{\mathcal{B}_{\kappa}\left(t-\frac{N-1}{2},\frac{N}{2}\right)}=\frac{\mathcal{B}_{\kappa}\left(t-j,\frac{1}{2}\right)}{\sum_{j=0}^{N-1}\mathcal{B}_{\kappa}\left(t-j,\frac{1}{2}\right)}$
(25)

we have both $0\leq\lambda_{j}\leq 1$, and $\sum_{j=0}^{N-1}\lambda_{j}=1$. By
replacing Eq. (25) in Eq. (24) we obtain Eq. (13) and i) follows. Similarly,
if we take

$\lambda_{j}=\frac{\Pi_{\kappa}\left(t-j,\frac{1}{2};N\right)}{\sum_{j=0}^{N-1}\Pi_{\kappa}\left(t-j,\frac{1}{2};N\right)}$
(26)

we, again, have $0\leq\lambda_{j}\leq 1$, and $\sum_{j=0}^{N-1}\lambda_{j}=1$.
By replacing Eq. (26) in Eq. (24) we obtain Eq. (15) and ii) follows. ∎

The importance of the above theorem lies in that all deformed shapes within
the same $\kappa$-family are easily located in Euclidean space
$\mathbb{R}^{D}$: they are all found within the convex hull of the landmarks.
As we shall see in Sec. IV, Theorem III.1 finds an important application in
finding viable trajectories of complex dynamical systems and in designing
invariant sets.

The $\kappa$-families of shapes in Eqs. (13) and (15) are _nonlinear_ in $t$
and $\kappa$ but are _linear_ functions (combinations) of the landmarks. This
has the following important implication: if $\mathbf{T}$ is any transformation
(matrix) sending each landmark $\mathbf{r}_{j}$ to
$\mathbf{T}\cdot\mathbf{r}_{j}$ (with $\cdot$ denoting the inner product),
then the $\kappa$ family of shapes transforms as
$\mathbf{r}_{\kappa}(t)\to\mathbf{T}\cdot\mathbf{r}_{\kappa}(t)$. Thus, if,
for example, $\mathbf{T}$ is an isometry transformation, the shapes
$\mathbf{r}_{\kappa}(t)$ and $\mathbf{T}\cdot\mathbf{r}_{\kappa}(t)$ for a
fixed $\kappa$ are congruent.

##  IV Application to dynamical systems

A smooth dynamical system

$\dot{\mathbf{y}}=\mathbf{F}(\mathbf{y})$ (27)

can be discretized in time as

$\mathbf{y}(t+\delta)\approx\mathbf{y}(t)+\delta\mathbf{F}(\mathbf{y}(t))$
(28)

In general, we have, from this latter equation

$\mathbf{y}(t+(n+1)\delta)\approx\mathbf{y}(t+n\delta)+\delta\mathbf{F}(\mathbf{y}(t+n\delta))$
(29)

Therefore, starting from $t=0$ and ending in time $t=N\delta$, we can obtain
$N$ suitable landmarks for the trajectory of the dynamical system up to
resolution $\delta$ as
$\left(\mathbf{r}_{0},\mathbf{r}_{1},\mathbf{r}_{2},\ldots,\mathbf{r}_{N-1}\right):=\left(\mathbf{y}(0),\mathbf{y}(\delta),\mathbf{y}(2\delta),\ldots,\mathbf{y}((N-1)\delta)\right)$
and the trajectory can be embedded in a $\kappa$-family of open shapes by
means of Eq. (13) where the $\mathbf{r}_{j}$’s are iteratively obtained from
Eq. (29) as

$\mathbf{r}_{j}=\mathbf{r}_{j-1}+\delta\mathbf{F}(\mathbf{r}_{j-1})$ (30)

starting from an initial condition $\mathbf{r}_{0}$.

We can also obtain periodic, closed shapes, over these same $N$ landmarks from
Eq. (15). Any trajectory of the smooth dynamical system can thus be embedded
in a $\kappa$-family of shapes by taking $\delta$ sufficiently small so that
the landmarks obtained constitute a faithful representation of the trajectory
up to precision $\delta$. As an example, in Sec. V.5 we construct
$\kappa$-families of shapes for the Lorenz attractor.

If the dynamical system is a discrete map, instead of Eq. (27) we have that
the dynamics is governed at discrete times $t=j$ by

$\mathbf{y}_{j}=\mathbf{G}(\mathbf{y}_{j-1})$ (31)

A set of $N$ iterations starting from an initial condition $\mathbf{y}_{0}$
can be made with this map so that $N$ landmarks
$\mathbf{r}_{j}=\mathbf{y}_{j}$ are generated, which exactly coincide with the
trajectory of the discrete map. The trajectory jumps discontinuously at
discrete times, but can be embedded in a _continuous_ trajectory by means of
an appropriate $\kappa$-family of shapes (type I) or (type II) by using Eqs.
(13) and (15) above. In Sec. V.1 we show, as example, open $\kappa$-families
of shapes for the logistic map.

By using the concept of $\eta$-family of scenes defined by Eqs. (II) and (II),
it is possible to generalize the above costructions to spatially extended
systems as partial differential equations, coupled map lattices and cellular
automata.

The extension of the above embeddings to non-autonomous dynamical systems is
straigthforward. Of major interest in those complex systems (in the presence
of controls and perturbations) is the notion of positive invariance Blanchini1
; Blanchini2 ; Nagumo . A set $S$ is said to be positively invariant with
respect to a dynamical system if any trajectory $\mathbf{y}(t)$ of the
dynamical system that is well defined for any $t>0$ and that starts with
initial condition $\mathbf{y}_{0}\in S$ satisfies $\mathbf{y}(t)\in S$ for
$t>0$. A set that contains a stable limit cycle (closed periodic orbit) or a
stable fixed point that attract all initial conditions in $S$, is clearly
positively invariant.

A broader notion of positive invariance is crucial in so-called viability
theory Aubin1 ; Aubin2 . Many complex natural and artificial systems,
organizations, and networks do not evolve deterministically nor stochastically
Aubin1 ; Aubin2 . These systems are approached in a general way considering
the trajectories $\mathbf{r}(t)$ (system states) to be constrained to belong
to a set $S$ instead of being governed by systems of differential or
stochastic equations. Indeed, the main requirement is positive invariance for
the trajectories defined analytically or experimentally, so that one has

$\displaystyle\mathbf{r}(t)$ $\displaystyle\in$ $\displaystyle S.$ (32)

An important consequence of Theorem III.1 in this context is that if $S$ is
convex then we can always augment such a complex system given by Eq. (32) with
a control parameter $\kappa$ so that for all $\kappa>0$ we have, as well,

$\displaystyle\mathbf{r}_{\kappa}(t)$ $\displaystyle\in$ $\displaystyle S$
(33)

with $\mathbf{y}_{\kappa}(t)$ being given by Eq. (13) or Eq. (15). Indeed, as
a direct corollary of Theorem III.1 applied to dynamical systems and
positively invariant sets, we have the following.

######  Theorem IV.1.

Let $S$ be any convex set that is positively invariant with respect to a
dynamical system with globally defined trajectories $\mathbf{y}(t)\in S$
($\forall t>0$) starting from initial conditions $\mathbf{y}_{0}\in S$. Let
$\mathbf{r}_{j}$, $j=0,\ldots,N-1$ be $N$ points taken on any of these
trajectories. Then, all points in the curve $\mathbf{r}_{\kappa}(t)$ obtained
from Eq. (13) satisfy $\mathbf{r}_{\kappa}(t)\in S$, $\forall\kappa>0$ and
$\forall t\in[0,N-1]$.

Any convex set $S$ includes as a subset the convex hull
$\text{conv}_{\mathbf{r}}$ of any set of points
$\\{\mathbf{r}_{j};j=0,\ldots,N-1\\}\subset S$. Then, another direct
application of Theorem III.1 to closed trajectories of any dynamical system
(continuous or discrete) is the following.

######  Theorem IV.2.

Let $\mathbf{y}(t)=\mathbf{y}(t+T)$ denote a closed periodic orbit of a
dynamical system with period $T$ and let $\mathbf{r}_{j}$, $j=0,\ldots,N-1$ be
$N$ points taken on the orbit. Then $\mathbf{r}_{\kappa}(t)$ given by Eq. (15)
is a closed curve with period $T$ and is contained in the convex hull _conv_ r
of the points taken on the trajectory of the dynamical system, i.e.
$\mathbf{r}_{\kappa}(t)\in$ _conv_ r, $\forall\kappa>0$ and $\forall t>0$.

##  V Examples

###  V.1 1D shapes: time series

In one dimension, the landmarks $\mathbf{r}_{0}$, $\mathbf{r}_{1}$, $\ldots$,
$\mathbf{r}_{N-1}$ become just a collection of real scalars $X_{0}$, $X_{1}$,
$\ldots$, $X_{N-1}$. It is straightforward to obtain from these landmarks and
Eqs. (13) and (15) $\kappa$-families of shapes. For open shapes (non-periodic
time series on a time window) we have

$\displaystyle r_{\kappa}(t)$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{N-1}X_{j}\frac{\mathcal{B}_{\kappa}\left(t-j,\frac{1}{2}\right)}{\mathcal{B}_{\kappa}\left(t-\frac{N-1}{2},\frac{N}{2}\right)}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad t\in[0,N-1],t\in\mathbb{R},$

and for periodic time series

$\displaystyle r_{\kappa}(t)$ $\displaystyle=$
$\displaystyle\frac{\sum_{j=0}^{N-1}X_{j}\Pi_{\kappa}\left(t-j,\frac{1}{2};N\right)}{\sum_{j=0}^{N-1}\Pi_{\kappa}\left(t-j,\frac{1}{2};N\right)}\qquad
t\in\mathbb{R}.\qquad$ (35)

Figure 1: Shapes $r_{\kappa}(t)$, calculated from Eq. (V.1) for the landmarks
(characteristic points) $y_{j}$ shown in the top left panel for the values of
the continuum parameter $\kappa$ indicated on the panels. The landmarks are
calculated by performing 149 iterations of the logistic map $y_{j}=\mu
y_{j-1}(1-y_{j-1})$, with $y_{0}=0.3$ and $\mu=3.5$. Figure 2: Periodic shapes
obtained from the landmarks $(X_{0},X_{1},\ldots,X_{4})=(1,4,2,2,1)$ and Eq.
(35) for the values of $\kappa$ indicated on the panels. In the left panel,
the landmarks are also shown.

As an example we consider the logistic map as a model providing the landmarks.
We put $X_{j}=y_{j}$ where $y_{j}$ is given by

$y_{j}=\mu y_{j-1}(1-y_{j-1})$ (36)

starting from a given $y_{0}$. If we perform 149 iterations of this map, we
obtain a sequence of 150 landmarks and we can directly construct a
$\kappa$-family of open shapes from these landmarks, using Eq. (13)

$\displaystyle r_{\kappa}(t)$ $\displaystyle=$ $\displaystyle
y_{0}\frac{\mathcal{B}_{\kappa}\left(t,\frac{1}{2}\right)}{\mathcal{B}_{\kappa}\left(t-\frac{149}{2},75\right)}$
$\displaystyle+\mu\sum_{j=1}^{149}y_{j-1}(1-y_{j-1})\frac{\mathcal{B}_{\kappa}\left(t-j,\frac{1}{2}\right)}{\mathcal{B}_{\kappa}\left(t-\frac{149}{2},75\right)}\qquad$
$\displaystyle\qquad\qquad\qquad\qquad\qquad t\in[0,149],t\in\mathbb{R}.$

In Fig. 1, $r_{\kappa}(t)$ obtained from Eq. (V.1) is plotted for different
values of $\kappa$, an the parameters $\mu=3.5$ and $y_{0}=0.3$ selected to be
in the chaotic regime. The landmarks (characteristic points) are also shown.
In every case, the shapes $r_{\kappa}(t)$ obtained are smooth and infinitely
differentiable. For $\kappa$ sufficiently low, the shapes interpolate the
landmarks through the $\epsilon$-shape with growing precision as $\kappa\to
0$. For $\kappa$ large, details within a scale $s=2\kappa$ are averaged out
and the resulting shapes have broader variations (the curve being always
smooth). As $\kappa\to\infty$, the shape collapses on the average value of the
landmarks for every $t$.

A possible use of the shape theory in 1D above sketched is the synthesis of
sound. Complex periodic waveforms can be simply designed by giving the
landmarks as input (these can also be read from recorded waveforms). No
Fourier decomposition is necessary to select the right harmonics. Then, by
using Eq. (15), an infinite family of periodic signals with the same frequency
can be constructed from the input landmarks.

An example of an invented periodic signal is shown in Fig. 2. By choosing the
five landmarks $(X_{0},X_{1},\ldots,X_{4})=(1,4,2,2,1)$ we can calculate any
shape within the $\kappa$-family by replacing them in Eq. (35) and putting
$N=5$. In Fig. 2 the shapes with $\kappa=0.01,0.1$ and $0.3$ are shown
together with the landmarks. We observe that five points together with the
parameter $\kappa$ is enough to specify an infinite family of qualitatively
related waveforms. Note that, because for $\kappa=0.01$ the shapes tend to
change more abruptly between the different landmarks, a huge number of Fourier
modes would be needed to reproduce such periodic signals, the Gibbs phenomenon
also playing an important role. No Gibbs or Runge phenomena are present in the
interpolation provided by the $\epsilon$-shape in our theory because there are
no oscillations of the $\epsilon$-shape between the landmark points under
interpolation. We remark that this interpolation is _not_ piecewise: the
shapes are all infinitely differentiable functions of both $t$ and $\kappa$.

###  V.2 2D shapes: planar curves

In two dimensions we have the vector
$\mathbf{r}_{\kappa}(t)=(x_{\kappa}(t),y_{\kappa}(t))$ that yields a plane
curve in parametric form and Eq. (13) for open shapes reduces to

$\displaystyle x_{\kappa}(t)$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{N-1}X_{j}\frac{\mathcal{B}_{\kappa}\left(t-j,\frac{1}{2}\right)}{\mathcal{B}_{\kappa}\left(t-\frac{N-1}{2},\frac{N}{2}\right)}$
(38) $\displaystyle y_{\kappa}(t)$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{N-1}Y_{j}\frac{\mathcal{B}_{\kappa}\left(t-j,\frac{1}{2}\right)}{\mathcal{B}_{\kappa}\left(t-\frac{N-1}{2},\frac{N}{2}\right)}$
$\displaystyle\qquad\qquad\qquad\qquad t\in[0,N-1],t\in\mathbb{R}$

where $(X_{j},Y_{j})=\mathbf{r}_{j}$ specifies the $j$-th landmark. For closed
shapes in the plane Eq. (15) is equivalent to

$\displaystyle x_{\kappa}(t)$ $\displaystyle=$
$\displaystyle\frac{\sum_{j=0}^{N-1}X_{j}\Pi_{\kappa}\left(t-j,\frac{1}{2};N\right)}{\sum_{j=0}^{N-1}\Pi_{\kappa}\left(t-j,\frac{1}{2};N\right)}$
(40) $\displaystyle y_{\kappa}(t)$ $\displaystyle=$
$\displaystyle\frac{\sum_{j=0}^{N-1}Y_{j}\Pi_{\kappa}\left(t-j,\frac{1}{2};N\right)}{\sum_{j=0}^{N-1}\Pi_{\kappa}\left(t-j,\frac{1}{2};N\right)}\qquad
t\in\mathbb{R}\quad$ (41)

where $(X_{j},Y_{j})=\mathbf{r}_{j}$ specifies the $j$-th landmark and we have
$(x_{\kappa}(t),y_{\kappa}(t))=\mathbf{r}_{\kappa}(t)$, as before.

An example of a $\kappa$-family of open shapes described by Eqs. (38) and
(V.2) is shown in Fig. 3 in which 50 landmarks are chosen randomly within the
unit square from the uniform distribution. For $\kappa=0.01$, the shape
interpolates among all the landmarks but, as $\kappa$ is increased, the shapes
within the family tend to shrink to the centroid of the landmarks. The convex
hull of the landmarks is shown in orange shadow and it is observed that for
all $\kappa>0$, the shapes are contained within the convex hull, as predicted
by Theorem III.1. If the shape is regarded as a trajectory of a complex
dynamical system, an infinite number of viable trajectories in the same set
can be generated by means of the above method, selecting appropriate subsets
of landmarks. As $\kappa$ is increased, the finest details are gradually
averaged out and, as a result, the shapes are smoothened, but they still
constitute different deformations that look qualitatively similar.

Figure 3: Shapes obtained from Eqs. (38) and (V.2) for 50 landmarks taken
randomly in the unit square from the uniform distribution. Shown in orange
shadow is the convex hull of the landmarks. Values of $\kappa$ are indicated
on the panels and the landmarks are also shown in the top leftmost panel.

We now discuss closed shapes in 2D. John von Neumann’s joke “With four
parameters I can fit an elephant, and with five I can make him wiggle his
trunk” Dyson  has motivated some work on how experimental data on the plane
can be fitted by parametric equations of planar curves Mayer ; Wei ;
Piantadosi . A most popular method is provided by expanding the $x(t)$ and
$y(t)$ coordinates of a closed contour (in this work $x_{\kappa}(t)$ and
$y_{\kappa}(t)$) in Fourier series Kuhl

$\displaystyle x(t)$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\left[A_{n}\cos\left(\frac{2\pi
nt}{T}\right)+B_{n}\sin\left(\frac{2\pi nt}{T}\right)\right]\qquad$ (42)
$\displaystyle y(t)$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\left[C_{n}\cos\left(\frac{2\pi
nt}{T}\right)+D_{n}\sin\left(\frac{2\pi nt}{T}\right)\right]\qquad$ (43)

where the coefficients $A_{n}$, $B_{n}$, $C_{n}$ and $D_{n}$ are related to
$x(t)$ and $y(t)$ by the integrals

$\displaystyle A_{n}$ $\displaystyle=$
$\displaystyle\frac{2}{T}\int_{0}^{T}x(t)\cos\left(\frac{2\pi
nt}{T}\right)\text{d}t,$ (44) $\displaystyle B_{n}$ $\displaystyle=$
$\displaystyle\frac{2}{T}\int_{0}^{T}x(t)\sin\left(\frac{2\pi
nt}{T}\right)\text{d}t,$ (45) $\displaystyle C_{n}$ $\displaystyle=$
$\displaystyle\frac{2}{T}\int_{0}^{T}y(t)\cos\left(\frac{2\pi
nt}{T}\right)\text{d}t,$ (46) $\displaystyle D_{n}$ $\displaystyle=$
$\displaystyle\frac{2}{T}\int_{0}^{T}y(t)\sin\left(\frac{2\pi
nt}{T}\right)\text{d}t.$ (47)

A calculation of these integrals can be done by assuming $N$ distinct
landmarks piecewisely joined by rectilinear segments by means of a chain code
Kuhl . Smooth shapes are then obtained by truncating the sums in Eqs. (42) and
(43) Kuhl . In this way, any closed shape in 2D can be described. This popular
method McLellan2  has been applied to biological shapes in animals Bierbaum ;
Diaz ; Ferson ; Rohlf  and plants Furuta ; Iwata ; McLellan ; Ohsawa ; White
and has been extended to open shapes in 3D by means of the discrete cosine
transform Zhou .

Figure 4: Plots of $y_{\kappa}(t)$ vs. $x_{\kappa}(t)$ obtained from Eqs. (40)
and (41)) for the landmarks shown in the top leftmost panel and the values of
$\kappa$ in the panels. The $\kappa$-family of shapes contains the silhouette
of the elephant and the centroid of the landmarks as its members (in the
limits $\kappa\to 0$ and $\kappa\to\infty$ respectively) as well as an
infinite family of shapes that qualitatively interpolate between both limiting
cases.

It is interesting to compare the Fourier method of Kuhl  with ours (embodied
in Eqs. (40) and (41)) since: _1) our method does not require the evaluation
of any coefficient since these are the landmarks themselves and
$x_{\kappa}(t)$ and $y_{\kappa}(t)$ are linear functions of them; 2) the
expansions in our method are always finite and equal to the finite number of
landmarks also for shapes with vertices and sharp edges; 3) between landmarks
the interpolation is always smooth and there are no oscillations; 4) no chain
code Freeman ; Bribiesca  is needed as in Kuhl  but merely the coordinates of
the landmarks in successive order; 5) the method is completely straightforward
and an infinite family of qualitatively related shapes is obtained by
continuously varying $\kappa$ between 0 and $\infty$_.

We note that only one parameter $\kappa$ is necessary to fit the data in our
method, the $\epsilon$-shape being this fitting. It has been suggested that,
indeed only one parameter is necessary to fit any arbitrary collection of data
Piantadosi . However, the fitting in Piantadosi  is extremely sensitive to the
value of the parameter. The shapes obtained with our method have not this
sensitivity: varying $\kappa$ continuously produces a set of qualitatively
related shapes and if changes in $\kappa$ are small, the changes in shape also
are.

In Fig. 4 we show a fitting of Eqs. (40) and (41)) to the shape of an
elephant. The 160 landmark points $(X_{j},Y_{j})$, $j=0,\ldots 159$ are chosen
in the contour of the elephant. Sufficiently small $\kappa$ values produce an
excellent fit of the elephant. As $\kappa$ is increased, the closed shape
gradually shrinks to the centroid of the landmarks. This centroid coincides
with the zero Fourier mode in the Fourier expansion method of Kuhl . It is,
however, to be noted that to apply the Fourier method first the shape must be
traced with rectilinear segments Kuhl  connecting landmarks by a chain code
Freeman ; Bribiesca . After that, Fourier coefficients need to be calculated
up to a certain number and inserted in the Fourier expansions to yield the
parametric equations of the shape. Our method only needs the landmarks
directly extracted from the contour and these lead directly to the parametric
equations of the shape thanks to the corresponding nonlinear
$\mathcal{B}_{\kappa}$-embeddings.

In the following we shall write the landmarks $(X_{0},Y_{0})$,
$(X_{1},Y_{1})$, $\ldots$, $(X_{N-1},Y_{N-1})$ more compactly in a rectangular
matrix as

$\left(\begin{array}[]{cccc}X_{0}&X_{1}&\ldots&X_{N-1}\\\
Y_{0}&Y_{1}&\ldots&Y_{N-1}\end{array}\right)$ (48)

The order of the landmarks and whether these are repeated or not is a defining
property of the resulting $\kappa$-family of shapes. In Fig. 5A we show four
shapes of the $\kappa$-family obtained from the landmarks

$\left(\begin{array}[]{ccc}0&4&7\\\ 0&3&1\end{array}\right)$ (49)

and for the values $\kappa=0.01$ (black curve), $\kappa=0.3$ (blue curve),
$\kappa=0.5$ (red curve) and $\kappa=1$ (green curve). By introducing the
landmarks in Eqs. (40) and (41)) we obtain the mathematical model in terms of
parametric curves for the whole $\kappa$-family

$\displaystyle x_{\kappa}(t)$ $\displaystyle=$
$\displaystyle\frac{4\Pi_{\kappa}\left(t-1,\frac{1}{2};3\right)+7\Pi_{\kappa}\left(t-2,\frac{1}{2};3\right)}{\sum_{j=0}^{2}\Pi_{\kappa}\left(t-j,\frac{1}{2};3\right)}\quad$
(50) $\displaystyle y_{\kappa}(t)$ $\displaystyle=$
$\displaystyle\frac{3\Pi_{\kappa}\left(t-1,\frac{1}{2};3\right)+\Pi_{\kappa}\left(t-2,\frac{1}{2};3\right)}{\sum_{j=0}^{2}\Pi_{\kappa}\left(t-j,\frac{1}{2};3\right)}\quad$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
t\in\mathbb{R}.$

We see that the $\epsilon$-shape is a triangle whose vertices are the
landmarks. As $\kappa$ is increased, shapes with broader round edges appear
and gradually shrink to the barycenter of the triangle as $\kappa$ is
continuously increased to infinity. Since Eqs. (50) and (V.2) are invariant
under the transformations $t\to t+3k$ $k\in\mathbb{Z}$ we find that any cyclic
permutation of the landmarks yields the same $\kappa$-family of shapes, i.e.
the choices of landmarks

$\left(\begin{array}[]{ccc}7&0&4\\\
1&0&3\end{array}\right)\quad\left(\begin{array}[]{ccc}4&7&0\\\
3&1&0\end{array}\right)$ (52)

correspond to the same $\kappa$-family.

Figure 5: Plots of $y_{\kappa}(t)$ vs. $x_{\kappa}(t)$ obtained from Eqs. (40)
and (41)) for the landmarks given by (A) Eq. (49), (B) Eq. (53),(C) Eq. (56)
with $P=6$, (D) Eq. (59) with $P=6$ and (E) Eq. (68). The values of $\kappa$
are $\kappa=0.01$ (black curves), $\kappa=0.3$ (blue curves), $\kappa=0.5$
(red curves) and $\kappa=1$ (green curves).

If some of the landmarks are repeated, the resulting $\kappa$-family is
different, even when parts of the outline can be the same. It is clear why:
repeating a landmark makes it to acquire more weight therefore displacing the
centroid. For example, if we consider the landmarks

$\left(\begin{array}[]{ccccc}0&4&0&7&0\\\ 0&3&0&1&0\end{array}\right)$ (53)

the landmark $(0,0)$ has thrice the weight of the other landmarks.
Furthermore, the connection between landmarks $(4,3)$ and $(7,1)$ dissappears.
From Eqs. (40) and (41)), the mathematical model of the resulting
$\kappa$-family is,

$\displaystyle x_{\kappa}(t)$ $\displaystyle=$
$\displaystyle\frac{4\Pi_{\kappa}\left(t-1,\frac{1}{2};5\right)+7\Pi_{\kappa}\left(t-3,\frac{1}{2};5\right)}{\sum_{j=0}^{4}\Pi_{\kappa}\left(t-j,\frac{1}{2};5\right)}$
(54) $\displaystyle y_{\kappa}(t)$ $\displaystyle=$
$\displaystyle\frac{3\Pi_{\kappa}\left(t-1,\frac{1}{2};5\right)+\Pi_{\kappa}\left(t-3,\frac{1}{2};5\right)}{\sum_{j=0}^{4}\Pi_{\kappa}\left(t-j,\frac{1}{2};5\right)}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
t\in\mathbb{R}.$

In Fig. 5B the curves obtained from Eqs.(54) and (V.2) are plotted for
$\kappa=0.01$ (black curve), $\kappa=0.3$ (blue curve), $\kappa=0.5$ (red
curve) and $\kappa=1$ (green curve). It is observed that the $\epsilon$-shape
in this case is no longer a triangle because the connection between the points
$(4,3)$ and $(7,1)$ is missing. Because of the higher weight of the origin,
the shapes shrink to a point that is closer to the origin than the barycenter
of the triangle in Fig 5A.

Any regular polygon defines a shape in which the vertices can be taken as
landmarks of a $\kappa$-family of shapes. Therefore, a polygon of $P$ sides
has landmarks

$\left(\begin{array}[]{ccccc}\cos\left(\frac{2\pi}{P}\right)&\cos\left(\frac{4\pi}{P}\right)&\ldots&\cos\left(\frac{2\pi(P-1)}{P}\right)&\cos\left(\frac{2\pi
P}{P}\right)\\\
\sin\left(\frac{2\pi}{P}\right)&\sin\left(\frac{4\pi}{P}\right)&\ldots&\sin\left(\frac{2\pi(P-1)}{P}\right)&\sin\left(\frac{2\pi
P}{P}\right)\end{array}\right)$ (56)

and Eqs. (40) and (41) become

$\displaystyle x_{\kappa}(t)$ $\displaystyle=$
$\displaystyle\frac{\sum_{j=0}^{P-1}\cos\left(\frac{2\pi(j+1)}{P}\right)\Pi_{\kappa}\left(t-j,\frac{1}{2};P\right)}{\sum_{j=0}^{P-1}\Pi_{\kappa}\left(t-j,\frac{1}{2};P\right)},$
(57) $\displaystyle y_{\kappa}(t)$ $\displaystyle=$
$\displaystyle\frac{\sum_{j=0}^{P-1}\sin\left(\frac{2\pi(j+1)}{P}\right)\Pi_{\kappa}\left(t-j,\frac{1}{2};P\right)}{\sum_{j=0}^{P-1}\Pi_{\kappa}\left(t-j,\frac{1}{2};P\right)}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad t\in\mathbb{R}$

In Fig. 5C the curves obtained from these equations are plotted for $P=6$,
$\kappa=0.01$ (black curve), $\kappa=0.3$ (blue curve), $\kappa=0.5$ (red
curve) and $\kappa=1$ (green curve). It is observed that as $\kappa$ is
increased, the shapes approach circumferences. This is obvious: the regularity
of the polygon obtained in the limit $\kappa\to 0$ (the $\epsilon$-shape),
shrinks to the center of the polygon and, as $\kappa$ is increased the shape
becomes uniformized because the details of the vertices are lost. We see that
in increasing $\kappa$ more symmetric shapes are generally obtained since one
passes from objects with discrete symmetries described by dihedral groups to
the continuous symmetry of the circumference.

If we join each vertex of the polygon to the center we also construct shapes
involving the same symmetry groups but with the appearance of stars. The
landmarks are now

$\left(\begin{array}[]{ccccccc}0&\cos\left(\frac{2\pi}{P}\right)&0&\cos\left(\frac{4\pi}{P}\right)&\ldots&0&\cos\left(\frac{2\pi
P}{P}\right)\\\
0&\sin\left(\frac{2\pi}{P}\right)&0&\sin\left(\frac{4\pi}{P}\right)&\ldots&0&\sin\left(\frac{2\pi
P}{P}\right)\end{array}\right)$ (59)

Figure 6: Plots of $y_{\kappa}(t)$ vs. $x_{\kappa}(t)$ obtained from Eqs. (40)
and (41) for the landmarks given by (A) Eq. (69) and (B) (70)). The values of
$\kappa$ are $\kappa=0.01$ (black curves), $\kappa=0.3$ (blue curves),
$\kappa=0.5$ (red curves) and $\kappa=1$ (green curves).

Eqs. (40) and (41) now become

$\displaystyle x_{\kappa}(t)$ $\displaystyle=$
$\displaystyle\frac{\sum_{j=0}^{P-1}\cos\left(\frac{2\pi(j+1)}{P}\right)\Pi_{\kappa}\left(t-2j-1,\frac{1}{2};2P\right)}{\sum_{j=0}^{2P-1}\Pi_{\kappa}\left(t-j,\frac{1}{2};2P\right)}\qquad$
(60) $\displaystyle y_{\kappa}(t)$ $\displaystyle=$
$\displaystyle\frac{\sum_{j=0}^{P-1}\sin\left(\frac{2\pi(j+1)}{P}\right)\Pi_{\kappa}\left(t-2j-1,\frac{1}{2};2P\right)}{\sum_{j=0}^{2P-1}\Pi_{\kappa}\left(t-j,\frac{1}{2};2P\right)}\qquad$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad t\in\mathbb{R}$

In Fig. 5D the curves obtained from these equations are plotted for $P=6$,
$\kappa=0.01$ (black curve), $\kappa=0.3$ (blue curve), $\kappa=0.5$ (red
curve) and $\kappa=1$ (green curve). It is observed that as $\kappa$ is
increased, the shapes approach again circumferences. However, for $\kappa$
sufficiently small, the shapes are not convex and their starry appearance
resemble the one of certain species of _Echinoderma_.

The shapes found within the $\kappa$-families and their deformations can be
controlled by means of appropriately selected and weighted landmarks.
Repeating a landmark after itself does not change the contour of the
$\epsilon$-shape but has the effect, as $\kappa$ is increased, of drawing the
shapes to a point nearer to the repeated landmark: the direction and amount of
deformation within the shapes of a $\kappa$-family as $\kappa$ is increased
can thus be _absolutely_ controlled.

Any connected planar graph can be used as a structure of landmarks or
$\epsilon$-shape. In Fig. 5E the same landmarks as in C and D are used but
repeated and connected diferently, as the $\epsilon$-shape reveals. The $N=10$
landmarks are now

$\displaystyle\left(\begin{array}[]{cccccccc}0&\cos\left(\frac{2\pi}{6}\right)&0&\cos\left(\frac{4\pi}{6}\right)&0&\cos\left(\frac{6\pi}{P}\right)&0&\cos\left(\frac{8\pi}{6}\right)\\\
0&\sin\left(\frac{2\pi}{6}\right)&0&\sin\left(\frac{4\pi}{6}\right)&0&\sin\left(\frac{6\pi}{P}\right)&0&\sin\left(\frac{8\pi}{6}\right)\end{array}\right.$
(64) $\displaystyle\qquad\qquad\qquad\left.\begin{array}[]{cc}\\\
\cos\left(\frac{10\pi}{6}\right)&\cos\left(\frac{12\pi}{6}\right)\\\
\sin\left(\frac{10\pi}{6}\right)&\sin\left(\frac{12\pi}{6}\right)\end{array}\right)$
(68)

and Eqs. (40) and (41) yield shapes as those in Fig. 5E.

Figure 7: Plots of $y_{\kappa}(t)$ vs. $x_{\kappa}(t)$ within the unit square
(axes not shown) obtained from Eqs. (38) and (V.2) for the landmarks obtained
after $Q=5$ iterations of the transformation in Eq. (7). The different values
of $\kappa$ are shown on the panels. Figure 8: Plots of $y_{\kappa}(t)$ vs.
$x_{\kappa}(t)$ within the unit square (axes not shown) obtained from Eqs.
(38) and (V.2) for the landmarks obtained after $Q=5$ iterations of the
transformation in Eq. (7). The different values of $\kappa$ are shown on the
panels. Each successive value of $\kappa$ is four times larger than the
previous one from left to right and top to bottom.

Let us assume that we do not know the parametric equations of a Bernoulli
lemniscate and we find a plot of such a curve and want a mathematical model
for it. Because of its symmetry we place its centroid at the origin. We can
then roughly select the vertices of a square as landmarks whose center of mass
is the origin and such that the periodic $\epsilon$-shape crosses the origin
two times through the diagonals of the square

$\left(\begin{array}[]{cccc}-1&1&1&-1\\\ -1&1&-1&1\end{array}\right)$ (69)

These crossings occur in going from landmark $(-1,\ -1)$ to $(1,\ 1)$ and in
going from $(1,\ -1)$ to $(-1,\ 1)$. The landmarks given by Eq. (69) can be
replaced in Eqs. (40) and (41)) yielding a mathematical model for the entire
$\kappa$-family of shapes corresponding to these landmarks. In Fig. 6A we plot
$y_{\kappa}(t)$ vs. $x_{\kappa}(t)$ obtained from these equations for several
$\kappa$ values. We observe that the $\kappa=1$ curve (green) reasonably
approximates a lemniscate.

Equal $\epsilon$-shapes may belong to different $\kappa$-families. If we now
consider the landmarks

$\left(\begin{array}[]{cccccc}-1&0&1&1&0&-1\\\
-1&0&-1&1&0&1\end{array}\right)$ (70)

the $\epsilon$-shape is identical as with the landmarks in Eq. (69) but as
$\kappa$ is varied, the shapes are different because they do not longer
contain the origin: the latter is not a point where the shape crosses itself.
This is shown in Fig. 6B: the $\epsilon$-shape is the same as in Fig. 6A but
the $\kappa$-family is different, as it is revealed when $\kappa$ is
increased.

###  V.3 Fractal shapes in 2D: Space-filling curves

Landmarks can also be specified recursively by means of cellular automata VGM1
; VGM2 ; VGM3 ; VGM7 ; Wolfram  substitution and Lindenmayer systems VGM4 ;
Wolfram ; Prusinkiewicz  and digit replacement techniques CHAOSOLFRAC ; VGM5 ;
VGM6 . These procedures, when acting on real numbers, naturally lead to
fractal structures CHAOSOLFRAC ; VGM5 ; VGM6 .

A famous example of a fractal curve in the plane is Hilbert’s curve Hilbert .
This curve, which constitutes an open shape, can be regarded as a substitution
system. Starting with the landmark $X_{0}=0$, $Y_{0}=0$, we iteratively
replace each previous landmark by four new landmarks following the
transformation

$\left(\begin{array}[]{c}X_{j}\\\
Y_{j}\end{array}\right)\to\left(\begin{array}[]{cccc}\frac{Y_{j}-1}{2}&\frac{X_{j}-1}{2}&\frac{X_{j}+1}{2}&\frac{1-Y_{j}}{2}\\\
\frac{X_{j}-1}{2}&\frac{Y_{j}+1}{2}&\frac{Y_{j}+1}{2}&\frac{-1-X_{j}}{2}\end{array}\right)$
(71)

Let $Q$ denote the number of iterations. For $Q=0$ we have only as landmark
the origin (initialization). For $Q=1$ we have the $N=4$ landmarks

$\left(\begin{array}[]{cccc}-\frac{1}{2}&-\frac{1}{2}&\frac{1}{2}&\frac{1}{2}\\\
-\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&-\frac{1}{2}\end{array}\right)$ (72)

For $Q=2$ we have $4^{2}=16$ landmarks which are obtained replacing in order
each of these landmarks following the prescription in Eq. (71). For $Q=n$ we
obtain by this process, $4^{n}$ landmarks. Hilbert’s curve is obtained by
joining these landmarks by straight line segments.

By using Eqs. (38) and (V.2) we can now construct a $\kappa$-family of fractal
curves having Hilbert’s curve as $\epsilon$-shape. In Fig. 7 we represent the
curves calculated from Eqs. (38) and (V.2) for the 1024 landmarks obtained
after $Q=5$ iterations and the values of $\kappa$ indicated on the panels.
When $\kappa$ is vanishingly small, the $\epsilon$-shape (Hilbert’s curve
after $Q=5$ iterations) is approached. When $\kappa$ increases, the smallest
details are averaged out and all straight edges and vertices become replaced
by curved contours. As $\kappa$ increases further, we observe the shapes that
we would have obtained with $Q=3$ iterations. Increasing $\kappa$ further then
seems like ‘going backwards in the iterative process’. In the limit
$\kappa\to\infty$ the shape collapses to its centroid, which corresponds to
$Q=0$ iterations.

In Fig. 8 we better illustrate this phenomenon. We observe that the curves are
self-similar for any value of $\kappa$: By multiplying $\kappa$ by 4 we move
backwards in the iteration process, obtaining the same shapes as if they were
zoomed in, or as if they were obtained by the landmarks after $Q=5$, $Q=4$,
$Q=3$ and $Q=2$ iterations (looking at the panels from left to right and top
to bottom). We note that the $\mathcal{B}_{\kappa}$ function has a scaling
property

$\mathcal{B}_{a\kappa}(x,y)=\mathcal{B}_{\kappa}(ax,ay)$ (73)

This property is responsible for the phenomenon observed. We thus note that
the smoothing parameter $\kappa$ is really a _scale_ parameter: as it is
increased it captures the features of a shape in a more broader scale. Since
the curves are self-similar we observe the same kind of structures as $\kappa$
is increased.

###  V.4 3D shapes

In three dimensions we have the vector
$\mathbf{r}_{\kappa}(t)=(x_{\kappa}(t),y_{\kappa}(t),z_{\kappa}(t))$ that
yields a plane curve in parametric form and Eq. (13) for open shapes reduces
to

$\displaystyle x_{\kappa}(t)$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{N-1}X_{j}\frac{\mathcal{B}_{\kappa}\left(t-j,\frac{1}{2}\right)}{\mathcal{B}_{\kappa}\left(t-\frac{N-1}{2},\frac{N}{2}\right)}$
(74) $\displaystyle y_{\kappa}(t)$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{N-1}Y_{j}\frac{\mathcal{B}_{\kappa}\left(t-j,\frac{1}{2}\right)}{\mathcal{B}_{\kappa}\left(t-\frac{N-1}{2},\frac{N}{2}\right)}$
(75) $\displaystyle z_{\kappa}(t)$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{N-1}Z_{j}\frac{\mathcal{B}_{\kappa}\left(t-j,\frac{1}{2}\right)}{\mathcal{B}_{\kappa}\left(t-\frac{N-1}{2},\frac{N}{2}\right)}$
$\displaystyle\qquad\qquad\qquad\qquad t\in[0,N-1],t\in\mathbb{R}$

where $(X_{j},Y_{j},Z_{j})=\mathbf{r}_{j}$ specifies the $j$-th landmark.

Figure 9: Plots of $z_{\kappa}(t)$ vs. $y_{\kappa}(t)$ and $x_{\kappa}(t)$ of
the curves obtained from Eqs. (77) to (79) for the landmarks given in Eq. (81)
and the values of $\kappa$ indicated on the figures. In the right column, the
shapes are also projected on the $x_{\kappa}-y_{\kappa}$ plane for better
visualization. The color code is related to the height $z_{\kappa}$ and ranges
from dark blue ($z_{\kappa}=0$) to yellow ($z_{\kappa}=1$).

For closed shapes in the plane Eq. (15) is equivalent to

$\displaystyle x_{\kappa}(t)$ $\displaystyle=$
$\displaystyle\frac{\sum_{j=0}^{N-1}X_{j}\Pi_{\kappa}\left(t-j,\frac{1}{2};N\right)}{\sum_{j=0}^{N-1}\Pi_{\kappa}\left(t-j,\frac{1}{2};N\right)}$
(77) $\displaystyle y_{\kappa}(t)$ $\displaystyle=$
$\displaystyle\frac{\sum_{j=0}^{N-1}Y_{j}\Pi_{\kappa}\left(t-j,\frac{1}{2};N\right)}{\sum_{j=0}^{N-1}\Pi_{\kappa}\left(t-j,\frac{1}{2};N\right)}$
(78) $\displaystyle z_{\kappa}(t)$ $\displaystyle=$
$\displaystyle\frac{\sum_{j=0}^{N-1}Z_{j}\Pi_{\kappa}\left(t-j,\frac{1}{2};N\right)}{\sum_{j=0}^{N-1}\Pi_{\kappa}\left(t-j,\frac{1}{2};N\right)}\qquad
t\in\mathbb{R}\quad$ (79)

The landmarks $(X_{0},Y_{0},Z_{0})$, $(X_{1},Y_{1},Z_{1})$, $\ldots$,
$(X_{N-1},Y_{N-1},Z_{N-1})$ can be written more compactly in a rectangular
matrix as

$\left(\begin{array}[]{cccc}X_{0}&X_{1}&\ldots&X_{N-1}\\\
Y_{0}&Y_{1}&\ldots&Y_{N-1}\\\ Z_{0}&Z_{1}&\ldots&Z_{N-1}\end{array}\right)$
(80)

As an example of 3D curve, we provide a mathematical model for a trefoil knot,
a closed curve which features prominently in the Lorenz attractor Kaufman .
The landmarks of any knot can be found from its planar representation by
selecting points on its contour. If a point lags at the bottom (resp. at the
front) in the planar representation, a value of $Z=0$ (resp. $Z=1$) is
adjoined to the coordinates $X$ and $Y$ read from the planar representation.
We can pick, e.g. $N=9$ landmarks in this way, outlining the periodic contour
of the shape

$\left(\begin{array}[]{cccccccccc}0.5&0.3&0.5&1&0.7&0.3&0&0.5&0.7\\\
0&0.4&0.8&0.8&0.4&0.4&0.8&0.8&0.4\\\ 1&0&1&1&0&1&1&0&1\end{array}\right).$
(81)

Eqs. (77) to (79) automatically provide a model for the $\kappa$-family of
closed shapes derived from these landmarks. In Fig. 9 we plot several shapes
that are members of this family. The shape of the trefoil knot shrinks towards
its centroid. At $\kappa>0.55$ aprox. the knot undoes and, therefore a
knot/unknot transition is observed. Complemented with structural, biological
information, the mathematical methods here presented may thus be of interest
in the modelling of knot/unknot transitions found in macromolecules and DNA
Wasserman ; Liu .

###  V.5 $\kappa$-family of shapes from Lorenz’ attractor

The celebrated Lorenz system

$\displaystyle\dot{x}$ $\displaystyle=$ $\displaystyle\sigma(y-x)$ (82)
$\displaystyle\dot{y}$ $\displaystyle=$ $\displaystyle x(\rho-z)-y$ (83)
$\displaystyle\dot{z}$ $\displaystyle=$ $\displaystyle xy-\beta z$ (84)

can be embedded in a $\kappa$-family of shapes as described in Sec. IV. The
landmarks $\mathbf{r}_{j}:=(X_{j},Y_{j},Z_{j})$ can be iteratively obtained
from Eq. (30) as

$\displaystyle X_{j}$ $\displaystyle=$ $\displaystyle
X_{j-1}+\delta\sigma\left(Y_{j-1}-X_{j-1}\right)$ (85) $\displaystyle Y_{j}$
$\displaystyle=$ $\displaystyle Y_{j-1}+\delta\left[X_{j-1}\left(\rho-
Z_{j-1}\right)-Y_{j-1}\right]$ (86) $\displaystyle Z_{j}$ $\displaystyle=$
$\displaystyle Z_{j-1}+\delta\left[X_{j-1}Y_{j-1}-\beta Z_{j-1}\right]$ (87)

starting from an initial condition $(X_{0},Y_{0},Z_{0})$. In Fig. 10
$z_{\kappa}(t)$ is shown together with $y_{\kappa}(t)$ and $x_{\kappa}(t)$
obtained from Eqs. (74) to (V.4) and the landmarks given by Eqs. (85) to (87)
for the initial condition $(X_{0},Y_{0},Z_{0})=(2.5704,\ 3.6945,\ 16.4286)$,
$N=1000$, $\delta=0.01$, $\rho=28$, $\sigma=10$, $\beta=8/3$ and the values of
$\kappa$ indicated over the panels. It is observed that as $\kappa$ is
decreased to 0, the $\epsilon$-shape approaches that of the Lorenz attractor.
When $\kappa$ is increased, deformed complex attractors are obtained. The
latter shrink to a point as $\kappa\to\infty$. The resulting, deformed
attractors, for $\kappa>0$ finite are always found within the convex hull of
the landmarks obtained from the Lorenz system.

Figure 10: Plots of $z_{\kappa}(t)$ vs. $y_{\kappa}(t)$ and $x_{\kappa}(t)$
obtained from Eqs. (74) to (V.4) and the landmarks given by Eqs. (85) to (87)
for the initial condition $(X_{0},Y_{0},Z_{0})=(2.5704,\ 3.6945,\ 16.4286)$,
$N=1000$, $\delta=0.01$, $\rho=28$, $\sigma=10$, $\beta=8/3$ and the values of
$\kappa$ indicated over the panels.

##  VI Conclusions

In this article, a general theory of shape has been presented and illustrated
with examples. The theory provides mathematical models for any shape that can
be found in the physical world and requires the specification of a finite set
of landmarks that fully characterize entire families of qualitatively related
shapes. The theory addresses all shapes that are path connected (all shapes of
interest in physics, engineering and the life sciences) although it may be
extended (by introducing additional parameters) to other topologically
interesting curves that are not path connected (as e.g. the Warsaw circle).
Disconnected shapes that are composed of path connected pieces can be treated
within the theory as _scenes_ (compound shapes).

The theory of shape developed in this article has been linked to viability
theory Aubin3 ; Aubin1 ; Aubin2  and invariant sets of dynamical systems
Blanchini1 ; Blanchini2 . When a shape is looked upon as a trajectory of a
complex dynamical system (possibly under the influence of unpredictable
perturbations) infinite families of viable trajectories can be constructed for
the dynamical system, from the knowledge of a finite number of points
(landmarks) on a measured trajectory. All viable trajectories generated as
$\kappa$-families of shapes are found in any convex set of the landmarks and,
specifically, in the convex hull of the latter (the convex set of minimal
size).

The mathematical models for the shapes correspond to nonlinear
$\mathcal{B}_{\kappa}$-embeddings JPHYSCOMPLEX . The latter structures are
nonlinear functions of the time $t$ and deformation $\kappa$ parameters, but
are linear functions of the landmarks. This makes straightforward the
application of transformations to the shapes (e.g. translations, rotations,
rescalings). Specifically, procrustean analysis can easily be carried on the
$\kappa$-families of shapes, after displacing the centroid to the origin and
rescaling the shapes appropriately Kendall1 ; Kendall2 . The former operation
is carried out subtracting the coordinates of the centroid to each of the
landmarks. The second operation proceeds by multiplying all the landmarks by a
suitable scalar to carry the shape to the confines of the unit interval,
square or box. It is not necessary to rotate the shapes within the same
$\kappa$-family to compare them because all of them share the same
orientation.

The theory explains how shapes can be hierarchically integrated in
increasingly complex structures (scenes) so that shape is _transferred_ to
more encompassing entities. The time unfolding that characterizes the shape is
entirely transferred. The theory also explains how structural information can
be retrieved at any level (the $\epsilon$-shapes or $\epsilon$-scenes, that
contain this information, are included as particular cases within the
$\kappa$-families derived from them). In this sense the theory satisfies
Leyton’s criteria for a succesful theory of shape Leyton . Furthermore, within
a same $\kappa$-family, shapes with larger $\kappa$ values are generally more
symmetrical that shapes with lower $\kappa$, the $\epsilon$-shape being the
least symmetrical shape within a $\kappa$-family. This allows to describe
within our theory Leyton’s processes in which symmetry-breaking serves as
memory storage. These processes are found in Leyton’s application of its
theory of shape to painting Leypain  and architecture Leyarch . Although a
previous version of Leyton’s theory Leymind  has been strongly criticized on
mathematical grounds and vagueness of the ideas Wagemans  that critique did
not incorporate the wreath product construction in Leyton , which in, our
view, constitutes a sound mathematical basis for Leyton’s theory of shape. Our
theory of shape presented in this manuscript yields independent support to
Leyton’s main basic idea that perception is nothing but the recovery of causal
history Leyton . Indeed if the parameter $\kappa$ in our theory is viewed as a
time variable, all shapes within a $\kappa$-family are causally related, as
they are perceptually, to the $\epsilon$-shape.

We can mention some interesting directions for further research. First of all,
we have described $\kappa$-families of shapes in Euclidean spaces but we can
also be interested in constraining the shapes so as to lie on a manifold Jupp
; Kim . Another interesting direction is investigating the relationship
between our theory and Leyton’s wreath product in Leyton  and the
relationships of $\kappa$-families with group actions, in general. Finally, a
very challenging but interesting problem, is to investigate the connection of
$\kappa$-families to continuous walks through Kendall shape spaces Kendall1 ;
Kendall2 ; Klingenberg .

## Acknowledgments

We have benefitted from fruitful conversations with Prof. José A. Manzanares.
Financial support under project No. PGC2018-097359-B-I00 from _Ministerio de
Ciencia, Innovación y Universidades_ (Spain) and the European Regional
Development Funds (FEDER) is also gratefully acknowledged.

## References

  * (1) P. Ball, _Shapes_ (Oxford University Press, Oxford UK, 2009).
  * (2) K. Borsuk, _Theory of Shape_ (Monografie Matematyczne, Warsaw, 1975).
  * (3) S. Mardešić, Thirty years of shape theory. Math. Comm. 2 (1997) 1-12.
  * (4) D. G. Kendall, D. Barden, T. K. Carne and H. Le,  Shape and Shape Theory (Wiley, New York, 1999).
  * (5) D. G. Kendall, Shape manifolds, procrustean metrics, and complex projective spaces. Bull. London Math. Soc. 16 (1984) 81-121.
  * (6) I.L. Dryden and K. V. Mardia, Statistical shape analysis, with applications in R, 2nd edn. (Wiley, Chichester, 2016).
  * (7) M. Leyton,  A generative theory of shape (Springer Verlag, Berlin, 2001).
  * (8) D. W. Thompson,  On growth and form (Cambridge University Press, Cambridge, UK, 1942).
  * (9) C. Alexander,  The nature of order. Vol. I-IV (CES Publishing, Berkeley CA, 2003).
  * (10) M. Leyton,  The structure of paintings (Springer Verlag, Berlin, 2006).
  * (11) M. Leyton,  Shape as Memory: A Geometric Theory of Architecture (De Gruyter, Berlin, 2006).
  * (12) A. Bejan,  Shape and structure: From Engineering to Nature (Cambridge University Press, Cambridge, UK, 2000).
  * (13) F. L. Bookstein,  Morphometric tools for landmark data (Cambridge University Press, Cambridge UK, 1991).
  * (14) H. Iwata and Y. Ukai. SHAPE: A computer program package for quantitative evaluation of biological shapes based on elliptic Fourier descriptors. J. Hered. 93 (2002) 384-385.
  * (15) L. da Fontoura Costa and R. M. Cesar,  Shape classification and analysis: Theory and Practice (CRC Press, Boca Raton, FL, 2009).
  * (16) J. P. Aubin. _Mutational and morphological analysis. Tools for shape evolution and morphogenesis_. (Springer, New York, 1999).
  * (17) J. P. Aubin. _Viability theory_. (Birkhäuser, Basel, 1991).
  * (18) J. P. Aubin, A. M. Bayen, P. Saint-Pierre. _Viability theory. New directions_. (Springer, Berlin, 2011).
  * (19) F. Blanchini. Set invariance in control. Automatica, 35 (1999) 1747-1767.
  * (20) F. Blanchini and S. Miani. _Set-Theoretic Methods in Control_. (Birkhäuser, Basel, 2008).
  * (21) M. Nagumo, Über die lage der integralkurven gewöhnlicher differentialgleichungen, J. Phys. Soc. Jpn. 24 (1942) 551-559.
  * (22) F. Takens. Detecting strange attractors in turbulence. In D. A. Rand and L.-S. Young (ed.). _Dynamical Systems and Turbulence_ , Lecture Notes in Mathematics, vol. 898. (1981) (Springer-Verlag, Berlin, 1981) pp. 366-381.
  * (23) F. P. Kuhl and C. R. Giardina, Elliptic Fourier features of a closed contour. Comput. Graph. Image Process. 18 (1982) 236-258.
  * (24) J.-P. Antoine, D. Barache, R. M. Cesar Jr. and L. da Fontoura Costa, Shape characterization with the wavelet transform, Signal Processing 62 (1997) 265-290.
  * (25) S. Osowski and D. D. Nghia, Fourier and wavelet descriptors for shape recognition using neural networks - a comparative study, Pattern Recognition, 35 (2002) 1949-1957. https://doi.org/10.1016/S0031-3203(01)00153-4
  * (26) T. Pavlidis, _Structural Pattern Recognition_ (Springer, New York, 1977).
  * (27) V. García-Morales, Nonlinear embeddings: Applications to analysis, fractals and polynomial root finding. Chaos Sol. Fract. 99 (2017) 312.
  * (28) V. García-Morales, Unifying vectors and matrices of different dimensions through nonlinear embeddings. J. Phys. Complex. 1 (2020) 025008.
  * (29) D. Hilbert, Über die stetige Abbildung einer Linie auf ein Flächenstück. Mathematische Annalen 38 (1891) 459-460.
  * (30) L. H. Kauffman,  Knots and Physics (World Scientific, Singapore, 1994).
  * (31) H. Freeman. On the encoding of arbitrary geometric configurations, IRE Transactions on Electronic Computers EC- 10(1961) 260-268.
  * (32) E. Bribiesca, A new chain code, Pattern Recognition 32 (1999) 235-251.
  * (33) V. García-Morales, Universal map for cellular automata. Phys. Lett. A ,376 (2012) 2645.
  * (34) V. García-Morales, Symmetry analysis of cellular automata. Phys. Lett. A, 377 (2013) 276.
  * (35) V. García-Morales, A new approach to fuzzy sets: Application to the design of nonlinear time series, symmetry-breaking patterns, and non-sinusoidal limit-cycle oscillations. Chaos Sol. Fract. 128 (2019) 191.
  * (36) V. García-Morales, From deterministic cellular automata to coupled map lattices. J. Phys. A.: Math. Theor. 49 (2016) 295101.
  * (37) R. T. Rockafellar,  Convex analysis (Princeton University Press, Princeton, 1970), p. 12.
  * (38) F. Dyson, A meeting with Enrico Fermi. Nature (London) 427(6972) (2004) 297.
  * (39) J. Mayer, K. Khairy and J. Howard, Drawing an elephant with four complex parameters. Am. J. Phys. 78 (2010) 648-649.
  * (40) J. Wei, Least-squares fitting of an elephant. CHEMTECH 5 (1975) 128-129.
  * (41) S. T. Piantadosi, One parameter is always enough, AIP Advances 8 (2018) 095118.
  * (42) T. McLellan and J. A. Endler, The relative success of some methods for measuring and describing the shape of complex objects. Syst. Biol. 47 (1998) 264-281.
  * (43) R. M. Bierbaum and S. Ferson, Do symbiotic pea crabs decrease growth rate in mussels? Biol. Bull., 170 (1986) 51-61.
  * (44) G. Diaz, A. Zuccarelli, I. Pelligra, and A. Ghiani, Elliptic Fourier analysis of cell and nuclear shapes. Comp. Biomed. Res. 22 (1989) 405-414.
  * (45) S. Ferson, F. J. Rohlf, and R. K. Koehn, Measuring shape variation of two-dimensional outlines. Syst. Zool. 34 (1985) 59-68.
  * (46) F. J. Rohlf and J. W. Archie, A comparison of Fourier methods for the description of wing shape in mosquitoes (Ritera culicidae). Syst. Zool. 33 (1984) 302-317.
  * (47) N. Furuta, S. Ninomiya, S. Takahashi, H. Ohmori, and Y. Ukai, Quantitative evaluation of soybean (Glycine max L., Merr.) leaflet shape by principal component scores based on elliptic Fourier descriptor. Breed Sci. 45 (1995) 315-320.
  * (48) H. Iwata, S. Niikura, S. Matsuura, Y. Takano, and Y. Ukai, Evaluation of variation of root shape of Japanese radish (_Raphanus sativus_ L.) based on image analysis using elliptic Fourier descriptors. Euphytica 102 (1998) 143-149.
  * (49) T. McLellan, The roles of heterochrony and heteroblasty in the diversification of leaf shapes in _Begonia dregei_ (Begoniaceae). Am. J. Bot. 80 (1993) 796-804.
  * (50) R. Ohsawa, T. Tsutsumi, H. Uehara, H. Namai, and S. Ninomiya, Quantitative evaluation of common buckwheat (_Fagopyrum esculentum_ Moench) kernel shape by elliptic Fourier descriptor. Euphytica 101 (1998) 175-183.
  * (51) R. White, H. C. Rentice, and T. Verwist. Automated image acquisition and morphometric description. Can. J. Bot. 66 (1988) 450-459.
  * (52) S. Zhou, B Li, and H. Nie, Parametric fitting and morphometric analysis of 3D open curves based on discrete cosine transform. Zoomorphology (2021). https://doi.org/10.1007/s00435-021-00520-w.
  * (53) S. Wolfram,  A New Kind of Science (Wolfram Media Inc., Champaign, IL, 2002).
  * (54) V. García-Morales, Origin of complexity and conditional predictability in cellular automata, Phys. Rev. E, 88 (2013) 042814.
  * (55) V. García-Morales, Diagrammatic approach to cellular automata and the emergence of form with inner structure, Commun. Nonlinear Sci. Numer. Simulat. 63 (2018) 117-124.
  * (56) V. García-Morales, Substitution Systems and Nonextensive Statistics, Physica A 440 (2015) 110-117.
  * (57) P. Prusinkiewicz and J. Hanan,  Lindenmayer systems, Fractals, and Plants (Springer Verlag, Berlin, 1980).
  * (58) V. García-Morales, The $p\lambda n$ fractal decomposition: Nontrivial partitions of conserved physical quantities, Chaos Sol. Fract. 83 (2016) 27.
  * (59) V. García-Morales, Digit replacement: A generic map for nonlinear dynamical systems, Chaos 26 (2016) 093109.
  * (60) V. García-Morales, Fractal surfaces from simple arithmetic operations, Physica A 447 (2016) 535-544.
  * (61) S. A. Wasserman and N. R. Cozzarelli, Biochemical topology: Applications to DNA recombination and replication, Science 232 (1986) 951-960.
  * (62) Z. Liu, R. W. Deibler, H. S. Chan and L. Zechiedrich, The why and how of DNA unlinking, Nucleic Acids Research, 37, (2009) 661-671 doi:10.1093/nar/gkp041
  * (63) M. Leyton, _Symmetry, causality, mind_ (MIT Press, Cambridge, MA, 1992).
  * (64) M. Hendrickx and J. Wagemans, A Critique of Leyton’s Theory of Perception and Cognition. Review of “Symmetry, Causality, Mind”, by Michael Leyton. J. Math. Psychol. 43 (1999) 314-345.
  * (65) P.E. Jupp and J.T. Kent, Fitting smooth paths to spherical data. J. R. Stat. Soc. Series C 36 (1987) 34-46.
  * (66) K. R. Kim, I. L. Dryden, H. Le, and K. E. Severn. Smoothing splines on Riemannian manifolds, with applications to 3D shape space. J. R. Stat. Soc. Series B. 83 (2021) 108-132. https://doi.org/10.1111/rssb.12402
  * (67) C. P. Klingenberg, Walking on Kendall’s Shape Space: Understanding Shape Spaces and Their Coordinate Systems, Evolutionary Biology 47 (2020) 334-352.