# Pebbles, graphs and equilibria: higher order shape descriptors for
sedimentary particles

Balázs Ludmány<EMAIL_ADDRESS>ELKH=̄BME Morphodynamics Research
Group  Dept. of Control Engineering and Information Technology, Budapest
University of Technology and Economics, 2. Magyar tudosok Blvd., H=̄1117,
Budapest  Gábor Domokos  ELKH=̄BME Morphodynamics Research Group  Dept. of
Morphology and Geometric Modeling, Budapest University of Technology and
Economics 1=̄3. Műegyetem rakpart, H=̄1111, Budapest

###### Abstract

While three/̄dimensional measurement technology is spreading fast, its
meaningful application to sedimentary geology still lacks content. Classical
shape descriptors (such as axis ratios, circularity of projection) were not
inherently three/̄dimensional, because no such technology existed. Recently a
new class of three/̄dimensional descriptors, collectively referred to as
_mechanical descriptors_ has been introduced and applied for a broad range of
sedimentary particles. _First order_ mechanical descriptors (registered for
each pebble as a pair $\\{S,U\\}$ of integers), refer to the respective
_numbers_ of stable and unstable static equilibria and can be reliably
detected by hand experiments. However, they have limited ability of
distinction as the majority of coastal pebbles fall into primary class
$\\{S,U\\}=\\{2,2\\}$. _Higher order_ mechanical descriptors offer a more
refined distinction. However for the extraction of these descriptors
(registered as _graphs_ for each pebble) hand measurements are not an option
and even computer/̄based extraction from 3D scans offers a formidable
challenge. Here we not only describe and implement an algorithm to perform
this task, but also apply it to a collection of 271 pebbles with various
lithologies, illustrating that the application of higher order descriptors is
a viable option for geomorphologists. We also show that the so-far uncharted
connection between the two known secondary descriptors, the so-called
_Morse=̄Smale graph_ and the _Reeb/̄graph_ can be established via a _third
order descriptor_ which we call the master graph.

##### Keywords

shape descriptor, sedimentary particle, 3D scan, mechanical equilibrium,
Reeb/̄graph, Morse=̄Smale graph

##### Acknowledgments

Support of the NKFIH Hungarian Research Fund, grants 134199 and of the NKFIH
Fund TKP2021 BME/̄NVA, carried out at the Budapest University of Technology
and Economics, is kindly acknowledged.

###### Contents

  1. 1 Introduction: Motivation and main goals
  2. 2 Basic concepts
    1. 2.1 Shape catalogs
    2. 2.2 Classical descriptors
    3. 2.3 Mechanical shape descriptors and natural classes
  3. 3 Classification of smooth, convex shapes
    1. 3.1 Primary equilibrium classes
    2. 3.2 Reeb/̄graph
    3. 3.3 Morse=̄Smale graph
    4. 3.4 Master graph
  4. 4 Classification of convex polyhedra
    1. 4.1 Reeb/̄graph
    2. 4.2 Morse=̄Smale graph
    3. 4.3 Master graph
  5. 5 Equilibrium classes as shape catalogs
  6. 6 Classification of natural shapes
    1. 6.1 Example on a single pebble
    2. 6.2 Cataloging a population of pebbles
  7. 7 Summary
  8. A Visualization of the algorithm’s output

##  1 Introduction: Motivation and main goals

The shape of sedimentary particles carries an infinite amount of information
and we know that some portion of this is highly relevant for detecting the
provenance of the particle [6, 25, 29]. However, picking this relevant portion
may not always be trivial: this is the observer’s prerogative, manifested in
the choice of _shape descriptors_. This choice is a trade/̄off between the
observer’s ability to reliably measure the chosen shape descriptor and the
observer’s desire to extract maximal information. _Classical descriptors_ ,
such as axis ratios and the isoperimetric ratio (which we discuss in
subsection 2.2) rely on hand measurement and two/̄dimensional image analysis.
They were picked by geomorphologists in the absence of computerized
three/̄dimensional tools, based on the mentioned trade/̄off. One, recently
introduced set of shape descriptors, called _first order mechanical
descriptors_ [13, 30], registered for each particle as a pair of integers, can
already be regarded as three/̄dimensional descriptors but still relies on hand
measurements. First order mechanical descriptors proved to be particularly
informative [6, 11, 25] in describing the provenance of particles (see
subsection 2.3.2, and see also Figure 3.) Their generalizations, also known as
_second order mechanical descriptors_ [8, 9, 22, 24] registered for each
particle as a graph (either a so-called Morse=̄Smale graph or a so-called
Reeb/̄graph, see subsection 2.3.2, see also Figure 4), also appear to be
promising tools in sedimentary geomorphology. However, they are fully
three/̄dimensional descriptors and their measurement has been prohibitive
until now and they could not be applied in either laboratory or field studies.

The challenge of measuring second order mechanical shape descriptors is
three/̄fold:

  1. 1.

_Mathematical challenge:_ There exist at least two different kinds of
secondary descriptors: Morse=̄Smale graphs [8, 9, 22] and Reeb/̄graphs [24].
The connection between these two secondary descriptors has not been determined
until now: that is, it was not clear how the two classification schemes are
related, whether one of them may be regarded as a refinement of the other or
not.

  2. 2.

_Algorithmic challenge:_ The extraction of second order descriptors from 3D
datasets is far from trivial. While related problems have been solved in image
processing on surfaces defined in orthogonal coordinates [14], the spherical
version of this method (to be used on particle surfaces) has not been
developed.

  3. 3.

_Technological challenge:_ Obtaining 3D (scanned) datasets for sedimentary
particles in a fast and reliable manner appeared to be, until very recently,
quite challenging. 3D scanning technologies did not offer the option to
measure an object quickly and reliably on the full spherical horizon.

The last mentioned (technological) challenge appears to be resolved: as 3D
measurement technology is spreading fast [26, 19, 28], it may become soon the
de facto standard also in sedimentary geomorphology [27, 15]. Encouraged by
these developments, our paper takes aim at the first two challenges. In
particular we offer the following:

  1. 1.

_Mathematical results:_ In subsections 3.4 and 4.3 we introduce a _third order
mechanical descriptor_ which we call the _master graph_ which establishes a
meaningful connection between second order mechanical descriptors, the
Reeb/̄graph and the Morse=̄Smale graph. We show that neither of them can be
derived from the other, but both can be derived from the master graph. We will
also give a complete third/̄order description of the geologically most
relevant primary class $\\{S,U\\}=\\{2,2\\}$, containing the majority of all
coastal pebbles (see also Table 1 and Figure 5). In particular, in subsection
2.3.2 we will formulate Lemma 1, claiming that there exist 3 tertiary classes
in the primary class $\\{S,U\\}=\\{2,2\\}$ and we prove this claim in Section
5.

  2. 2.

_Algorithmic results and application_ : In Section 4 we describe a reliable
tool to determine higher order mechanical descriptors based on 3D point clouds
obtained from scans. Here we will consider all three relevant types of graphs:
Morse=̄Smale graphs, Reeb/̄graphs and the master graph. We illustrate our
algorithm in the Supplementary material where we show both first, second and
third/̄order mechanical descriptors for 271 scanned pebbles of various
lithologies. On these 271 pebbles we identified 29 primary equilibrium
classes, 69 distinct Reeb/̄graphs, 62 distinct Morse=̄Smale graphs and 115
distinct master graphs. In particular, despite the fact that over 50 pebbles
belong to the primary class $\\{2,2\\}$, inside it we only identified one
Reeb/̄graph, one Morse=̄Smale graph and one single master graph.

Despite the formidable difficulties of their measurement, higher order
mechanical descriptors appear to be an interesting and inviting tool for
geomorphology: they are naturally encoded in the pebble shape, that is, when
we use them in the description of the pebble, we do not add any arbitrary,
man/̄made information. They carry deep, essential information on the shape and
its evolution and thus they might help to uncover new, surprising connections
between pebbles and pebble populations. It is not a coincidence that these
concepts have been applied in image processing and morphology. In this paper
we offer the above mentioned results and algorithmic tools as the first step
towards the geological application of these deep, natural shape descriptors.

The structure of the paper is the following: In Section 2 we give intuitive
definitions of the basic concepts and state the above mentioned Lemma 1. In
Section 3 we discuss the previous concepts more rigorously on smooth, convex
shapes. In Section 4 we interpret the same concepts in the context of convex
polyhedra as models of scanned particles. In Section 5 we prove Lemma 1 and
other mathematical results. Section 6 describes our algorithmic results, in
particular, we show how we can extract higher order classes from natural
shapes based on 3D scans.

##  2 Basic concepts

###  2.1 Shape catalogs

If the value of a shape descriptor may assume an interval of real numbers
(such as the value of roundness) then we refer to it as a _continuous
descriptor_ , whereas if it defines discrete classes (i.e. its set of values
is discrete) then we call it a _shape catalog_ (Zingg classes, discussed
below, are an example). Catalogs (not necessarily of shapes) are fundamental
tools of scientific progress and they have been applied broadly in physics
(conductors/semiconductors/insulators), chemistry (periodic table) and biology
(taxonomy). The main advantage of a shape catalog, compared to a continuous
shape descriptor, is that recorded datasets are exact, unambiguous and they
can be easily interpreted, compared and understood. Catalogs may be either
_natural_ or _artificial_ , depending on whether classes are separated based
on some natural, or some man/̄made condition. Catalogs may be, depending on
the number of classes, either _finite_ or _infinite_. Catalogs may be either
_complete_ or _incomplete_ , depending on whether each class contains shapes
or not. Catalogs may be either _biased_ or _uniform_ , depending on the
statistical distribution of natural shapes among classes (in biased catalogs
this distribution is non/̄uniform). In case of shape catalogs it is easy to
see that from the geophysical point of view we seek natural, biased
classifications (whether or not they are complete may be more of mathematical
interest).

While we did not apply higher order mechanical descriptors in field studies
yet, still, by relying on a small laboratory dataset of 52 pebbles we argue
that their application carries substantial potential for geophysical insight
as they offer _biased natural catalogs_ of sedimentary shapes, meaning they
offer naturally defined, discrete classification schemes where the majority of
natural shapes is contained in very few classes. Such strong bias can not only
motivate the search for particular (rare) natural shapes, it can also offer
clues about the underlying evolution process. The first such strongly biased
natural catalog is associated with first/̄order mechanical shape descriptors
and led to the discovery of a fundamental monotonic trend in natural shape
evolution, the monotonic decrease of static balance points [6, 25]. Below we
briefly review classical shape descriptors as well as mechanical shape
descriptors and explain the concept of biased catalogs in more detail.

###  2.2 Classical descriptors

The most established shape descriptors are, without doubt, _axis ratios_ [31]
which, for an ellipsoid with axes $a>b>c$ may be written as
$y^{1}=c/b,y^{2}=b/a$ and for non/̄ellipsoidal shapes an approximating
ellipsoid is considered. An alternative geophysical shape descriptor is
_roundness_ [18] which is commonly measured as the isoperimetric ratio $0\leq
I\leq 1$ of the pebble’s contour [29, 25]. We will refer to the axis ratios
$y^{1},y^{2}$ and the isoperimetric ratio $I$ as _classical descriptors_.
Classical descriptors are, by definition, real numbers defined on a continuous
domain, so, in the sense defined in subsection 2.1, they are _continuous
descriptors_. Such descriptors admit the comparison of sample averages,
however, they do not immediately provide a classification or catalog for
natural shapes which could be based only on integer/̄type descriptors. While
we are not aware of any existing catalog for the isoperimetric ratio $I$, in
case of axis ratios it is apparent that there is a need for such catalogs: by
introducing the thresholds at $y^{1}=y^{2}=2/3$, Zingg [31] created the first
such system, subdividing all shapes into the four classes called _discs,
spheres, blades_ and _rods_.

Zingg classes $y^{1}<2/3$ $y^{1}>2/3$ $y^{2}>2/3$ Class I
DISC Class II
SPHERE $y^{2}<2/3$ Class III
BLADE Class IV
ROD

Figure 1: The Zingg catalog

Zingg’s catalog, illustrated in Figure 1 is finite (it defines 4 classes) and
it is complete, as each class contains geometric shapes (in fact, each class
contains ellipsoids). On the other hand, the Zingg classification is
artificial, as the threshold $2/3$ is an arbitrary choice. One can, of course,
study generalized Zingg catalogs where this threshold is being varied [13],
however, there exits infinitely many generalized Zingg catalogs and it is not
clear, which one should be used. Although artificial, the Zingg classification
still has the advantage to offer some bias: in coastal environments _blades_
and _discs_ appear to be dominant, admitting conclusions about the effects of
friction [7]. We remark that we also computed the Zingg classes for the
laboratory dataset, shown in Online Resource 1.

###  2.3 Mechanical shape descriptors and natural classes

####  2.3.1 The primary mechanical classification

The _primary mechanical classification_ , introduced in [30] is based on the
number of different types of _equilibrium points_ , that is, positions where
the body is at rest when supported on a horizontal surface, under gravity.
When pushed gently from any direction, the body sitting on a _stable_
equilibrium returns to its original position, while it tips over from an
_unstable_ equilibrium. We denote the respective numbers for stable and
unstable equilibria by $S,U$ and we also note that 3/̄dimensional objects also
have $H$  _saddle_ /̄type equilibria, where the behavior depends on the
direction of the push. The mathematical background of the equilibrium points
is the analysis of the _radial distance function_ $r_{K}$ measured from the
center of gravity $o$ of the body $K$. For planar objects this is a function
$r_{K}(\varphi)$ of the single polar angle $\varphi$ while in 3 dimensions we
have $r_{K}(\varphi,\theta)$ depending on two angles. Stable, unstable and
saddle points of the body correspond to the minima, maxima and saddles of this
function, respectively. These concepts are illustrated in Figure 2 for an
ellipse in two/̄dimensions and for an ellipsoid in three/̄dimensions. For the
latter, with main axes $a>b>c$ we can use the following parameterization in
the orientation depicted in Figure 1:

$\displaystyle x(\varphi,\theta)$
$\displaystyle=\tfrac{b}{2}\sin\theta\cos\varphi$ $\displaystyle
y(\varphi,\theta)$ $\displaystyle=\tfrac{c}{2}\sin\theta\sin\varphi$
$\displaystyle z(\varphi,\theta)$ $\displaystyle=\tfrac{a}{2}\cos\theta$

where $0\leq\varphi<2\pi$ and $0\leq\theta\leq\pi$. The distance from the
ellipsoid’s center of gravity $o$ is then the Euclidean distance:

$r_{\text{ell}}(\varphi,\theta)=\scriptstyle\sqrt{(x(\varphi,\theta)-o_{x})^{2}+(y(\varphi,\theta)-o_{y})^{2}+(z(\varphi,\theta)-o_{z})^{2}}$
(1)

The function $r_{\text{ell}}$ has maxima at $(0,0,\pm\frac{a}{2})$, saddles at
$(\pm\frac{b}{2},0,0)$ and minima at $(0,\pm\frac{c}{2},0)$. We are going to
return to this example in subsection 3.1 where we give a more rigorous
description of the equilibria of convex surfaces defined by smooth functions.
We are also giving a more precise definition of equilibria on convex
polyhedra, another well studied subset of convex bodies later in Section 4.
You can also see Figure 11 with the equilibria of a regular tetrahedron
marked.

Figure 2: Mechanical equilibria as stationary points of the radial distance
function $r$ a) In two/̄dimensions we have $r=r(\varphi)$ and stable and
unstable equilibria appear alternating, in pairs, so we have $S=U$ b) In
three/̄dimensions we have $r=r(\varphi,\theta)$. The respective numbers
$S,U,H$ for stable, unstable and saddle/̄type equilibria satisfy the
Poincaré=̄Hopf formula $S+U-H=2$

The numbers of different types of equilibria are related by the Poincaré=̄Hopf
formula $S+U-H=2$ [23] so it is sufficient to record $S$ and $U$ and we call
the pair $\\{S,U\\}$ the _primary equilibrium class_ of the body [30, 9, 6].
This means that the ellipsoid described by Eq. 1 is in the primary class
$\\{2,2\\}$. The primary mechanical classification system is illustrated in
Figure 3.

Figure 3: The primary mechanical catalog, showing examples of shapes with $i$
stable and $j$ unstable static balance points

Unlike the Zingg system, the primary mechanical catalog is infinite, and it is
also natural, since we do not make any arbitrary choices when assigning the
primary class $\\{S,U\\}$ to a particular shape, the class is _encoded in the
shape itself_. We also know that this catalog is complete, meaning no primary
class is empty [30]. This mathematical fact, however, is not related to the
statistical distribution of natural particles: datasets of beach pebbles
classified according to the primary mechanical catalog show very strong bias,
as approximately 70% of all beach pebbles appear in class
$\\{S,U\\}=\\{2,2\\}$ [13], making $\\{2,2\\}$ the dominant primary class.
This fact, along with analysis of the statistical distribution is a key
geophysical evidence supporting the theory that the total number $N=S+U+H$ of
equilibria is monotonically decreasing in natural abrasion [6].

This bias is also present in the dataset with 271 pebbles we based our results
on. Their distribution in the primary classes is presented in Table 1. We will
have special focus on the class $\\{2,2\\}$ which contains 52 pebbles in our
dataset.

Table 1: Pebbles used in the experiment

S U | 1 | 2 | 3 | 4 | 5 | 6 | 7 | sum
---|---|---|---|---|---|---|---|---
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
2 | 0 | 52 | 32 | 7 | 2 | 0 | 0 | 93
3 | 0 | 13 | 28 | 10 | 3 | 1 | 0 | 55
4 | 0 | 15 | 28 | 19 | 3 | 1 | 0 | 66
5 | 0 | 2 | 10 | 10 | 12 | 1 | 1 | 36
6 | 0 | 3 | 5 | 2 | 3 | 0 | 0 | 13
7 | 0 | 0 | 0 | 3 | 1 | 1 | 1 | 6
8 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 2
sum | 0 | 85 | 103 | 51 | 24 | 6 | 2 | 271
(a) Number of pebbles in each primary equilibrium class

S U | 2 | 3 | 4 | 5
---|---|---|---|---
2 |  |  |  |
3 |  |  |  |
4 |  |  |  |
5 |  |  |  |
(b) A pebble from each class in the highlighted region

While the strong bias in the primary mechanical classification is most
telling, it also has the drawback that it does not offer any clue on shapes
_inside_ any of the primary classes, most notably, inside the dominant class
$\\{S,U\\}=\\{2,2\\}$.

####  2.3.2 Higher order mechanical classifications

The trivial drawback of catalogs is that they do not distinguish between
shapes in one class. This can be remedied by introducing finer, higher order
catalogs. In case of mechanical descriptors this idea led to study, beyond the
number, also the _relative locations_ of equilibrium points. We can have two
alternative approaches to this task: we can either apply a natural, discrete
decomposition to the _range_ of the distance function $r=r(\varphi,\theta)$,
or, we can apply a natural, discrete decomposition of its _domain_. To
decompose the range, we will use saddle points and to decompose the domain, we
will use isolated integral curves of the gradient. In the first case we arrive
at a graph defining a natural _hierarchy_ among equilibria, in the second case
we arrive at a graph defining a natural _arrangement_ among equilibria. We
describe both graphs below.

The hierarchy among equilibria is based on the value of the radial distance
function, and this information is carried by the _Reeb/̄graph_ $R(K)$
associated with the body $K$ [1]. Each point of an edge in the Reeb/̄graph
$R(K)$ corresponds to a connected component of the level set
$r_{K}=\text{constant}$. We will define the Reeb/̄graph more rigorously for
smooth functions in subsection 3.2, where we also show how the ellipsoid of
Eq. 1 is degenerate due to its symmetry. A similar issue arises with the
regular tetrahedron, an issue we will discuss in subsection 4.1 along with the
definition of the Reeb/̄graph for polyhedra. Slightly moving the reference
point $o$ off all 3 symmetry planes of the ellipsoid results in the
non/̄degenerate Reeb/̄graph shown in Figure 4(a). However, this is not the
only Reeb/̄graph in the primary class $\\{2,2\\}$. The number $R(S,U)$ of
distinct Reeb/̄graphs in each primary class grows exponentially [1, 24].
Reeb/̄graphs define a finite, complete natural catalog inside any primary
equilibrium class and of course they define an infinite catalog if we consider
all primary classes. We call this the _R/̄secondary mechanical classification_
scheme. Whether and to what extent R/̄secondary classification is biased, we
will discuss below.

The information on the spatial arrangement of equilibria can be described by
using the isolated integral curves of the gradient vector field $\nabla r_{K}$
[16], which define, as edges, the so/̄called Morse=̄Smale graph $M(K)$
associated with the body $K$ [9]. The vertices of the Morse=̄Smale graph are
the static equilibrium points. Morse=̄Smale graphs have various equivalent
representations, to which we return in subsection 3.3 where we define the
Morse=̄Smale graph for smooth functions. Each of these representations can be
enumerated and thus can be associated with an integer label, the so/̄called
canonical code [2]. A polyhedral version of the Morse=̄Smale graph also
exists, which well be discussed in subsection 4.2.

The Morse=̄Smale graph belonging to the ellipsoid in the previous example is
shown in Figure 4(b). However, this is not the only possible Morse=̄Smale
graph in the primary class $\\{S,U\\}=\\{2,2\\}$. The number $M(S,U)$ of
distinct Morse=̄Smale graphs in each primary class grows approximately with
the exponent $p=S+U$ [17]. Similarly to Reeb/̄graphs, Morse=̄Smale graphs
define a finite, complete natural catalog inside any primary equilibrium class
and they define an infinite catalog if we consider all primary classes. We
call this the _M/̄secondary mechanical classification_ scheme. Similarly to
Reeb/̄graphs, we will discuss whether the M/̄secondary classification is
biased.

(a) Reeb/̄graph

(b) Morse=̄Smale graph

Figure 4: Secondary mechanical descriptors associated with the tri/̄axial
ellipsoid given in Eq. 1 (a) Reeb/̄graph with center of mass $o$ slightly
offset from symmetry planes (b) Morse=̄Smale graph, center of mass $o$ can be
located either at the intersection of symmetry planes or with small offset

These R/̄secondary and M/̄secondary classification schemes are independent in
the sense that identifying the class of a body in one system does not locate
it in the other system. To bridge the gap we will introduce the _master graph_
$G(K)$ associated with the body $K$, which carries information on both the
integral curves and the level sets of the radial distance function and from
which both the M/̄secondary and the R/̄secondary classification can be
obtained. The master graph is going to be defined separately for smooth
surfaces in subsection 3.4 and for polyhedra in subsection 4.3. We will call
the scheme defined by the master graph the _tertiary mechanical
classification_ , defining up to $M(S,U)\cdot R(S,U)$ tertiary classes in the
primary class $\\{S,U\\}$. We will denote their exact number by $G(S,U)$.
Inside any primary class $\\{S,U\\}$, the tertiary scheme defines a natural,
finite catalog. However, unlike any previous catalog, the tertiary
classification is incomplete as there exists pairs of Reeb/̄graphs and
Morse=̄Smale graphs in the same primary class which can not belong to the same
object. We will illustrate the tertiary classification scheme on the primary
class $\\{S,U\\}=\\{2,2\\}$ and explain its significance. In particular, in
Section 5 we will prove

######  Lemma 1.

$G(2,2)=3$

For an example see Figure 5 showing the possible secondary and tertiary
classes in the primary class with 2 stable and 2 unstable points. A more
detailed view of this primary class and its higher order subclasses will be
presented later in Table 2.

Figure 5: Division of the primary class $\\{2,2\\}$ into higher order
subclasses. The class of the ellipsoid used in previous examples is
highlighted under each classification scheme

While these higher order classifications are most promising, the main obstacle
in their application was that they cannot be reliably identified by hand
experiments and there was no reliable algorithm and no computer/̄based tool to
extract them from natural shapes. The first main result presented in this
article is an algorithm computing all three previously mentioned graphs using
the 3D scan of the particle.

We have investigated the primary class with the highest number of naturally
occurring pebbles, the class of bodies with 2 stable and 2 unstable points.
Every such pebble in our collection fell into the same tertiary class. This
suggests that the natural abrasion process is heavily skewed towards certain
classes, just like in the case of the primary classes.

##  3 Classification of smooth, convex shapes

###  3.1 Primary equilibrium classes

In subsection 2.3 we have introduced the radial distance function
$r_{K}(\varphi,\theta):S^{2}\to\mathbb{R}$, and also stated how its extrema
correspond to equilibria of the convex body $K$. Next we examine the case
where $r_{K}$ is at least twice continuously differentiable. In this case the
non/̄degenerate _critical point $x\in S^{2}$_ of the gradient field $\nabla
r_{K}$ corresponds to a static equilibrium point of the body: a minimum,
maximum or saddle point corresponds to a stable, unstable or saddle/̄type
equilibrium point, respectively [30]. Non/̄critical points are called
_regular_. We denote the type of equilibrium point with a superscript when
relevant, $x^{s},x^{u}$ and $x^{h}\in S^{2}$ are stable, unstable and saddle
points, respectively. The capital letters $S,U$ and $H\in\mathbb{N}$ still
represent the total number of these equilibria for $K$.

The smooth function $r_{K}$ is called _Morse_ if all its critical points are
non/̄degenerate. In this case the numbers of different equilibria are all
finite because the function’s domain is compact. The Poincaré=̄Hopf theorem
establishes the relationship between these values: $H=S+U-2$. If the _critical
value_ $r_{K}(x)$ for every critical point $x$ is distinct too then the
function is called an _excellent Morse function_ [24].

For example, let us return to the ellipsoid introduced in Eq. 1. All of its
equilibria are non/̄degenerate, but the ones of the same type are at equal
distances from the center of gravity. This makes it a Morse function but not
an excellent one. Choosing the value of $o$ such that none of its coordinates
is zero breaks the symmetry, making $r_{\text{ell}}$ an excellent Morse
function. Figure 6 shows the visible equilibrium points for $o=(0.2,0.1,0.3)$.
An additional minimum, maximum and saddle point is hidden behind the
ellipsoid.

Figure 6: All visible equilibria

The _primary equilibrium class $\\{i,j\\}$_ ($i,j=0,1,\dots$) contains all
convex bodies with $S=i$ stable equilibria and $U=j$ unstable equilibria.
Bodies with 1 stable point are called _mono/̄stable_ , bodies with 1 unstable
point are called _mono/̄unstable_ and bodies in the class $\\{1,1\\}$ are
called _mono/̄monostatic_. Classifying an object whose radial distance
function is Morse is trivial, but we will show in the next section that
primary equilibrium classes are not limited to these.

Now we define two important tools – introduced in subsection 2.3.2 as second
order mechanical descriptors – for Morse functions specifically, the
Reeb/̄graph and the Morse=̄Smale graph.

###  3.2 Reeb/̄graph

A _level set_ of a real valued function is the set of points where the
function takes on a particular value ${d\in\mathbb{R}}$. In our case these are
the points at distance $d$ from the point $o$. A level set of $r_{K}$ may
consist of several closed curves, called _contour lines_. Contour lines are
unique at regular points. Contour lines at a stable or an unstable point
consist of that single point. We will call a contour line of $r_{K}$
containing a saddle point $x^{s}$ the _saddle contour line of $x^{s}$_. An
excellent Morse function’s saddle contour lines always contain a single saddle
point due to the critical values being distinct. Therefore the saddle contour
line of $x^{s}$ has a single self/̄intersection at $x^{s}$, as shown in Figure
7.

Figure 7: The saddle contour line of $x^{h}_{1}$

The number of contour lines is not constant throughout the codomain of
$r_{K}$. New contour lines appear at critical values of minima, disappear at
critical values of maxima, and split or merge at critical values of saddles.
To capture this evolution of level sets, first an equivalence relation
$\sim_{r_{K}}$ is defined for two points $p,q\in S^{2}$ such that
$p\sim_{r_{K}}q$ whenever $p$ and $q$ belong to the same contour line of
$r_{K}$. The _Reeb/̄graph of $r_{K}$_ is then the quotient space
$S^{2}/\sim_{r_{K}}$ equipped with the quotient topology. This graph’s
vertices correspond to the previously mentioned points of change in the number
of contour lines: every leaf corresponds to a minimum or maximum, the rest of
the vertices correspond to saddles. If $r_{K}$ is excellent Morse then its
Reeb/̄graph is a tree with inner vertices of degree 3 [1]. Figure 4(a) shows
the Reeb/̄graph of the ellipsoid in Figure 6.

Due to the one/̄to/̄one correspondence from its vertices to critical points,
the Reeb/̄graph of an object also determines its primary equilibrium class. On
the other hand, two objects in the same primary class might not have
isomorphic Reeb/̄graphs. For example see the first row of Table 2 showing the
different Reeb/̄graphs in the primary equilibrium class $\\{2,2\\}$. An
_R/̄secondary equilibrium class_ contains all convex bodies with isomorphic
Reeb graphs.

###  3.3 Morse=̄Smale graph

By definition at any regular point $p$ the gradient vector $\nabla r_{K}$ is
non/̄zero. Following the vector an _integral curve_ $c:\mathbb{R}\to S^{2}$ is
traced out, which is the longest possible curve through $p$ whose derivatives
agree with the gradient $\nabla r_{K}$ [14].

If $r_{K}$ is Morse, then every regular point belongs to one and only one
integral curve, two integral curves are either disjoint or exactly the same.
An integral curve starts at a critical point and ends at a critical point but
does not contain them. The two critical points at the ends of an integral
curve are distinct because the function’s value is strictly ascending along
the curve. The critical point with lower critical value is called the _origin_
while the other one is the _destination_ of the integral curve. The origin and
destination is either minimum=̄maximum, minimum=̄saddle or saddle=̄maximum.
Figure 8 gives an example for the first one.

Figure 8: One of the integral curves from $x^{s}_{1}$ to $x^{u}_{1}$

The _descending (ascending) manifold_ of a maximum (minimum) critical point
$x$ is the union of $x$ and all integral curves with $x$ as their destination
(origin). Only minimum=̄maximum integral curves belong to both a descending
and an ascending manifold, an integral curve is _isolated_ otherwise. Every
saddle point is the origin of exactly two isolated integral curves as well as
the destination of exactly two isolated integral curves. Figure 9 shows the
isolated integral curves of the ellipsoid in previous examples.

Figure 9: Isolated integral curves

The cells of the _Morse=̄Smale complex of $r_{K}$_ are the connected
components of the intersections between descending and ascending manifolds.
Every cell of a Morse=̄Smale complex is a quadrangle bounded by isolated
integral curves connecting critical points in the following order: saddle,
maximum, saddle, minimum [14]. We call the 1/̄skeleton of a Morse=̄Smale
complex the _Morse=̄Smale graph_.

The vertices in the Morse=̄Smale graph of $r_{K}$ correspond to critical
points and the edges correspond to isolated integral curves. Every vertex
corresponding to a saddle is of degree 4 and it is connected to exactly two
vertices corresponding to stable points and two vertices corresponding to
unstable points. There are no edges in the graph other than these. The
Morse=̄Smale graph is a 3/̄colored quadrangular graph. You can see the
Morse=̄Smale graph corresponding to Figure 9 in Figure 4(b). The _quasi/̄dual_
is an alternative, equivalent representation of the Morse=̄Smale graph, which
is constructed from the original graph by adding stable=̄unstable diagonal
edges and removing all saddle/̄type vertices. The quasi/̄dual of the
Morse=̄Smale graph is a two/̄colored quadrangular graph [9]. This
representation has fewer vertices and edges, which makes it more suitable for
visualization. For this reason, we used it in the pebble catalog published as
Online Resource 1.

A secondary classification method was introduced in [9] based on the
isomorphism classes of Morse=̄Smale graphs – that is, shapes that belong to
the same secondary class are in the same primary class as well. The authors
also proved that for every combinatorially possible Morse=̄Smale graph a
smooth convex body exists. The Reeb/̄graph also has the first property, two
shapes with isomorphic Reeb graphs belong to the same primary class. On the
other hand, the Reeb/̄graph associated with a function does not uniquely
define the Morse=̄Smale graph associated with the same function, or vice
versa.

An _M/̄secondary equilibrium class_ contains all convex bodies with isomorphic
Morse=̄Smale graphs. Objects in the same M/̄secondary class belong to the same
primary class. However, bodies in the same M/̄secondary class do not
necessarily belong to the same R/̄secondary class or vice versa. See Table 2
for examples in the $\\{2,2\\}$ primary class. In the next section we
introduce a tertiary classification scheme where bodies in the same tertiary
class belong to the same primary, M/̄secondary and R/̄secondary class.

###  3.4 Master graph

The Morse=̄Smale graph encompasses information about integral curves, the
lines of fastest ascend. The Reeb/̄graph on the other hand portrays contour
lines, paths of constant function value. We have introduced the so called
_master graph_ in subsection 2.3.2 that describes both aspects. This
subsection will focus on the master graph of Morse functions specifically. The
key is the relationship between two types of significant curves: the isolated
integral curves and the saddle contour lines.

######  Definition 1.

The _intersection point_ $y\in S^{2}$ is a point where an isolated integral
curve of $\nabla r_{K}$ and a saddle contour of $r_{K}$ intersect.

######  Definition 2.

Let $x_{1}$ and $x_{2}$ be critical points with an isolated integral curve $c$
between them. Let $y_{1},\dots,y_{n}$ be all the intersection points on $c$
($r_{K}(y_{1})<\dots<r_{K}(y_{n})$). Since $r_{K}$ is monotonic along $c$, the
ordering along $c$ will agree with the ordering by function values. An
_isolated set_ is the ordered set of points $x_{1}-y_{1}-y_{2}-\dots-
y_{n-1}-y_{n}-x_{2}$. Two adjacent points in this sequence are called
_gradient neighbors_.

######  Definition 3.

A saddle point $x^{h}$ and an intersection point $y$ are _contour neighbors_
if there is a saddle contour line of $r_{K}$ that contains both $x^{h}$ and
$y$.

######  Definition 4.

In the _master graph_ a vertex corresponds to every critical point and every
intersection point. Edges run between every gradient neighbor and every
contour neighbor.

We call edges between gradient neighbors _gradient edges_. We call the path
corresponding to an isolated set an _isolated path_ which is made up of
gradient edges. We call edges between contour neighbors _contour edges_.
Figure 10 shows an example.

Figure 10: A master graph ($x^{u}_{1},x^{u}_{2}$: unstable points,
$x^{h}_{1},x^{h}_{2}$: saddle points, $x^{s}_{1},x^{s}_{2}$: stable points,
orange dots: intersection points, solid lines: gradient edges and solid lines
of the same color constitute an isolated path, dashed lines: contour edges)

We may obtain the Morse=̄Smale complex from the master graph by removing every
contour edge and replacing isolated paths with single edges. We may obtain the
Reeb/̄graph by contracting the contour edges and unifying parallel edges.
Figure 4 shows the Morse=̄Smale complex and Reeb/̄graph that belong to the
master graph in Figure 10.

Table 2: Secondary and tertiary classes in the $\\{2,2\\}$ primary class. Rows correspond to M/̄secondary classes, columns correspond to R/̄secondary classes. Intersection of row $i$ and column $j$ shows the master graph belonging to the $i$/̄th M/̄secondary and $j$/̄th R/̄secondary class. MS Reeb |
---|---
| empty

Using the master graph we can define a third level in our classification
hierarchy.

######  Definition 5.

A _tertiary equilibrium class_ contains all convex bodies with isomorphic
master graphs.

Bodies in the same tertiary class belong to the same primary, M/̄secondary and
R/̄secondary classes. Table 2 shows how there could be two different master
graphs in the same M/̄secondary and R/̄secondary classes.

##  4 Classification of convex polyhedra

In this section we examine the case where the radial distance function is not
smooth, rather its image is the convex polyhedron $P$. Just like introduced in
subsection 2.3.1, the value of the function still measures the distance from
the point $o$ which is usually chosen as the center of gravity of $P$. The
function is continuous just like in the previous section, but the gradient
$\nabla r_{P}$ exists only in the interior of the faces of $P$. If the point
$p$ is in the interior of the edge $e$, then at $p$ only the directional
derivative along $e$ exists. However, for each face adjacent to $e$ we can get
the gradient of the function at $p$ that measures the distance from $o$ for
the whole plane of the face. In [22] we named these vectors _candidate
gradients at $p$_ if they are tangential to the polyhedron $P$. Candidate
gradients were defined the same way at vertices too. We called points where at
least one non/̄zero candidate gradient exists _regular_ , and also proved that
at these points there is a unique candidate gradient with maximal length that
we called the _extended gradient $\nabla^{\text{ext}}r_{P}$_.

The concept of equilibrium points and non/̄degeneracy of convex polyhedra are
already established in the literature [12]. We say that $x\in P$ is an
_equilibrium point_ of $P$ (with respect to $o$) if the plane $H$ through $x$
and perpendicular to $[o,x]$ supports $P$ at $x$. In this case $x$ is
_non/̄degenerate_ if $H\cap P$ is the (unique) face, edge or vertex of $P$
that contains $x$ in its relative interior. We have shown in [22] that a point
$x\in P$ is an equilibrium point if and only if there is no candidate gradient
at $x$. The function $r_{P}$ is _polyhedral Morse_ if all its equilibrium
points are non/̄degenerate. The function is _excellent polyhedral Morse_ if
the equilibrium points are at distinct distances from $o$.

Let us take a regular tetrahedron as an example. If we set $o$ in its center
of gravity then it has a stable equilibrium point on all of its faces, a
saddle on all of its edges and an unstable equilibrium in all of its vertices.
All of these are non/̄degenerate, but all equilibria of the same kind has the
same distance from $o$. Similarly to the ellipsoid previously, we can move $o$
such that the distance from it becomes an excellent polyhedral Morse function.
See Figure 11 for reference.

Figure 11: Visible equilibrium points of the example tetrahedron

An approach of defining critical points directly on polyhedra already exists
in the literature [3] where the function is the distance from a reference
plane. This function is piecewise linear and all of its critical points fall
on vertices. This setup is a good fit for processing all kinds of datasets
from terrains to models for 3D printing, which makes it widely used. A review
of existing algorithms for computing critical points and the Morse=̄Smale
graph in such a case is available in [5]. These are not applicable to the
not/̄piecewise/̄linear radial distance function $r_{P}$ discussed in the
current article but inspired aspects of our solution.

###  4.1 Reeb/̄graph

Subsection 3.2’s definitions of the contour lines and the Reeb/̄graph apply to
$r_{P}$ as well, but their properties might differ. Contour lines are
continuous, closed curves, made up of circular arcs connecting at
non/̄differentiable points. Contour lines are unique at regular points.
Contour lines containing a stable or unstable point do not include other
points. If $r_{P}$ is excellent polyhedral Morse then a saddle contour line
contains a single saddle only. Saddle contour lines containing the
non/̄degenerate saddle $x^{h}$ on the edge $e$ contain a circular arc (which
might be a complete circle) on each face adjacent to $e$ that is tangent to
$e$ at $x^{h}$. See Figure 12 for an example.

Figure 12: The saddle contour line of one of the saddle points

The Reeb/̄graph of an excellent polyhedral Morse function $r_{P}$ is a tree,
its vertices correspond to equilibrium points. Leaves correspond to stable and
unstable, inner vertices correspond to saddle points. Inner vertices are of
degree 3.

###  4.2 Morse=̄Smale graph

The other key concept introduced in [22] beside the extended gradient was the
_ascending curve_. It traces out a path following the extended gradient,
similarly to integral curves, but relaxes the requirement on the derivative.
An ascending curve through a regular point is the longest possible curve whose
right hand derivatives agree with the extended gradient
$\nabla^{\text{ext}}r_{P}$. An ascending curve is a continuous open polygon.
See Figure 13 for example.

Every regular point belongs to at least one ascending curve, two ascending
curves can merge, but cannot cross or split. An ascending curve starts at an
equilibrium point and ends at an equilibrium point, containing the latter but
not the first one. The two equilibrium points at the ends of an ascending
curve are distinct because the function’s value is strictly ascending along
the curve. The equilibrium point with lower critical value is called the
_origin_ while the other one is the _destination_ of the ascending curve. The
origin and destination is either stable=̄unstable, stable=̄saddle or
saddle=̄unstable.

Figure 13: A stable=̄unstable ascending curve

The _descending (ascending) polyhedral manifold_ of an unstable (stable)
equilibrium point $x$ is the union of $x$ and all ascending curves with $x$ as
their destination (origin). Only stable=̄unstable ascending curves can belong
to both a descending and an ascending polyhedral manifold, an ascending curve
is _isolated_ otherwise. Every saddle point is the origin of exactly two
isolated ascending curves as well as the destination of exactly two isolated
ascending curves. See Figure 14 for an example.

Figure 14: Isolated ascending curves

The cells of the _Morse=̄Smale complex of $P$_ are the connected components of
the intersections between descending and ascending polyhedral manifolds. Every
cell of the Morse=̄Smale complex is a quadrangle bounded by isolated integral
curves connecting equilibrium points in the following order: saddle, unstable,
saddle, stable. We call the 1/̄skeleton of a Morse=̄Smale complex the
_Morse=̄Smale graph_.

The vertices in the Morse=̄Smale graph of $P$ correspond to equilibrium points
and the edges correspond to isolated ascending curves. Every vertex
corresponding to a saddle is of degree 4 and it is connected to exactly two
vertices corresponding to stable points and two vertices corresponding to
unstable points. There are no edges in the graph other than these. The
Morse=̄Smale graph is a 3/̄colored quadrangular graph.

###  4.3 Master graph

The master graph of a Morse function encompasses the information of isolated
integral curves and saddle contours. In the polyhedral case discussed in this
subsection it describes the relationship of isolated ascending curves and
saddle contours.

######  Definition 6.

The _intersection point $y\in S^{2}$_ is a point where an isolated ascending
curve of $\nabla^{\text{ext}}r_{P}$ and a saddle contour of $r_{P}$ intersect.

######  Definition 7.

Let $x_{1}$ and $x_{2}$ be critical points with an isolated ascending curve
$a$ between them. Let $y_{1},\dots,y_{n}$ be all the intersection points on
$a$ ($r_{P}(y_{1})<\dots<r_{P}(y_{n})$). An _isolated set_ is the ordered set
of points $x_{1}-y_{1}-y_{2}-\dots-y_{n-1}-y_{n}-x_{2}$. Two adjacent points
in this sequence are called _gradient neighbors_.

######  Definition 8.

A saddle point $x^{h}$ and an intersection point $y$ are _contour neighbors_
if there is a saddle contour line that contains both $x^{h}$ and $y$.

Two isolated ascending curves might have merged at a point $p$ and both
intersect a saddle contour at $y$ ($r_{P}(p)<r_{P}(y)$). We treat these two
isolated ascending curves as separate even if they share some of their points.
In the master graph we will have two separate vertices corresponding to $y$ as
well.

######  Definition 9.

An intersection point $y$ is called _$n$ /̄fold_ if it is the intersection of
$n$ isolated ascending curves and a saddle contour at the same point.

######  Definition 10.

In the _polyhedral master graph_ a vertex corresponds to every critical point
and $n$ vertices correspond to every $n$/̄fold intersection point. Every
vertex is connected to one and only one of the vertices corresponding to each
of its gradient neighbors. Edges run between contour neighbors too.

The steps of creating the Reeb/̄graph and Morse=̄Smale graph from the
polyhedral master graph are identical to the smooth case.

##  5 Equilibrium classes as shape catalogs

So far we have defined the primary, secondary and tertiary equilibrium classes
for both smooth surfaces and polyhedra. Now we can more clearly formulate the
mathematical challenge outlined in Section 1. We do this by exploring more
details of the classification system in general. In subsection 2.3.2 we
mentioned that $G(S,U)\leq R(S,U)\cdot M(S,U)$ for any primary class
$\\{S,U\\}$. For example, the equality holds for trivial cases like the
classes $\\{1,1\\}$, $\\{1,2\\}$ or $\\{2,1\\}$ with a single Reeb/̄graph, a
single Morse=̄Smale graph and therefore a single master graph.

Our goal in this section is to show by proving Lemma 1 that there is at least
one primary class where the equality does not hold. We start by constructing
polyhedra in 3 tertiary classes, providing a lower bound on their number.
First, let us classify the polyhedron with the lowest possible number of
vertices and faces. A tetrahedron with 2 stable and 2 unstable equilibria is
presented in [12]. We use the following lemma in the classification process.

######  Lemma 2.

A non=̄monostatic tetrahedron’s radial distance function cannot have parallel
isolated ascending curves in its Morse=̄Smale graph.

Before proving Lemma 2, we provide some necessary background. In [22] we
defined an edge of a polyhedron as _followed_ , if the extended gradient is
parallel to the edge at every single one of its points, otherwise we called it
_crossed_. The names come from the behavior of an ascending curve through
interior points of an edge: in the first case they follow the edge, in the
second case they cross it. We have proven that this behavior is uniform at
every single point of a given edge. In the following we take advantage of
these two observations:

  1. 1.

Every edge connected to an unstable point is followed.

  2. 2.

On convex polyhedra, a saddle=̄stable isolated ascending curve crosses only
crossed edges.

###### Proof.

Consider the tetrahedron with vertices $A,B,C$ and $D$.

First we prove that two isolated ascending curves between the saddle
$x_{1}^{h}$ and the unstable point $x_{1}^{u}$ cannot exist. Let us consider
the case where the tetrahedron has one more unstable point $x_{2}^{u}$, and
assign them to vertices as $x_{1}^{u}=A$ and $x_{2}^{u}=B$. The two isolated
ascending curves connecting $x_{1}^{h}$ to $x_{1}^{u}$ form a cycle on the
tetrahedron. It must be 3 edges long as any shorter would not be a cycle and
any longer would contain both unstable points, or in other words, the two
isolated ascending curves run along the boundary of a face. This can only be
the face $ACD$ as every other face contains the other unstable vertex $B$.
According to the quadrangle lemma [14], there is a stable point $x_{1}^{s}$ on
the face $ACD$ connected to $x_{1}^{h}$ by an isolated ascending curve. The
radial distance function is strictly monotone ascending from $x_{1}^{h}$ to
$x_{1}^{u}$ along both isolated ascending curves running along the edges of
$ACD$. However, any face containing a stable point contains at least 2 local
minima when looking at the distance function along its boundary curve. This
contradiction means that parallel $x_{1}^{h}-x_{1}^{u}$ isolated ascending
curves cannot exist on tetrahedra with only 2 unstable points in total. The
statement holds for 3 or 4 unstable points as well, because every face would
contain at least 2 of them on its boundary, which makes parallel isolated
ascending curves impossible.

Next we prove that two isolated ascending curves between the saddle
$x_{1}^{h}$ and the stable point $x_{1}^{s}$ cannot exist either. Let us
consider the case where the tetrahedron has one more stable point $x_{2}^{s}$.
The parallel isolated ascending curves connecting $x_{1}^{h}$ to $x_{1}^{s}$
form a cycle and run on all 3 faces of the tetrahedron other than the one
containing $x_{2}^{s}$. The same logic applies as previously: a cycle on less
faces would not be possible, a cycle on more faces would have to contain
$x_{2}^{s}$ too. The cycle encircles one of the vertices of the tetrahedron on
its own, which must be unstable according to the quadrangle lemma. This leads
to a contradiction though, because every edge connected to an unstable vertex
must be _followed_ , but one of the $x_{1}^{h}-x_{1}^{s}$ isolated ascending
curves also _crossed_ them. The extension to 3 or 4 stable points is also
similar to the unstable case: adding more of these points would make it
impossible to form a cycle of parallel isolated ascending curves on the faces
of the tetrahedron. ∎

As a consequence, the previously cited tetrahedron is in the first row of
Table 2. In the next step we construct polyhedra for the bottom row of the
table. Start with a body that has parallel isolated ascending curves between
one of its saddle and its single stable point already, the monostable
polyhedron of Conway and Guy [4]. It has 1 stable, 3 saddle and 4 unstable
points, see these in Figure 15.

Figure 15: Monostable polyhedron of Conway and Guy. The stable point is shown
in green, saddles are shown in blue, unstable points are shown in red. The
figure is not proportional, for example $r=1,a=0.1,b=30$ results in a
monostable polyhedron

We merge the two pairs of unstable points and the saddle between them at both
ends of the polyhedron. Next we raise the two resulting unstable points above
their current plane by a small amount as shown in Figure 16. The resulting
polyhedron is in the class $\\{2,2\\}$. We end up in different cells of our
table depending on how much we move the edge with the saddle on the bottom.
Using the notation of the previous two figures, if $c=0$, then there is two
isolated ascending curves between the two saddle points, resulting in a
degenerate master graph. If $c$ is greater than 0 but close to it, then the
polyhedron is in the first column. If $c$ is less then $a+b$ but close to it,
then the body is in the second column. If $c$ is close to neither 0 nor $a+b$
then one of the stable points vanishes, the polyhedron leaves the class
$\\{2,2\\}$.

Figure 16: Polyhedron with 2 stable and 2 unstable equilibria

We have seen that in a non/̄degenerate Reeb/̄graph, every saddle point has at
least one of its 3 neighbors at higher distance and at least one at lower
distance from the center of gravity $o$. This means that there can only be 2
non/̄isomorphic Reeb/̄graphs in $\\{2,2\\}$, both shown in Table 2. The same
primary class also has 2 Morse=̄Smale graphs [9]. However, the number of
master graphs inside this class is not 4 but only 3.

###### Proof of Lemma 1.

We have shown that $G(2,2)\geq 3$ by providing a polyhedron in 3 of its
tertiary classes.

Let us use the notation of Table 2, and assume that there exists a master
graph in the cell at the first row and second column. According to the
Reeb/̄graph, the saddle contour through $x_{1}^{h}$ divides the surface into 3
parts, one containing the stable point $x_{2}^{s}$, one containing the
unstable point $x_{1}^{u}$ and one containing the rest. Notice that the other
stable point $x_{1}^{s}$ is in this last part.

Considering the neighbors of the saddle $x_{1}^{h}$ in the Reeb/̄graph, only 1
of them has a lower value of $r_{K}$ than $r_{K}(x_{1}^{h})$, namely the
stable point $x_{2}^{s}$. This means that both stable=̄saddle isolated
integral curves fall within the same one of the 3 previously mentioned parts
bounded by the saddle contour of $x_{1}^{h}$. Because this part contains
$x_{2}^{s}$ only, both curves are connected to this stable point. These two
parallel edges contradict our assumption on the Morse=̄Smale graph, leading to
no master graph in the cell at the first row and second column of Table 2. ∎

##  6 Classification of natural shapes

Our last goal is the classification of natural shapes, which leads us to the
solution of the algorithmic challenge mentioned in Section 1. The equilibria
of a pebble can be measured by hand, but this method is highly dependent on
the person’s abilities carrying out the experiment. Manual measurement of
boulders gets more difficult if not outright impossible as their size
increases. We utilize 3D scanning technology to achieve consistency in the
results and to get around size limitations.

In such scenarios equilibria appear on two separate scales: the global value
$N$ corresponds to the approximation of the particle’s convex hull by a
sufficiently smooth surface while the local value $N^{\triangle}$ corresponds
to the polyhedral approximation (with faces of maximal diameter $\triangle$)
acquired by 3D scanning. The main obstacle is that in general
$\lim_{\triangle\to 0}N^{\triangle}>N$ [10, 11].

Local equilibria appear in _flocks_ which are spatially localized around the
locations of global equilibria. This poses a considerable problem for
measurements: while only $N^{\triangle}$ is directly available from any 3D
scanned dataset, the physically relevant quantity is $N$. To obtain $N$ based
on $N^{\triangle}$, one needs some artificial “blurring” of the data.

Edelsbrunner et.al. achieved this by defining a simplifying operation on the
Morse=̄Smale graph called _cancellation_ , which eliminates two adjacent
critical points [14]. First we describe the original operation, then present
our extension of it to the master graph. There are two possible combinations
of critical points to be cancelled: a minimum and a saddle or a saddle and a
maximum, but they are symmetrical. Let $x^{h}$ be the saddle and $x^{u}_{1}$
the unstable vertex of the canceled pair while $x^{u}_{2}$ the other unstable
vertex connected to $x^{h}$ ($x^{u}_{1}\neq x^{u}_{2}$). The cancellation
combines the three vertices into $x^{u}_{2}$ by removing every edge connected
to $x^{h}$ and merging the $x^{u}_{1}$ and $x^{u}_{2}$ vertices.

The basic idea is the same on the master graph, but only the two cancelled
vertices must be connected directly, the remaining stable or unstable point
can be connected to the cancelled saddle through an isolated path. This
requires some extra bookkeeping in the following definition to end up with a
valid master graph after cancellation.

######  Definition 11.

Let $x^{h}$ be a saddle and $x^{u}_{1}$ an unstable point such that there is
an $x^{h}-x^{u}_{1}$ edge in the master graph. Let $x^{u}_{2}$ be the other
unstable point that is connected to $x^{h}$ through an isolated path
($x^{u}_{1}\neq x^{u}_{2}$), let the intersection points on this path be
$y_{1},\dots,y_{n}$. The _cancellation of $x^{h}$_ merges $x^{u}_{1}$ and
$x^{h}$ into $x^{u}_{2}$. The critical value in the remaining vertex is
$r_{K}(x^{u}_{2}):=\max\\{r_{K}(x^{u}_{1}),r_{K}(x^{u}_{2})\\}$. The steps are
the following:

  1. 1.

Remove every intersection along the two isolated paths originating from
$x^{h}$ and ending in a stable vertex.

  2. 2.

Let $D$ be the set of the third/̄to/̄last vertices on every isolated path
ending in $x^{u}_{1}$. Every isolated path ending in $x^{u}_{1}$ – except for
the $x^{h}-x^{u}_{1}$ edge – is at least 3 vertices long because they
originate from a saddle, end in $x^{u}_{1}$ and intersect the contour line
through $x^{h}$.

  3. 3.

Remove $x^{u}_{1}$ and all of its neighbors.

  4. 4.

Copy the $y_{1},\dots,y_{n}$ vertices and the edges connected to them $|D|-1$
times and connect a unique copy of $y_{1}$ to every vertex in $D$. These new
edges are not contour edges.

Figure 17 shows an example.

(a) Part of a master graph. Dashed: contour edge, squiggly: isolated path with
internal vertices omitted

(b) The graph after step 1. $D=\\{y^{\prime}_{2},y^{*}_{2}\\}$

(c) The graph after step 3.

(d) The resulting graph

Figure 17: Canceling the saddle $x^{h}$

Now that we defined the basic simplification step, the next question becomes
that which of the equilibria we should cancel and in what order. We draw our
inspiration from the same article as before, where the authors utilized the
level set of the surface. They called a critical point _positive_ if it
created a new contour line and _negative_ if it destroyed one while ascending
in the codomain of the distance function. Minima are positive, maxima are
negative and saddles can be either positive or negative. Every negative saddle
was paired with the preceding positive minimum and every negative maximum was
paired with the preceding positive saddle. They defined the _persistence_ of a
pair of equilibria as the difference in function value between the two points.
Saddles were then canceled from lower to higher persistence until the desired
Morse=̄Smale complex was reached.

These steps are easily adapted to the master graph, as it contains the
necessary information from the level sets of the surface. At any given point,
the saddle with the lowest persistence is connected directly to a stable or an
unstable point. There is always at least one such edge in the graph. We choose
the next equilibria to be canceled by iterating over all the remaining edges
of the graph looking for the saddle=̄stable or saddle=̄unstable edge having
the lowest difference in the value of $r_{K}$ at its vertices. We cancel the
saddles until we get the closest possible to the number of stable and unstable
points measured by hand.

###  6.1 Example on a single pebble

In this section we show the steps the algorithm takes to classify a single
pebble. First it was established via manual measurements that this pebble has
2 stable and 3 unstable equilibrium points, placing it in the $\\{2,3\\}$
primary class. Next the pebble was scanned using a 3D scanner, resulting in
the polyhedron denoted from now on by $P_{\text{ex}}$. Determining the higher
order equilibrium classes consists of three main steps:  1) constructing the
master graph of $P_{\text{ex}}$,  2) sorting the saddles and canceling them up
to the point where the master graph has only 2 stable and 3 unstable points,
3) creating the Reeb/̄graph and Morse=̄Smale graph from the master graph.  In
the rest of this section we will discuss these steps in detail. The source
code of the algorithm’s implementation is also available [20].

####  6.1.1 Construction of the master graph of $\boldsymbol{P_{\text{ex}}}$

In the very first step the algorithm goes over all the edges of
$P_{\text{ex}}$ and determines if they contain a saddle point. There are 25
such edges with saddle points on them in this example. This results in a list
of $(e_{i},x^{h}_{i})$ pairs where $e_{i}$ is the edge containing the saddle
point $x^{h}_{i}$ and $i$ goes from 1 to 25.

The next step is tracing the saddle contour line containing the saddle
$x^{h}_{i}$ for every $i\in[1,25]$. As established in Section 4.1, contour
lines are closed, continuous curves made up of circular arcs. Every arc is
either a full circle on a single face or has its endpoints on edges of
$P_{\text{ex}}$. We can therefore trace a contour line in an iterative fashion
by always finding the next arc connected to one of the endpoints of a partial
contour line until the curve is closed.

We have also established previously that the saddle contour line containing
any $x^{h}_{i}$ has an arc on each face adjacent to $e_{i}$. Any of these two
can be the starting partial contour line in the previous paragraph’s iterative
method. The resulting saddle contour lines are shown in blue in Figure 20. You
can also take a closer look at one of the flocks in Figure 18.

Once we have all the saddle contours, the next step is tracing the isolated
ascending curves. Every saddle point is the origin of 2 such curves and the
destination of 2 such curves. We start from the point $x^{h}_{i}$ for every
$i\in[1,25]$ and trace all 4 curves with an iterative algorithm, one segment
at a time. One curve is constructed toward each endpoint of $e_{i}$ in the
direction of the extended gradient, and one curve is constructed toward each
face adjacent to $e_{i}$ in the direction opposite to the extended gradient.
For both types of curves the last iteration stops at an unstable or stable
point, respectively. The two figures mentioned in the previous paragraph show
these curves as well.

Figure 18: A flock of equilibria. Stable, unstable and saddle points are
represented by green, red and blue dots respectively. Saddle contour lines are
shown in blue, green lines are stable=̄saddle, red lines are saddle=̄unstable
ascending curves

The creation of the master graph starts by adding the vertices $x^{h}_{i}$ for
every $i\in[1,25]$. We refer to the vertices of the master graph by the same
name we used for their corresponding points on $P_{\text{ex}}$. The graph is
extended every time an isolated ascending curve is traced, take for example
the one between $x^{h}_{1}$ and $x^{u}$ in Figure 18. The $x^{h}_{1}-x^{u}$
curve intersects the saddle contours of $x^{h}_{2},x^{h}_{3}$ and $x^{h}_{4}$,
therefore we add intersection points denoted by $y_{2},y_{3}$ and $y_{4}$ to
the graph. Every intersection point is then connected to the saddle with the
same index by a contour edge. If this is the first time we encounter $x^{u}$
we assign a new vertex to it in the master graph. The last step is adding
gradient edges along the $x^{h}_{1}-y_{2}-y_{3}-y_{4}-x^{u}$ path.

####  6.1.2 Cancellation of saddles

At this point we have the master graph of the polyhedron $P_{\text{ex}}$ with
14 stable and 13 unstable vertices. We sort the saddles and apply
cancellations as described in Section 6 until we get as close to the hand
measurements as possible. In this particular case we can get a graph exactly
in the $\\{2,3\\}$ primary class. Figure 19(a) shows the master graph after
the cancellations.

It is important to mention that cancellation works on the master graph, it
does not modify the polyhedron itself. After any number of cancellations, the
contour lines passing through the remaining saddles can still be visualized on
the polyhedron. On the other hand, there is no one/̄to/̄one correspondence
from edges of the master graph to isolated ascending curves on the surface of
$P_{\text{ex}}$ anymore. The method we use to circumvent this issue is as
follows (using the notations of definition 11): every isolated ascending curve
with its destination at the unstable point $x^{u}_{1}$ is extended with the
segments of both the $x^{h}-x^{u}_{1}$ and the $x^{h}-x^{u}_{2}$ isolated
ascending curves before cancellation. The resulting curve is not strictly
ascending anymore, but in our experience it gives a good intuition of where
the corresponding isolated integral curve on the original body would be.
Figure 21 shows this kind of visualization of the master graph of
$P_{\text{ex}}$ after the last cancellation.

####  6.1.3 Creation of the Reeb/̄graph and Morse=̄Smale graph

The steps to get the Reeb/̄graph and the Morse=̄Smale graph from the master
graph have already been described in Section 3.4. The resulting graphs for
$P_{\text{ex}}$ can be seen in Figure 19.

(a) Master graph. Contour edges are in blue, green edges are on a
stable=̄saddle isolated path, red edges are on a saddle=̄unstable isolated
path

(b) Reeb/̄graph

(c) Morse=̄Smale graph. Green edges represent stable=̄saddle isolated integral
curves, red edges represent saddle=̄unstable isolated integral curves

Figure 19: Graphs of $P_{\text{ex}}$. Vertices in the same position in all 3
subfigures correspond to the same equilibrium point. Stable, saddle, unstable
and intersection points are represented by green, blue, red and orange points,
respectively

###  6.2 Cataloging a population of pebbles

In the previous example we purposefully selected a pebble where – after
several cancellations – the number of computed stable and unstable equilibria
perfectly matched the hand measurements. This is not always the case. Take for
instance a pebble that was placed in the class $\\{3,4\\}$, with its master
graph having 4 stable and 4 unstable vertices at an intermediate step of the
algorithm. The number of stable points is off by 1, and the next cancellation
can either decrease this or decrease the number of unstable points. In the
first case we get a perfect match, but in the second case both values would be
off by 1, resulting in a worse match.

We have to keep in mind that despite all their best efforts, the people
measuring the pebbles by hand can make mistakes. Consequently a perfect match
is not inherently possible or even desirable in every single case. Following
the previous paragraph’s logic, we computed the master graph that is off by
the smallest amount for 271 pebbles in our inventory. More formally, after
every cancellation we calculated the absolute difference of the manual and the
computer measurements in both the number of stable and the number of unstable
points, and selected the master graph where the sum of these two absolute
differences was the lowest.

A full list of every single pebble with a photo and a host of manually
measured and computed values of both classical and mechanical shape
descriptors is available in Online Resource 1. The 3D scan of every pebble was
also published in a repository [21] with the LaTeX source code of the
aforementioned document and the scripts compiling it. Be aware that some of
the pebbles could not be photographed or their material could not be
identified, in this case the corresponding field is left empty. The visual
representation of graphs in the document are for illustration purposes only,
our program outputs an alphanumerical encoding of every graph which can be
used for actual classification.

Table 3 gives the number of pebbles with a given difference in the number of
computed stable/unstable points compared to the hand measurements. In 185 out
of 271 cases ($68.27\%$) the computed master graph perfectly matched the
measured primary class of the corresponding pebble. There are no pebbles where
both values are positive because further cancellations always get results
closer to the hand measurements. There are also no cases where both values are
negative because further cancellations would only increase the sum of their
absolute values.

Table 3: The number of pebbles with a given difference in the number of computed stable/unstable points compared to the hand measurements stable difference | unstable difference | sum
---|---|---
-1 | 0 | 1 | 2 | 3
-3 | 0 | 0 | 0 | 0 | 0 | 0
-2 | 0 | 0 | 1 | 1 | 0 | 2
-1 | 0 | 27 | 4 | 4 | 0 | 35
0 | 4 | 185 | 32 | 10 | 3 | 234
1 | 0 | 0 | 0 | 0 | 0 | 0
sum | 4 | 212 | 37 | 15 | 3 | 271

The most important point is that 248 out of 271 pebbles ($92.51\%$) is off by
at most 1 stable or 1 unstable point compared to the manual measurements. With
the primary classification being this close we are confident in the computed
secondary and tertiary classes as well.

In all 52 cases where the manual measurement put a pebble in the primary class
$\\{2,2\\}$, the computer did so as well. Our most important observation is
that all of them belong to the same tertiary – and thus R/̄secondary and
M/̄secondary – equilibrium class, namely the one containing all the tetrahedra
inside of this primary class.

Based on the mathematical results presented in Section 5 and the measurement
data in this subsection, the higher order mechanical descriptors give a
finite, complete, biased natural catalog inside the primary class $\\{2,2\\}$.
Two of these adjectives outline the direction of further research:  a) from
the mathematical point, the completeness of the tertiary classification is
still an open question,  b) from the geomorphological point, the bias of the
higher order classification can be further examined.

##  7 Summary

In this paper we showed that higher order mechanical descriptors are a viable
option for the description of sedimentary particles. These descriptors, given
as graphs, carry essential, naturally encoded three/̄dimensional information
on the shape. Despite the fact that their mathematical existence was known,
the challenges connected with their extraction and identification proved to
be, until now, prohibitive.

First order mechanical descriptors, defined by the respective numbers of
stable and unstable static balance points of the scalar, radial distance
function $r=r(\varphi,\theta)$ measured from the center of mass $o$, have
already proven their utility in geomorphology. Second order descriptors carry
deeper information on the _relative position_ of these points by using
natural, discrete decompositions of the radial distance function
$r=r(\varphi,\theta)$. The decomposition of its range by saddle points leads
to the concept of Reeb/̄graphs while the decompositon of its domain by
isolated integral curves of the gradient leads to the concept of Morse=̄Smale
graphs. While both second-order descriptors have been discussed before in the
mathematical literature, still, their mutual relationship remained unclear.

Since 3D measurement technology is becoming increasingly accessible and
standard, obtaining 3D datasets is not a problem any more. However, 3D data
itself, if not coupled with geometric ideas, does not solve the basic question
of _how to describe shapes_. Encouraged by these technological developments,
in this paper we presented the geometric background of the mentioned second-
order descriptors, provided an algorithm to reliably extract them from scanned
3D point clouds. By introducing a _third-order descriptor_ , called the master
graph, we established the relationship between Reeb/̄graphs and Morse=̄Smale
graphs.

To illustrate the feasibility of the application, we created a catalog of 271
scanned pebbles where we performed these measurements and we also provided the
source code for the implementation of our algorithm as well as the 3D datasets
of the aforementioned pebbles.

We hope that our paper gives a signal to geomorphologists that new,
geometrically inspired tools for fully three/̄dimensional shape analysis are
now available and ready to deploy.

##  Appendix A Visualization of the algorithm’s output

Figure 20: Saddle contours (blue), saddle=̄unstable ascending curves (red) and
stable=̄saddle ascending curves (green) of the polyhedron $P_{\text{ex}}$

Figure 21: Saddle contours (blue), saddle=̄unstable ascending curves (red) and
stable=̄saddle ascending curves (green) after cancellations up to the primary
class measured by hand

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