# Fractal Calculus to Derive Fractal Frenet Equations for Fractal Curves

Alireza Khalili Golmankhaneh1, Palle E. T. Jørgensen 2, Dimiter Prodanov 3
1 Department of Physics, Urmia Branch,
Islamic Azad University, Urmia 63896,West Azerbaijan, Iran
<EMAIL_ADDRESS>
2 Department of Mathematics,The University of Iowa
Iowa City, IA 52242-1419, USA
<EMAIL_ADDRESS>
3 ITSDP, IICT, Bulgarian Academy of Sciences, Sofia, Bulgaria.
EHS and NERF, Interuniversity Microelectronics Center (Imec), 3001 Leuven,
Belgium
<EMAIL_ADDRESS>

###### Abstract

This paper introduces the concept of Fractal Frenet equations, a set of
differential equations used to describe the behavior of vectors along fractal
curves. The study explores the analogue of arc length for fractal curves,
providing a measure to quantify their length. It also discusses fundamental
mathematical constructs, such as the analogue of the unit tangent vector,
which indicates the curve’s direction at different points, and the analogue of
curvature vector or fractal curvature vector, which characterizes its
curvature at various locations. The concept of torsion, describing the
twisting and turning of fractal curves in three-dimensional space, is also
explored. Specific examples, like the fractal helix and the fractal snowflake,
illustrate the application and significance of the Fractal Frenet equations.

††footnotetext:  MSC2020:28A80,54F50,53A04††Corresponding author Alireza
Khalili Golmankhaneh

##  1 Introduction

Fractal geometry has emerged as a powerful mathematical framework for
understanding and describing complex natural patterns that exhibit self-
similarity across different scales and dimensions [1, 2, 3, 4, 5, 6]. These
mesmerizing geometric shapes can be found in various natural occurrences, such
as snowflakes, Romanesco broccoli, mountain tops, clouds, tree branches,
lightning, river deltas, state borders, buildings, and crystals [7]. Their
captivating property of endless repetition and similarity at different levels
of magnification has led to widespread fascination and application in diverse
scientific disciplines.

Fractals often possess fractional dimensions, exceeding their topological
dimensions, making them indefinitely complicated and challenging to analyze
using traditional Euclidean geometry [8, 9]. Their non-differentiability and
lack of integrability have pushed the study of fractal curves towards measures
like the Hausdorff measure [10]. As a result, the exploration of fractal
curves and their behaviors requires the development of specialized
mathematical tools beyond standard calculus [11].

The significance of fractals extends beyond mathematics and reaches into
various scientific domains, including biology, chemistry, earth sciences,
physics, and technology [12]. In physics, fractal geometry has been employed
to model physical phenomena in non-integer spaces [13, 14]. Furthermore,
fractal patterns have been experimentally detected, from quantum observations
to relativistic light aberration effects [15].

Through the application of mathematical methodologies such as fractal
geometry, probability theory, and analysis, significant strides have been made
in understanding complex processes [16, 17]. In particular, the study of heat
transfer in fractal materials involves the use of continuous models with non-
integer dimensional spaces [18, 19, 20, 21, 22, 23, 24]. Measure theory plays
a pivotal role in defining derivation and integration on fractal sets,
providing a systematic approach to measure and analyze these structures, and
facilitating the development of integral concepts tailored to fractals [25,
26, 27, 28, 29, 30, 31].

To address the challenges of dealing with fractal curves and to harness their
potential applications, researchers have formulated fractal calculus as a
generalization of ordinary calculus, designed to incorporate fractals [32, 33,
34]. This extension of calculus has paved the way for investigating stochastic
processes, differential equations, and anomalous diffusion on fractal
structures [35, 36, 37, 38, 39, 40, 41, 42, 43].

Focusing on nonstandard Lagrangians as their origin, the study delved into
Finsler-Randers manifolds and fractal electrodynamics, revealing their
quadratic damping geodesic characteristics [44, 45].

It is demonstrated that there exist Hilbert spaces, originating from the
Hausdorff measures, that permit the existence of multiresolution wavelets
[46].

Moreover, fractal Fourier and Laplace transforms of local and non-local
fractal derivatives have been introduced and utilized to solve fractal
differential equations [47, 48, 34]. These mathematical developments have
opened up new avenues for understanding the behavior of fractal curves and
analyzing their intricate properties. Fractal calculus is applied to fractal
interpolation functions and Weierstrass functions, both of which have the
potential to exhibit non-differentiable and non-integrable behaviors within
the realm of ordinary calculus [49].

In this paper, we provide a comprehensive overview of fractal geometry,
calculus, and their applications in various scientific disciplines. We review
the concept of $F^{\alpha}$-Calculus on fractal curves, explore the analogue
of arc length, unit tangent vector, and fractal curvature vector, and discuss
the concept of torsion. Additionally, we introduce the Fractal Frenet
equations, a set of differential equations that describe the behavior of
specific vectors along a fractal curve. By delving into these mathematical
constructs, we aim to shed light on the captivating and enigmatic world of
fractal geometry and its relevance in the scientific realm.

##  2 Review of $F^{\alpha}$-Calculus on Fractal Curves

The adaptation of calculus principles to fractal curves through
$F^{\alpha}$-calculus provides a powerful mathematical framework for
understanding and analyzing functions on these intricate mathematical
constructs. This section highlights the significance of $F^{\alpha}$-calculus
in dealing with the complexities of fractal geometry and lays the groundwork
for further exploration and application in various scientific and engineering
domains. In the following, we want to explain how to parameterize a self-
similar fractal curve in two dimensions [33].

Let $\mathfrak{F}$ be a self-similar fractal curve, and let
$Q_{i},i=0,...,n-1$ be linear transformations composed of rotations and
scalings. Each $Q_{i}$ can be expressed as:

$Q_{i}=s_{i}\begin{pmatrix}\cos\theta_{i}&-\sin\theta_{i}\\\
\sin\theta_{i}&\cos\theta_{i}\end{pmatrix},$ (1)

subject to the condition:

$\sum_{i=0}^{n-1}Q_{i}(\mathbf{r})=\mathbf{r},$ (2)

where $\mathbf{r}$ is a vector and $0<s_{i}<1$ for $i=0,...,n-1$. Then, the
fractal curve $\mathfrak{F}$ is defined by the iterative equation:

$S_{j}(\mathbf{r})=\sum_{i=0}^{j-1}Q_{i}(\mathbf{r}_{0})+Q_{j}(\mathbf{r}),\quad
j=0,...,n-1,$ (3)

where $\mathbf{r}_{0}$ is a fixed vector. The limit set of this iterative
equation will form a curve since the way $S_{j}$ are constructed from $Q_{i}$.
To parameterize the fractal curve $\mathfrak{F}$, consider $\lfloor mt\rfloor$
as the integer part of $mt$, and define:

$\mathbf{u}(t)=\mathbf{u}_{x}(t)\hat{i}+\mathbf{u}_{y}(t)\hat{j}=\sum_{i=0}^{\lfloor
mt\rfloor-1}Q_{i}(\mathbf{r}_{0})+Q_{\lfloor mt\rfloor}(\mathbf{u}(mt-\lfloor
mt\rfloor)),\quad 0\leq t\leq 1,$ (4)

which provides a parametric representation of $\mathfrak{F}$. The
parameterization and the measurement concepts described here allow us to
analyze and quantify properties of the self-similar fractal curve
$\mathfrak{F}$ in two dimensions. Next, we introduce some definitions related
to the fractal curve $\mathfrak{F}$ and its calculus:

######  Definition 1.

For a set $\mathfrak{F}$ and a subdivision $P_{[a,b]}$, where $a<b$ within the
interval $[a_{0},b_{0}]$, we define:

$\sigma^{\alpha}[\mathfrak{F},P]=\sum_{i=0}^{n-1}\frac{|\mathbf{u}(t_{i+1})-\mathbf{u}(t_{i})|^{\alpha}}{\Gamma(\alpha+1)},$

where $|.|$ denotes the Euclidean norm on $\mathbb{R}^{n}$ and
$P_{[a,b]}=\\{a=t_{0},...,t_{n}=b\\}$.

######  Definition 2.

For a given $\delta>0$ and $a_{0}\leq a\leq b\leq b_{0}$, the coarse-grained
mass is defined as:

$\gamma_{\delta}^{\alpha}(\mathfrak{F},a,b)=\inf_{\\{P_{[a,b]}:|P|\leq\delta\\}}\sigma^{\alpha}[\mathfrak{F},P],$

where $|P|$ represents the maximum length of the subintervals in the
subdivision $P$.

######  Definition 3.

For $a_{0}\leq a<b<b_{0}$, the mass function
$\gamma^{\alpha}(\mathfrak{F},a,b)$ is defined as the limit of the coarse-
grained mass $\gamma_{\delta}^{\alpha}(\mathfrak{F},a,b)$ as $\delta$
approaches zero. Mathematically, this can be expressed as:

$\gamma^{\alpha}(\mathfrak{F},a,b)=\lim_{\delta\rightarrow
0}\gamma_{\delta}^{\alpha}(\mathfrak{F},a,b).$ (5)

The mass function $\gamma^{\alpha}(\mathfrak{F},a,b)$ quantifies the fractal
properties of the set $\mathfrak{F}$ within the interval $[a,b]$, and it
serves as a crucial measure for characterizing self-similar fractal curves. As
the value of $\delta$ becomes smaller, the coarse-grained mass
$\gamma_{\delta}^{\alpha}(\mathfrak{F},a,b)$ provides a finer approximation to
the actual mass $\gamma^{\alpha}(\mathfrak{F},a,b)$. Taking the limit as
$\delta$ tends to zero allows us to obtain the precise mass value for the
fractal set within the given interval.

######  Definition 4.

The rise function of a fractal curve $\mathfrak{F}$, denoted as
$S_{\mathfrak{F}}^{\alpha}(v)$, is defined as follows:

$S_{\mathfrak{F}}^{\alpha}(v)=\left\\{\begin{array}[]{ll}\mathfrak{M}^{\alpha}(\mathfrak{F},p_{0},v),&v\geq
p_{0};\\\
-\mathfrak{M}^{\alpha}(\mathfrak{F},v,p_{0}),&v<p_{0},\end{array}\right.$ (6)

Here, $v\in[a_{1},a_{2}]$, and $p_{0}=a_{1}$ is an arbitrary and fixed number.
The function $S_{\mathfrak{F}}^{\alpha}(v)$ represents the mass of the fractal
curve $\mathfrak{F}$ up to the point $v$.
The rise function is useful for understanding how the mass of the fractal
curve accumulates as we traverse along the curve. It provides a measure of the
distribution of mass at different points of the curve and helps in
characterizing the scaling behavior of $\mathfrak{F}$ with respect to the mass
function $\mathfrak{M}^{\alpha}(\mathfrak{F},a,b)$. The rise function is an
important tool in the study of self-similar fractals and their properties.

######  Definition 5.

The staircase function, denoted by $J(\theta)$, is defined as follows:

$J(\theta)=S_{\mathfrak{F}}^{\alpha}(\textbf{u}^{-1}(\theta)),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \theta\in\mathfrak{F}.$ (7)

Here, $\textbf{u}^{-1}(\theta)$ represents the inverse of the mapping function
$\textbf{u}(t)$ for $\theta$ belonging to the fractal set $\mathfrak{F}$. The
staircase function $J(\theta)$ provides a way to associate points from the
fractal curve $\mathfrak{F}$ with their corresponding mass values represented
by the staircase function $S_{\mathfrak{F}}^{\alpha}(t)$.

Since $S_{\mathfrak{F}}^{\alpha}(t)$ is a strictly increasing function, it is
invertible, ensuring that the mapping $\textbf{u}^{-1}(\theta)$ is well-
defined and one-to-one. This means that each point on the fractal curve
$\mathfrak{F}$ is uniquely linked to its corresponding mass value through the
staircase function.

The staircase function is a valuable tool in analyzing the relationship
between the geometrical properties of the fractal curve and its mass
distribution. It allows us to explore the distribution of mass along the
fractal curve and study how different parts of the curve contribute to the
overall mass of the fractal.

######  Definition 6.

The $\gamma$-dimension of the fractal curve $\mathfrak{F}$, denoted by
$\textmd{dim}_{\gamma}(\mathfrak{F})$, can be defined as follows:

$\displaystyle\textmd{dim}_{\gamma}(\mathfrak{F})$
$\displaystyle=\inf\\{\alpha:\gamma^{\alpha}(\mathfrak{F},a,b)=0\\}$
$\displaystyle=\sup\\{\alpha:\gamma^{\alpha}(\mathfrak{F},a,b)=\infty\\}.$ (8)

The $\gamma$-dimension provides a measure of the scaling behavior of the
fractal set $\mathfrak{F}$ with respect to the mass function
$\gamma^{\alpha}(\mathfrak{F},a,b)$. It helps in quantifying the self-
similarity and the fractal nature of $\mathfrak{F}$ within a specific interval
$[a,b]$ in $\mathbb{R}^{n}$.

######  Definition 7.

Let $\mathfrak{F}\subset\mathbb{R}^{n}$ be a fractal curve, and let
$f:\mathfrak{F}\rightarrow\mathbb{R}$ be a real-valued function. Given a point
$\theta\in\mathfrak{F}$, a number $l$ is considered to be the limit of $f$
through points of $\mathfrak{F}$ if, for any given positive value $\epsilon$,
there exists a corresponding $\delta>0$ such that:

$\theta^{\prime}\in\mathfrak{F},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
|\theta^{\prime}-\theta|<\delta\quad\Rightarrow\quad|f(\theta^{\prime})-l|<\epsilon.$
(9)

If such a number $l$ exists and satisfies the above condition, it is denoted
as the fractal limit of $f(\theta)$ and is represented as:

$l=\mathfrak{F}_{-}\lim_{\theta^{\prime}-\theta}f(\theta)$ (10)

This definition implies that the function $f(\theta)$ has a well-defined limit
at the point $\theta$ on the fractal curve $\mathfrak{F}$ if, as points
$\theta^{\prime}$ on $\mathfrak{F}$ get arbitrarily close to $\theta$, the
corresponding values of $f(\theta^{\prime})$ get arbitrarily close to $l$. In
essence, it extends the concept of a limit from ordinary calculus to functions
defined on fractal curves.

######  Definition 8.

A function $f:\mathfrak{F}\rightarrow\mathbb{R}$ is considered to be
$\mathfrak{F}$-continuous at a point $\theta\in\mathfrak{F}$ if the following
condition holds:

$f(\theta)=\mathfrak{F}_{-}\lim_{\theta^{\prime}\rightarrow\theta}f(\theta^{\prime}).$
(11)

In other words, $f(\theta)$ is $\mathfrak{F}$-continuous at $\theta$ if its
value at $\theta$ is equal to the fractal limit of $f(\theta^{\prime})$ as
$\theta^{\prime}$ approaches $\theta$ within the fractal curve $\mathfrak{F}$.
This definition characterizes a specific type of continuity suitable for
functions defined on fractal curves.

######  Definition 9.

For a function $f:\mathfrak{F}\rightarrow\mathbb{R}$ and
$t_{1},t_{2}\in[a_{0},b_{0}]$ with $t_{1}\leq t_{2}$, we define the upper and
lower limits of $f$ over the interval $C(t_{1},t_{2})$ as follows:

$M[f,C(t_{1},t_{2})]=\sup_{\theta\in C(t_{1},t_{2})}f(\theta),$ (12)
$m[f,C(t_{1},t_{2})]=\inf_{\theta\in C(t_{1},t_{2})}f(\theta).$ (13)

######  Definition 10.

Let $S_{\mathfrak{F}}^{\alpha}(t)$ be finite for
$t\in[a,b]\subset[a_{0},b_{0}]$. Consider a subdivision $P$ of $[a,b]$ with
points ${t_{0},...,t_{n}}$. The upper and lower $\mathfrak{F}^{\alpha}$-sums
for the function $f$ over the subdivision $P$ are given respectively by:

$U^{\alpha}[f,\mathfrak{F},P]=\sum_{i=0}^{n-1}M[f,C(t_{i},t_{i+1})]\left[S_{\mathfrak{F}}^{\alpha}(t_{i+1})-S_{\mathfrak{F}}^{\alpha}(t_{i})\right],$
(14)
$L^{\alpha}[f,\mathfrak{F},P]=\sum_{i=0}^{n-1}m[f,C(t_{i},t_{i+1})]\left[S_{\mathfrak{F}}^{\alpha}(t_{i+1})-S_{\mathfrak{F}}^{\alpha}(t_{i})\right].$
(15)

It is evident from Equations (14) and (15) that

$U^{\alpha}[f,\mathfrak{F},P]\geq L^{\alpha}[f,\mathfrak{F},P].$ (16)

######  Definition 11.

Let $\mathfrak{F}$ be such that $S_{\mathfrak{F}}^{\alpha}$ is finite on
$[a,b]$. For a function $f\in B(\mathfrak{F})$, the lower and upper
$\mathfrak{F}^{\alpha}$-integrals of $f$ on the section $C(a,b)$ are defined
as follows:

$\underline{\int_{C(a,b)}}f(\theta)d_{\mathfrak{F}}^{\alpha}\theta=\sup_{P_{[a,b]}}L^{\alpha}[f,\mathfrak{F},P],$
(17)
$\overline{\int_{C(a,b)}}f(\theta)d_{\mathfrak{F}}^{\alpha}\theta=\inf_{P_{[a,b]}}U^{\alpha}[f,\mathfrak{F},P].$
(18)

These integrals represent the supremum and infimum, respectively, of the lower
and upper $\mathfrak{F}^{\alpha}$-sums of $f$ over all possible subdivisions
of $[a,b]$. The lower integral captures the largest sum of function values
over the subdivisions, while the upper integral captures the smallest sum of
function values over the subdivisions.

######  Definition 12.

If $f\in B(\mathfrak{F})$, we say that $f$ is
$\mathfrak{F}^{\alpha}$-integrable on $C(a,b)$ if the lower and upper
$\mathfrak{F}^{\alpha}$-integrals of $f$ on $C(a,b)$ are equal, i.e.,

$\underline{\int_{C(a,b)}}f(\theta)d_{\mathfrak{F}}^{\alpha}\theta=\overline{\int_{C(a,b)}}f(\theta)d_{\mathfrak{F}}^{\alpha}\theta,$
(19)

and the common value is referred to as the $\mathfrak{F}^{\alpha}$-integral:

$\int_{C(a,b)}f(\theta)d_{\mathfrak{F}}^{\alpha}\theta.$ (20)

This integral represents the sum of function values along the fractal curve
$\mathfrak{F}$ on the section $C(a,b)$ with respect to the
$\mathfrak{F}^{\alpha}$-measure. If this common value exists, then $f$ is
considered $\mathfrak{F}^{\alpha}$-integrable on $C(a,b)$.

######  Definition 13.

Consider a fractal curve $\mathfrak{F}$. The
$\mathfrak{F}^{\alpha}$-derivative of a function $f$ at a point
$\theta\in\mathfrak{F}$ is defined as follows:

$\frac{d^{\alpha}_{\mathfrak{F}}f}{d^{\alpha}_{\mathfrak{F}}\theta}=D_{\mathfrak{F},\theta}^{\alpha}f(\theta)=\mathfrak{F}-\lim_{\theta^{\prime}\rightarrow\theta}\frac{f(\theta^{\prime})-f(\theta)}{J(\theta^{\prime})-J(\theta)},$
(21)

provided the limit exists. Here, $J(\theta)$ represents the staircase function
associated with the fractal curve $\mathfrak{F}$ as defined earlier. This
derivative extends the concept of ordinary differentiation to functions
defined on fractal curves, taking into account the non-smooth and self-
replicating nature of these curves.

######  Theorem 1.

Suppose $f\in B(\mathfrak{F})$ is an $\mathfrak{F}$-continuous function on the
section $C(a,b)$, and let $g:\mathfrak{F}\rightarrow\mathbb{R}$ be defined as
follows:

$g(\textbf{u}(t))=\int_{C(a,t)}f(\theta)d_{\mathfrak{F}}^{\alpha}\theta,$ (22)

for all $t\in[a,b]$, where $\textbf{u}(t)$ is the parameterization of the
fractal curve $\mathfrak{F}$ given earlier. Then, it follows that:

$D_{\mathfrak{F}}^{\alpha}g(\theta)=f(\theta).$ (23)

In other words, the $\mathfrak{F}^{\alpha}$-derivative of the function $g$ is
equal to the original function $f$ on the fractal curve $\mathfrak{F}$. This
result highlights the connection between the fractal integral and derivative,
indicating that integration and differentiation can be understood in the
context of fractal curves.

######  Theorem 2.

Suppose $f:\mathfrak{F}\rightarrow\mathbb{R}$ is an
$\mathfrak{F}^{\alpha}$-differentiable function, and
$h:\mathfrak{F}\rightarrow\mathbb{R}$ is $\mathfrak{F}$-continuous such that
$h(\theta)=D_{\mathfrak{F}}^{\alpha}f(\theta)$. Then, we have:

$\int_{C(a,b)}h(\theta)d_{\mathfrak{F}}^{\alpha}\theta=f(\textbf{u}(b))-f(\textbf{u}(a)).$
(24)

This result shows that the integral of the product of an
$\mathfrak{F}$-continuous function $h$ and an
$\mathfrak{F}^{\alpha}$-differentiable function $f$ over the section
$C(t,t^{\prime})$ of the fractal curve $\mathfrak{F}$ is equal to the
difference in the values of the function $f$ at the endpoints of the section.
The theorem establishes a relationship between the fractal integral and the
$\mathfrak{F}^{\alpha}$-derivative and provides a way to compute the integral
using the derivative of the function.

##  3 Analogue of Arc Length of Fractal Curve

This section delves into the definition of the analogue of arc length for
fractal curves, as well as the concepts of unit tangent vector, fractal
curvature, and torsion. These essential mathematical constructs offer unique
insights into the intricate properties of fractal curves, allowing us to
navigate the dimensional complexities inherent in these mesmerizing
geometrical shapes. Consider a fractal curve denoted by
$\textbf{u}=\textbf{u}(t)$, where $t$ lies within the interval
$I=[a_{0},b_{0}]$. In the context of fractal geometry, we define the analogue
of arc length, denoted by $s_{\mathfrak{F}}$, for this curve as follows:

$s_{\mathfrak{F}}=s_{\mathfrak{F}}(t)=\int_{C(t_{0},t)}\left|\frac{d^{\alpha}_{\mathfrak{F}}\textbf{u}(t)}{d^{\alpha}_{\mathfrak{F}}t}\right|d^{\alpha}_{\mathfrak{F}}t$
(25)

Moreover, we define the derivative of the fractal arc length, denoted by
$d^{\alpha}_{\mathfrak{F}}s_{\mathfrak{F}}/d^{\alpha}_{\mathfrak{F}}t$, as
[50]:

$\frac{d^{\alpha}_{\mathfrak{F}}s_{\mathfrak{F}}}{d^{\alpha}_{\mathfrak{F}}t}=\left|\frac{d^{\alpha}_{\mathfrak{F}}\textbf{u}(t)}{d^{\alpha}_{\mathfrak{F}}t}\right|$
(26)

Here, $\alpha$ denotes the fractal order and $t>t_{0}$. It’s important to note
that the fractal curve $\textbf{u}(t)$ takes on the value $\theta$.

######  Example 1.

Let’s consider a fractal helix described by the following equation:

$\textbf{u}(t)=a\cos(S_{F}^{\alpha}(t))\hat{e}_{1}+a\sin(S_{F}^{\alpha}(t))\hat{e}_{2}+bS_{F}^{\alpha}(t)\hat{e}_{3}.$
(27)

The analogue of arc length for this fractal helix is given by:

$s_{\mathfrak{F}}=\int_{C(t_{0},t)}(a^{2}+b^{2})^{1/2}d^{\alpha}_{\mathfrak{F}}t=(a^{2}+b^{2})^{1/2}S_{F}^{\alpha}(t)\equiv(a^{2}+b^{2})^{1/2}J(\theta)$
(28)

where $S_{F}^{\alpha}(0)=0$.

Figure 1: Graph displaying the fascinating fractal helix Figure 2: Graph
illustrating the analogue of arc length

Figure 1, intricate illustration reveals the captivating beauty of a coil
pattern that replicates itself at different scales, exhibiting intricate
geometric intricacies. Embodying the mesmerizing realm of fractal geometry,
the helix showcases its distinctive attributes through an infinite sequence of
self-replicating coils, resulting in an enchanting array of mesmerizing shapes
and forms.
Figure 2, visual representation depicts the concept of measuring distances
along a curve, capturing the relationship between the length of the curve and
the variable defining its position. By plotting the analogue of arc length, we
gain valuable insights into how the curve’s length changes concerning the
parameter or variable that governs its shape. This graphical portrayal offers
a clear and intuitive understanding of the concept, enabling us to explore the
intricate behavior of curves in relation to their respective arc lengths.

######  Remark 1.

We assume the following notations for convenience:

$\dot{\textbf{u}}(\theta)=\frac{d^{\alpha}_{\mathfrak{F}}\textbf{u}(\theta)}{d^{\alpha}_{\mathfrak{F}}\theta},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\ddot{\textbf{u}}(\theta)=\frac{d^{2\alpha}_{\mathfrak{F}}\textbf{u}(t)}{d^{\alpha}_{\mathfrak{F}}\theta^{2}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\textbf{u}^{\prime}(t)=\frac{d^{\alpha}_{\mathfrak{F}}\textbf{u}(t)}{d^{\alpha}_{\mathfrak{F}}t},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\textbf{u}^{\prime\prime}(t)=\frac{d^{2\alpha}_{\mathfrak{F}}\textbf{u}(t)}{d^{\alpha}_{\mathfrak{F}}t^{2}}.$
(29)

######  Definition 14.

The concept denoting the "analogue of the unit tangent vector" for a fractal
curve is as follows: If $\dot{\textbf{u}}(\theta)$ represents the derivative
of the fractal curve concerning $\theta$, then
$\textbf{t}=\textbf{t}(\theta)=\dot{\textbf{u}}$ is recognized as the
equivalent of the unit tangent vector pertaining to the fractal curve.

Alternatively, the tangent vector linked with the fractal curve, labeled as
$\textbf{t}_{\mathfrak{F}}$, is articulated as:

$\textbf{t}_{\mathfrak{F}}=\textbf{t}_{\mathfrak{F}}(\theta)=\frac{d^{\alpha}_{\mathfrak{F}}\textbf{u}(\theta)}{d^{\alpha}_{\mathfrak{F}}\theta}.$
(30)

This tangent vector conforms to the class $C^{\alpha}$, indicating its
qualities of continuity and differentiability.

######  Example 2.

Consider the fractal helix given by the equation:

$\textbf{u}(t)=a\cos(S_{\mathfrak{F}}^{\alpha}(t))\hat{e}_{1}+b\sin(S_{\mathfrak{F}}^{\alpha}(t))\hat{e}_{2}+bS_{\mathfrak{F}}^{\alpha}(t)\hat{e}_{3},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ a\neq 0,\leavevmode\nobreak\ b\neq
0.$ (31)

Then, the derivative of this fractal curve with respect to $t$ is given by:

$\frac{d^{\alpha}_{\mathfrak{F}}\textbf{u}(t)}{d^{\alpha}_{\mathfrak{F}}t}=\frac{1}{\Gamma(\alpha+1)}(-a\sin(S_{\mathfrak{F}}^{\alpha}(t))\hat{e}_{1}+b\cos(S_{\mathfrak{F}}^{\alpha}(t))\hat{e}_{2}+b\chi_{\mathfrak{F}}\hat{e}_{3}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ .$ (32)

where $\chi_{\mathfrak{F}}$ represents characteristic function. Consequently,
the magnitude of the derivative is given by:

$\left|\frac{d^{\alpha}_{\mathfrak{F}}\textbf{u}(t)}{d^{\alpha}_{\mathfrak{F}}t}\right|=\frac{1}{\Gamma(\alpha+1)}(a^{2}+b^{2})^{1/2}.$
(33)

Hence, the analogue of the unit tangent vector $\textbf{t}_{\mathfrak{F}}$ for
the fractal curve can be expressed as:

$\textbf{t}_{\mathfrak{F}}=\frac{d^{\alpha}_{\mathfrak{F}}\textbf{u}(\theta)}{d^{\alpha}_{\mathfrak{F}}\theta}=\frac{\frac{d^{\alpha}_{\mathfrak{F}}\textbf{u}(t)}{d^{\alpha}_{\mathfrak{F}}t}}{\left|\frac{d^{\alpha}_{\mathfrak{F}}\textbf{u}(t)}{d^{\alpha}_{\mathfrak{F}}t}\right|}=\frac{1}{(a^{2}+b^{2})^{1/2}}(-a\sin(S_{\mathfrak{F}}^{\alpha}(t))\hat{e}_{1}+b\cos(S_{\mathfrak{F}}^{\alpha}(t))\hat{e}_{2}+b\chi_{\mathfrak{F}}\hat{e}_{3}).$
(34)

######  Definition 15.

The fractal vector, denoted as $\textbf{k}_{\mathfrak{F}}$, or alternatively,
the analogue of curvature vector/fractal curvature vector on $\mathfrak{F}$,
is defined as follows:

$\textbf{k}_{\mathfrak{F}}=\dot{\textbf{t}}_{\mathfrak{F}}=\frac{d^{\alpha}_{\mathfrak{F}}\textbf{t}_{\mathfrak{F}}(\theta)}{d^{\alpha}_{\mathfrak{F}}\theta}=\frac{d^{\alpha}_{\mathfrak{F}}\textbf{t}_{\mathfrak{F}}}{d^{\alpha}_{\mathfrak{F}}t}\frac{d^{\alpha}_{\mathfrak{F}}t}{d^{\alpha}_{\mathfrak{F}}\theta}=\frac{d^{\alpha}_{\mathfrak{F}}\textbf{t}_{\mathfrak{F}}}{d^{\alpha}_{\mathfrak{F}}t}\bigg{/}\frac{d^{\alpha}_{\mathfrak{F}}\theta}{d^{\alpha}_{\mathfrak{F}}t}=\frac{d^{\alpha}_{\mathfrak{F}}\textbf{t}_{\mathfrak{F}}}{d^{\alpha}_{\mathfrak{F}}t}\bigg{/}\bigg{|}\frac{d^{\alpha}_{\mathfrak{F}}\textbf{u}}{d^{\alpha}_{\mathfrak{F}}t}\bigg{|}$
(35)

This vector characterizes the analogue of curvature or fractal curvature on
the fractal curve $\mathfrak{F}$.

######  Definition 16.

The magnitude of the fractal curvature is denoted by
$\kappa_{\mathfrak{F}}=|\textbf{k}_{\mathfrak{F}}(\theta)|$ and is referred to
as the radius of fractal curvature at $\theta$. Furthermore, its reciprocal,
represented by $\rho_{\mathfrak{F}}=\frac{1}{\kappa_{\mathfrak{F}}}$, is also
called the radius of fractal curvature at $\theta$.

######  Example 3.

Considering the fractal snowflake with the following equation:

$\textbf{u}(t)=(a\cos(S_{\mathfrak{F}}^{\alpha}(t))\hat{e}_{1}+a\sin(S_{\mathfrak{F}}^{\alpha}(t))\hat{e}_{2}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ a>0,$ (36)

we find that its fractal derivative with respect to $t$ is given by:

$\frac{d^{\alpha}_{\mathfrak{F}}\textbf{u}(t)}{d^{\alpha}_{\mathfrak{F}}t}=\frac{1}{\Gamma(\alpha+1)}(-a\sin(S_{\mathfrak{F}}^{\alpha}(t))\hat{e}_{1}+a\cos(S_{\mathfrak{F}}^{\alpha}(t))\hat{e}_{2}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $ (37)

with a magnitude of:

$\bigg{|}\frac{d^{\alpha}_{\mathfrak{F}}\textbf{u}(t)}{d^{\alpha}_{\mathfrak{F}}t}\bigg{|}=\frac{a}{\Gamma(\alpha+1)}.$
(38)

Hence, the analogue of the unit tangent vector, $\textbf{t}{\mathfrak{F}}$, is
given by:

$\textbf{t}_{\mathfrak{F}}=(-\sin(S_{\mathfrak{F}}^{\alpha}(t))\hat{e}_{1}+\cos(S_{\mathfrak{F}}^{\alpha}(t))\hat{e}_{2})$
(39)

and the fractal curvature vector, $\textbf{k}_{\mathfrak{F}}$, can be
expressed as:

$\textbf{k}_{\mathfrak{F}}=-\frac{1}{a}(\cos(S_{\mathfrak{F}}^{\alpha}(t))\hat{e}_{1}+\sin(S_{\mathfrak{F}}^{\alpha}(t))\hat{e}_{2})$
(40)

######  Example 4.

Let us consider the fractal curve described by:

$\textbf{u}(t)=S_{\mathfrak{F}}^{\alpha}(t)\hat{e}_{1}+\frac{1}{3}S_{\mathfrak{F}}^{\alpha}(t)^{3}\hat{e}_{2}$
(41)

Upon calculating the fractal derivative with respect to $t$, we obtain:

$\frac{d^{\alpha}_{\mathfrak{F}}\textbf{u}(t)}{d^{\alpha}_{\mathfrak{F}}t}=\frac{1}{\Gamma(\alpha+1)}(e_{1}+S_{\mathfrak{F}}^{\alpha}(t)^{2}\hat{e}_{2}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $ (42)

where the magnitude of this derivative is given by:

$\bigg{|}\frac{d^{\alpha}_{\mathfrak{F}}\textbf{u}(t)}{d^{\alpha}_{\mathfrak{F}}t}\bigg{|}=\frac{1}{\Gamma(\alpha+1)}(1+S_{\mathfrak{F}}^{\alpha}(t)^{4})^{1/2}.$
(43)

As a result, the analogue of the unit tangent vector,
$\textbf{t}_{\mathfrak{F}}$, is expressed as:

$\textbf{t}_{\mathfrak{F}}=\frac{1}{(1+S_{\mathfrak{F}}^{\alpha}(t)^{4})^{1/2}}(\hat{e}_{1}+S_{\mathfrak{F}}^{\alpha}(t)^{2}\hat{e}_{2})$
(44)

and the fractal curvature vector, $\textbf{k}_{\mathfrak{F}}$, is given by:

$\textbf{k}_{\mathfrak{F}}=-2S_{\mathfrak{F}}^{\alpha}(t)(1+S_{\mathfrak{F}}^{\alpha}(t)^{4})^{-2}(S_{\mathfrak{F}}^{\alpha}(t)^{2}\hat{e}_{1}-\hat{e}_{2})$
(45)

######  Definition 17.

We define the unit vector in the direction of $\textbf{k}_{\mathfrak{F}}$ as
follows:

$\textbf{n}_{\mathfrak{F}}=\frac{\textbf{k}_{\mathfrak{F}}(\theta)}{|\textbf{k}_{\mathfrak{F}}(\theta)|}$
(46)

This unit vector points in the same direction as $\textbf{k}_{\mathfrak{F}}$
and has a magnitude of 1, ensuring its normalization.

######  Definition 18.

The unit fractal binormal vector is defined as:

$\textbf{b}_{\mathfrak{F}}(\theta)=\textbf{t}_{\mathfrak{F}}(\theta)\times\textbf{n}_{\mathfrak{F}}(\theta)$
(47)

This vector is obtained by taking the cross product of the unit tangent vector
$\textbf{t}_{\mathfrak{F}}$ and the unit vector
$\textbf{n}_{\mathfrak{F}}(\theta)$ that points in the direction of the
fractal curvature $\textbf{k}_{\mathfrak{F}}(\theta)$. The resulting vector,
$\textbf{b}_{\mathfrak{F}}(\theta)$, is orthogonal to both
$\textbf{t}_{\mathfrak{F}}(\theta)$ and $\textbf{n}_{\mathfrak{F}}(\theta)$
and represents the binormal direction of the fractal curve at $\theta$.

######  Example 5.

Consider the fractal curve described by:

$\textbf{u}(t)=a\cos(S_{\mathfrak{F}}^{\alpha}(t))\hat{e}_{1}+b\sin(S_{\mathfrak{F}}^{\alpha}(t))\hat{e}_{2}+bS_{\mathfrak{F}}^{\alpha}(t)\hat{e}_{3},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ a\neq 0,\leavevmode\nobreak\ b\neq
0.$ (48)

Then, the unit tangent vector $\textbf{t}_{\mathfrak{F}}$, and the fractal
curvature vector $\textbf{k}_{\mathfrak{F}}$ are given by:

$\textbf{t}_{\mathfrak{F}}=(a^{2}+b^{2})^{-1/2}(-a\sin(S_{\mathfrak{F}}^{\alpha}(t))\hat{e}{1}+a\cos(S_{\mathfrak{F}}^{\alpha}(t))\hat{e}{2}+b\hat{e}{3})$
(49)
$\textbf{k}_{\mathfrak{F}}=-\frac{a}{a^{2}+b^{2}}(\cos(S_{\mathfrak{F}}^{\alpha}(t))\hat{e}{1}+\sin(S_{\mathfrak{F}}^{\alpha}(t))\hat{e}{2})$
(50)

The unit vector $\textbf{n}_{\mathfrak{F}}$ in the direction of
$\textbf{k}_{\mathfrak{F}}$ is given by:

$\textbf{n}_{\mathfrak{F}}=-(\cos(S_{\mathfrak{F}}^{\alpha}(t))\hat{e}{1}+\sin(S_{\mathfrak{F}}^{\alpha}(t))\hat{e}{2})$
(51)

And the unit fractal binormal vector $\textbf{b}_{\mathfrak{F}}$ is obtained
as:

$\textbf{b}_{\mathfrak{F}}=(a^{2}+b^{2})^{-1/2}(b\sin(S_{\mathfrak{F}}^{\alpha}(t))\hat{e}{1}-b\cos(S_{\mathfrak{F}}^{\alpha}(t))\hat{e}{2}+a\hat{e}{3})$
(52)

The equation of the fractal binormal line at $t=t_{0}$ is then given by:

$\displaystyle y=\textbf{u}(t_{0})+k\textbf{b}_{\mathfrak{F}}(t_{0})$
$\displaystyle=(a\cos(S_{\mathfrak{F}}^{\alpha}(t_{0}))+kb(a^{2}+b^{2})^{-1/2}\sin(S_{\mathfrak{F}}^{\alpha}(t_{0}))\hat{e}_{1}+(a\sin(S_{\mathfrak{F}}^{\alpha}(t_{0}))-kb(a^{2}+b^{2})^{-1/2}\cos(S_{\mathfrak{F}}^{\alpha}(t_{0}))\hat{e}_{2}$
$\displaystyle+(bS_{\mathfrak{F}}^{\alpha}(t_{0}))+ak(a^{2}+b^{2})^{-1/2})\hat{e}_{3},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ -\infty<k<\infty.$ (53)

######  Definition 19.

The fractal torsion of the fractal curve at $\theta$, denoted by
$\tau_{\mathfrak{F}}(\theta)$, is defined as follows:

$\tau_{\mathfrak{F}}(\theta)=-\frac{d^{\alpha}_{\mathfrak{F}}\textbf{b}_{\mathfrak{F}}}{d^{\alpha}_{\mathfrak{F}}\theta}\cdot\textbf{n}_{\mathfrak{F}}$
(54)

Here, $\textbf{n}_{\mathfrak{F}}$ represents the unit fractal binormal vector,
and
$d^{\alpha}_{\mathfrak{F}}\textbf{b}_{\mathfrak{F}}/d^{\alpha}_{\mathfrak{F}}\theta$
is the fractal derivative of the unit fractal binormal vector with respect to
$\theta$.

######  Example 6.

Let’s reconsider the fractal helix defined by:

$\textbf{u}(t)=a\cos(S_{\mathfrak{F}}^{\alpha}(t))\hat{e}_{1}+b\sin(S_{\mathfrak{F}}^{\alpha}(t))\hat{e}_{2}+bS_{\mathfrak{F}}^{\alpha}(t)\hat{e}_{3},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ a\neq 0,\leavevmode\nobreak\ b\neq
0.$ (55)

As obtained previously, the unit fractal binormal vector is given by:

$\textbf{b}_{\mathfrak{F}}=(a^{2}+b^{2})^{-1/2}(b\sin(S_{\mathfrak{F}}^{\alpha}(t))\hat{e}_{1}-b\cos(S_{\mathfrak{F}}^{\alpha}(t))\hat{e}_{2}+a\hat{e}_{3})$
(56)

The fractal torsion, denoted by $\tau_{\mathfrak{F}}$, is constant and can be
calculated as follows:

$\dot{\textbf{b}}_{\mathfrak{F}}=\frac{d^{\alpha}_{\mathfrak{F}}\textbf{b}_{\mathfrak{F}}}{d^{\alpha}_{\mathfrak{F}}\theta}=\frac{d^{\alpha}_{\mathfrak{F}}\textbf{b}_{\mathfrak{F}}}{d^{\alpha}_{\mathfrak{F}}t}\bigg{/}\bigg{|}\frac{d^{\alpha}_{\mathfrak{F}}\textbf{u}}{d^{\alpha}_{\mathfrak{F}}t}\bigg{|}=(a^{2}+b^{2})^{-1}(b\cos(S_{\mathfrak{F}}^{\alpha}(t))\hat{e}_{1}+b\sin(S_{\mathfrak{F}}^{\alpha}(t))\hat{e}_{2})$
(57)

$\displaystyle\tau_{\mathfrak{F}}$
$\displaystyle=-\frac{d^{\alpha}_{\mathfrak{F}}\textbf{b}_{\mathfrak{F}}}{d^{\alpha}_{\mathfrak{F}}\theta}\cdot\textbf{n}_{\mathfrak{F}}$
$\displaystyle=-\frac{1}{\Gamma(\alpha+1)(a^{2}+b^{2})}(b\cos(S_{\mathfrak{F}}^{\alpha}(t))\hat{e}_{1}+b\sin(S_{\mathfrak{F}}^{\alpha}(t))\hat{e}_{2})\cdot(-\cos(S_{\mathfrak{F}}^{\alpha}(t))\hat{e}_{1}-\sin(S_{\mathfrak{F}}^{\alpha}(t))\hat{e}_{2})$
$\displaystyle=\frac{b}{\Gamma(\alpha+1)(a^{2}+b^{2})}$ (58)

Depending on the sign of $b$, the fractal helix can be classified as a right-
handed curve if $b>0$ or a left-handed fractal helix curve if $b<0$.

The following theorem holds for a point on the fractal curve $\textbf{u}(t)$
where $\textbf{k}_{\mathfrak{F}}\neq 0$:

######  Theorem 3.

At such a point, the fractal torsion $\tau_{\mathfrak{F}}$ can be expressed
as:

$\tau_{\mathfrak{F}}=\frac{|\textbf{u}^{\prime}\textbf{u}^{\prime\prime}\textbf{u}^{\prime\prime\prime}|}{|\textbf{u}^{\prime}\times\textbf{u}^{\prime\prime\prime}|^{2}}$
(59)

where $\textbf{u}^{\prime}$, $\textbf{u}^{\prime\prime}$, and
$\textbf{u}^{\prime\prime\prime}$ represent the first, second, and third
derivatives of the fractal curve $\textbf{u}(t)$ with respect to $t$.

######  Definition 20.

The fractal tangent vectors along a fractal curve $C$ give rise to a new
fractal curve $\Upsilon$ on the fractal sphere snowflake of radius $1$
centered at the origin, as illustrated in Figure 3. This curve $\Upsilon$ is
referred to as the fractal spherical indicatrix of
$\textbf{t}_{\mathfrak{F}}$. If we consider $\textbf{u}=\textbf{u}(\theta)$ as
a natural parameterization of the fractal curve $C$, then
$\textbf{u}{1}=\textbf{t}_{\mathfrak{F}}(\theta)=\dot{\textbf{u}}(\theta)$
represents the fractal curve $\Upsilon$.

Figure 3: Graph of the 3D fractal snowflake

######  Example 7.

Consider the fractal helix curve described by:

$\textbf{u}(t)=a\cos(S_{\mathfrak{F}}^{\alpha}(t))\hat{e}{1}+b\sin(S{\mathfrak{F}}^{\alpha}(t))\hat{e}{2}+bS{\mathfrak{F}}^{\alpha}(t)\hat{e}_{3},\leavevmode\nobreak\
\leavevmode\nobreak\ a>0,b\neq 0.$ (60)

We find the following properties for the fractal helix curve and its fractal
spherical indicatrix:

$\displaystyle\textbf{t}_{\mathfrak{F}}$
$\displaystyle=(a^{2}+b^{2})^{-1/2}(-a\sin(S_{\mathfrak{F}}^{\alpha}(t))\hat{e}_{1}+a\cos(S_{\mathfrak{F}}^{\alpha}(t))\hat{e}_{2}+b\hat{e}_{3})$
$\displaystyle\textbf{n}_{\mathfrak{F}}$
$\displaystyle=-(\cos(S_{\mathfrak{F}}^{\alpha}(t))\hat{e}_{1}+\sin(S_{\mathfrak{F}}^{\alpha}(t))\hat{e}_{2})$
$\displaystyle\textbf{b}_{\mathfrak{F}}$
$\displaystyle=(a^{2}+b^{2})^{-1/2}(b\sin(S_{\mathfrak{F}}^{\alpha}(t))\hat{e}_{1}-b\cos(S_{\mathfrak{F}}^{\alpha}(t))\hat{e}_{2}+a\hat{e}_{3})$
(61)

Note that all these vectors have constant components with respect to
$\hat{e}_{3}$, resulting in fractal snowflakes about the $x_{3}$ axis in their
spherical images. The radii of fractal curvature for the fractal spherical
indicatrix of $\textbf{t}_{\mathfrak{F}}$, $\textbf{n}_{\mathfrak{F}}$, and
$\textbf{b}_{\mathfrak{F}}$ are respectively given by:

$\varrho_{\textbf{t}_{\mathfrak{F}}}=\frac{a^{2}}{(a^{2}+b^{2})^{1/2}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\varrho_{\textbf{n}_{\mathfrak{F}}}=1,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\varrho_{\textbf{b}_{\mathfrak{F}}}=\frac{b^{2}}{(a^{2}+b^{2})^{1/2}}$ (62)

##  4 Fractal Frenet Equation

In this section, we introduce the analogue of the Frenet equations for a
fractal curve. The Fractal Frenet equation is a set of differential equations
that describe the behavior of certain vectors along a fractal curve. The
Fractal Frenet equations can be expressed as follows:

######  Theorem 4.

Along a curve $\textbf{u}=\textbf{u}(\theta)$, the vectors
$\textbf{t}_{\mathfrak{F}}$, $\textbf{n}_{\mathfrak{F}}$, and
$\textbf{b}_{\mathfrak{F}}$ satisfy the following equations, which are known
as the Serret-Frenet equations of the fractal curve:

$\displaystyle\dot{\textbf{t}}_{\mathfrak{F}}$
$\displaystyle=\kappa_{\mathfrak{F}}\textbf{n}_{\mathfrak{F}}$
$\displaystyle\dot{\textbf{n}}_{\mathfrak{F}}$
$\displaystyle=-\kappa_{\mathfrak{F}}\textbf{t}_{\mathfrak{F}}+\tau_{\mathfrak{F}}\textbf{b}_{\mathfrak{F}}$
$\displaystyle\dot{\textbf{b}}_{\mathfrak{F}}$
$\displaystyle=-\tau_{\mathfrak{F}}\textbf{n}_{\mathfrak{F}}$ (63)

These equations describe the evolution of the fractal tangent vector
$\textbf{t}_{\mathfrak{F}}$, normal vector $\textbf{n}_{\mathfrak{F}}$, and
binormal vector $\textbf{b}_{\mathfrak{F}}$ along the fractal curve.

###### Proof.

To prove the second equation, we begin by fractal differentiating
$\textbf{n}_{\mathfrak{F}}=\textbf{b}_{\mathfrak{F}}\times\textbf{t}_{\mathfrak{F}}$.
This yields:

$\dot{\textbf{n}}_{\mathfrak{F}}=\dot{\textbf{b}}_{\mathfrak{F}}\times\textbf{t}_{\mathfrak{F}}+\textbf{b}_{\mathfrak{F}}\times\dot{\textbf{t}}_{\mathfrak{F}}=-\tau_{\mathfrak{F}}(\textbf{n}_{\mathfrak{F}}\times\textbf{t}_{\mathfrak{F}})+\textbf{b}_{\mathfrak{F}}\times(\kappa_{\mathfrak{F}}\textbf{n}_{\mathfrak{F}})$
(64)

Since $\textbf{n}_{\mathfrak{F}}\times\textbf{t}_{\mathfrak{F}}$ is orthogonal
to $\textbf{b}_{\mathfrak{F}}$, the cross product is zero. So, the equation
simplifies to:

$\dot{\textbf{n}}_{\mathfrak{F}}=\kappa_{\mathfrak{F}}(-\textbf{t}_{\mathfrak{F}})=-\kappa_{\mathfrak{F}}\textbf{t}_{\mathfrak{F}}$
(65)

Thus, we have obtained the second equation of the Serret-Frenet equations for
the fractal curve. ∎

######  Remark 2.

A noteworthy observation is that the Frenet equations can be conveniently
written in matrix form as:

$\displaystyle\dot{\textbf{t}}_{\mathfrak{F}}$
$\displaystyle=0\textbf{t}_{\mathfrak{F}}+\kappa_{\mathfrak{F}}\textbf{n}_{\mathfrak{F}}+0\textbf{b}_{\mathfrak{F}}$
$\displaystyle\dot{\textbf{n}}_{\mathfrak{F}}$
$\displaystyle=-\kappa_{\mathfrak{F}}\textbf{t}_{\mathfrak{F}}+\tau_{\mathfrak{F}}\textbf{b}_{\mathfrak{F}}$
$\displaystyle\dot{\textbf{b}}_{\mathfrak{F}}$
$\displaystyle=-\tau_{\mathfrak{F}}\textbf{n}_{\mathfrak{F}}$ (66)

This allows us to compactly represent the coefficients of
$\textbf{t}{\mathfrak{F}}$, $\textbf{n}{\mathfrak{F}}$, and
$\textbf{b}_{\mathfrak{F}}$ in a matrix:

$\left(\begin{array}[]{ccc}0&\kappa_{\mathfrak{F}}&0\\\
-\kappa_{\mathfrak{F}}&0&\tau_{\mathfrak{F}}\\\ 0&-\tau_{\mathfrak{F}}&0\\\
\end{array}\right)$ (67)

This matrix representation elegantly summarizes the relationships between the
tangent, normal, and binormal vectors along the fractal curve.

######  Example 8.

Consider the fractal logarithmic spiral described by the intrinsic equations
$\kappa_{\mathfrak{F}}=1/J(\theta)$ and $\tau_{\mathfrak{F}}=0$, where
$\theta>0$. We can further simplify the equations as follows:

$\frac{d^{\alpha}_{\mathfrak{F}}\phi}{d^{\alpha}_{\mathfrak{F}}t}=\kappa_{\mathfrak{F}}=1/J(\theta)$
(68)

By integrating the above expression, we obtain:

$\phi=\log(J(\theta))+C_{1}$ (69)

where $J(\theta)=\exp(\phi-C_{1})$ and $\kappa_{\mathfrak{F}}=\exp(-(\phi-
C_{1}))$. Setting $\phi-\pi/4=\psi$, we can express the vector u as:

$\textbf{u}=\int\frac{1}{\kappa_{\mathfrak{F}}}((\cos\phi)\hat{e}_{1}+(\sin\phi)\hat{e}_{2})d\phi+C_{2}=\frac{1}{\sqrt{2}}\exp(\psi)[\cos(\psi)\hat{e}_{1}+\sin(\psi)\hat{e}_{2}]$
(70)

Here, we have chosen specific values for constants $C_{1}$ and $C_{2}$,
specifically $C_{1}=\pi/4$ and $C_{2}=0$, to simplify the expression. This
represents the fractal logarithmic spiral in its natural representation.

##  5 Conclusion

The study of fractal curves and the Fractal Frenet equations provides a
fascinating and profound exploration into the world of fractal geometry. These
insights not only deepen our understanding of complex shapes but also have
potential applications in various fields, including mathematics, physics,
computer graphics, and beyond. As researchers continue to investigate and
analyze fractal curves, we can anticipate further discoveries and applications
that will enrich our understanding of the beauty and intricacy of the natural
world.

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# Advances in Self-Supervised Learning for Synthetic Aperture Sonar Data
Processing, Classification, and Pattern Recognition

Brandon W. Sheffield, IEEE Member, Frank E. Bobe III, Bradley Marchand,
Matthew S. Emigh, IEEE Member  Naval Surface Warfare Center Panama City
Division

###### Abstract

Synthetic Aperture Sonar (SAS) imaging has become a crucial technology for
underwater exploration because of its unique ability to maintain resolution at
increasing ranges, a characteristic absent in conventional sonar techniques.
However, the effective application of deep learning to SAS data processing is
often limited due to the scarcity of labeled data. To address this challenge,
this paper proposes MoCo-SAS that leverages self-supervised learning (SSL) for
SAS data processing, classification, and pattern recognition. The experimental
results demonstrate that MoCo-SAS significantly outperforms traditional
supervised learning methods, as evidenced by significant improvements observed
in terms of the F1-score. These findings highlight the potential of SSL in
advancing the state-of-the-art in SAS data processing, offering promising
avenues for enhanced underwater object detection and classification.

###### Index Terms:

Self-Supervised Learning, Computer Vision, Synthetic Aperture Sonar, Sidescan
Sonar

##  I Introduction

Acoustic sonar, particularly Synthetic Aperture Sonar (SAS), has emerged as a
vital technology in the domain of underwater imaging, primarily due to the
unique challenges that marine environments pose to traditional computer vision
techniques. The often unpredictable and adverse conditions of underwater
settings, characterized by poor lighting, turbidity, and suspended sediment,
have made it arduous for conventional computer vision methodologies, which
predominantly rely on optical camera imagery, to adapt effectively. Unlike
side scan sonar (SSS) or real aperture sonar (RAS), where resolution worsens
with range, SAS overcomes this limitation by coherently combining consecutive
sonar pings to synthetically lengthen the aperture, resulting in constant
resolution throughout the image[1, 2, 3, 4, 5].

In recent years, Deep Neural Networks (DNNs) have found favor over classical
machine learning methodologies due to their proficiency in autonomously
discovering features in data, thereby eliminating the need for manual feature
crafting by subject matter experts. However, a notable limitation associated
with DNNs is their dependence on substantial volumes of labeled data and
considerable computational power to learn robust features. In the context of
SAS, the scarcity of labeled data exacerbates this problem, making it
difficult to adapt DNNs to underwater imaging. Emerging as a promising
solution to this predicament is Self-Supervised Learning (SSL). The growing
availability of computational power and data has propelled SSL into
prominence, offering a methodology that allows models to learn features in
data without the need for labels. Despite its potential, the application of
SSL to SAS data processing, classification, and pattern recognition has
received limited attention.

This work is an iterative improvement on a previous work that demonstrated
that two popular SSL approaches in SAS data produced positive results for low-
labeled scenarios and the models were fine-tuned only in the last layer[6].
The aim of this paper is to demonstrate the potential of SSL when limited SAS-
labeled data exist, by proposing the MoCo-SAS framework. The SSL model is
trained on real-world SAS data to learn useful feature representations, and
once the model is pretrained on unlabeled data, it serves as a feature
extractor for downstream binary image classification using a Support Vector
Machine(SVM). The performance of this SSL model is further evaluated and
compared to its supervised learning counterpart.

##  II Background

Early work involving unsupervised learning in SAS data processing utilized
Latent Dirichlet Allocations (LDA) and Auto-Encoders (AE) to extract features
of objects in SAS images unsupervised[7].

Pre-training is another important aspect of SAS data processing. Multiple
works have shown that the use of a pre-trained CNN for feature extraction in
combination with a Support Vector Machine for classification improves
classification performance [8, 9, 10, 11]. In the work of Williams [12, 13],
the author explored the use of transfer learning with SAS image convolutional
neural networks for improved classification of underwater objects.

The paper titled ”Self-supervised Learning for Sonar Image Classification”[14]
by Preciado-Grijalva et al. discusses how SSL can be a powerful approach to
learn forward-look sonar (FLS) image representations without the need for
large labeled data sets. The paper investigates the potential of three SSL
methods (RotNet, Denoising Autoencoders, and Jigsaw) to learn high-quality
sonar image representation without the need for human labels.

##  III MoCo-SAS

Figure 1: Block diagram demonstrating the MoCo-SAS classification processing
chain. The process begins with the pretraining phase, where the model learns
to generate informative features from the unlabeled SAS data. These features
are then passed through a feature extractor, which transforms the raw SAS data
into high-level representations. These representations are then used in the
downstream tasks, such as classification using an SVM classifier.

The MoCo-SAS framework is built upon the Momentum Contrast (MoCo) approach
[15][16], which is a strategy for contrastive loss learning. To enhance the
model’s ability to learn robust features from unlabeled data during the
pretraining phase, we have integrated a data augmentation module specifically
tailored for Synthetic Aperture Sonar (SAS). MoCo was selected as the
foundation for this framework primarily due to its inclusion of a queue or
’memory bank.’ This queue stores numerous negative samples from previous
batches, thereby mitigating the substantial computational expense associated
with the need for large training batches when calculating the contrastive
loss.

MoCo-SAS is composed of a backbone network based on the ResNet[17]
architecture and a multi-layer perceptron projection head. The backbone
network extracts low-level features from the input data, while the projection
head maps these features to a latent space where contrastive learning is
performed. In this study, a custom version of the ResNet18 architecture shown
in Table I, is customized to handle 1-channel SAS imagery that is used for
both the MoCo-SAS backbone as well as a comparative version of supervised
learning in model evaluation. The final classification head of the standard
ResNet18 is removed, making the model suitable for downstream tasks. After
pretraining, the model acts as a feature extractor, transforming raw SAS data
into high-level representations that serve as input for an SVM classifier.

TABLE I: Custom ResNet-18 Architecture Layer (type) | Output Size | Number of Parameters | Configuration
---|---|---|---
Conv2d | 64x112x112 | 3200 | Kernel = 7, Stride = 2, Padding = 3
BatchNorm2d | 64x112x112 | 128 | Eps = 1e-05, Momentum = 0.1, Affine = True
ReLU | 64x112x112 | 0 | Inplace = True
MaxPool2d | 64x56x56 | 0 | Kernel = 3, Stride = 2, Padding = 1, Dilation = 1
… | … | … | …
AdaptiveAvgPool2d | 512x1x1 | 0 | Output Size = 1

##  IV Experimental Setup

The MoCo-SAS model is first pre-trained on a large unlabeled SAS data set
using the PyTorch Lightning framework [18]. The backbone network of the MoCo-
SAS model is a ResNet18 architecture.

An SVM from scikit-learn[19] is trained using the high-level features
extracted from the pretrained MoCo-SAS model. Training of this classifier and
the comparative ResNet18 supervised learning model is carried out using a
fraction of the labeled SAS data set, where the fraction is a variable
parameter ranging from 1% to 100% of the data set. The performance of the
MoCo-SAS framework is evaluated based on the F1-score of the SVM
classification model in the test data set and compared against a typical
supervised learning ResNet18 classification model that is initialized with
random weights. The comparison between MoCo-SAS and ResNet18 seeks to show
that initialization of pre-trained weights from unlabeled data is beneficial
for improving classification performance when there are limited labeled data.

The experiments are carried out using the Weights and Biases (WandB)[20] tool
for the tracking of the experiment. The results of each run, including the run
configuration and the performance metrics, are logged to WandB. The best run
is determined based on the highest F1-score achieved across all runs on the
test data set.

###  IV-A Data set

The high-resolution nature of Synthetic Aperture Sonar (SAS) data often
exceeds the processing capabilities of contemporary GPUs. This requires the
division of the imagery into smaller, manageable sections, often referred to
as tiles, snippets, or chips. To ease the burden of creating a data set
comprising snippets that contain objects on the sea floor, we employ the FRED
anomaly detector[21] which is a fusion of two simple and efficient detection
algorithms: the raw energy detector and the KL divergence Reed-Xialo (RX)
[22]. Each snippet, both from the low-frequency (LF) and high-frequency (HF)
ranges, is resized to a dimension of 224x224. These snippets are then stacked
to form a multiband SAS image with dimensions 2x224x224. The primary
experiments conducted in this study focus on the HF portion of the images, and
a later ablation study considers both single and dual frequency channels.

Data set | Number of Samples
---|---
Pretrain | 400,000
Train | 6,000
Test | 2,000
TABLE II: The pretrain data set is unlabeled while the train and test data
sets are labeled. Both the train and test data sets are balanced with positive
and negative instances.

###  IV-B Augmentations

This study applied a multi-view data augmentation strategy to SAS imagery for
pretraining the MoCo-SAS model. This strategy is crucial for learning robust
representations, as it encourages the model to identify and focus on the
invariant features of the same image under different transformations. However,
in the downstream training and test phases, only the horizontal flip
augmentation is applied.

The multiview data augmentation process generates two distinct views from the
same input image. Each view is produced by applying a probabilistic set of
transformations, including random horizontal flip, random resized crop, subtle
random rotation, random affine transformations, random Gaussian blur, and
speckle noise as can be seen in Figure 2. The latter introduces a type of
interference commonly found in radar systems, including SAS, thereby enhancing
the model’s resilience to such noise in real-world scenarios.

Figure 2: Before and after transformation of a SAS image with a crab trap that
has data augmentations applied. Augmentations must be carefully selected such
that the semantic content of the image remains the same.

###  IV-C Pretraining

The MoCo-SAS model is pre-trained on the pre-trained data set using the
hyperparameters in Table III. During pretraining, each mini-batch comprises
two views for each image, produced by the aforementioned data augmentation
process. The model learns to associate the two views of the same image (a
positive pair) while contrasting them from all other views (negative pairs) in
the minibatch. By employing this multiview data augmentation strategy, the
model learns to recognize and understand the underlying invariant features of
the SAS imagery, thereby achieving better performance in downstream tasks.

This process is repeated for multiple epochs, with the model’s parameters
being updated after each batch to minimize the contrastive loss. To prevent
overfitting and ensure efficient learning, early stopping is applied during
training. This stops the training process if there is no significant decrease
in loss for 10 consecutive epochs. With 8 Nvidia A6000 GPUs, training halted
at 6 hours and 39 minutes at 184 epochs as shown in Figure 3.

Figure 3: Loss curve of the pretraining phase. The x-axis represents the epochs, and the y-axis represents the loss. Hyperparameters | Values
---|---
Backbone | ResNet18
Channels | 1
Batch Size | 1536
Optimizer | AdamW
Scheduler | One Cycle LR
Loss | NTXent
Learning Rate | 0.3
Weight Decay | 0.0001
Memory Bank | 4096
Temperature | 0.07
Momentum | 0.99
Warmup Epochs | 10
TABLE III: MoCo-SAS hyperparameters that are used in the pretraining stage.

###  IV-D Metrics

The effectiveness of the proposed MoCo-SAS framework is assessed using three
widely applied evaluation criteria: Precision, Recall, and F1-score. These
metrics provide a comprehensive evaluation of the model’s performance,
quantifying its ability to accurately identify positive instances (objects)
and its capacity to limit false alarms (misclassifying clutter or negative
instances as positive).

Precision is defined as the proportion of true positives (TP) out of all
predicted positives. It measures the classification model’s ability to return
only relevant instances. Specifically, it calculates the fraction of instances
correctly identified as objects out of all instances that the model classified
as such.

$\text{Precision}=\frac{\text{True Positives}}{\text{True
Positives}+\text{False Positives}}$

Recall, also known as Sensitivity or True Positive Rate, is the ratio of true
positives (TP) to all actual positives. Measures the ability of the
classification model to identify all relevant instances. Specifically, it
calculates the fraction of actual objects that the model correctly identified.

$\text{Recall}=\frac{\text{True Positives}}{\text{True Positives}+\text{False
Negatives}}$

F1-score is the harmonic mean of Precision and Recall, providing a balanced
representation of these two metrics.

$\text{F1-score}=2\times\frac{\text{Precision}\times\text{Recall}}{\text{Precision}+\text{Recall}}$

In the experiments, the F1-score of the classification model is evaluated on
the test data set, as it provides a balanced measure of its performance in
terms of both Precision and Recall. In the context of the SAS object detection
task, both missing an object (a false negative) and falsely identifying
clutter as an object of interest (a false positive) can carry substantial
implications.

##  V Results and Discussion

To validate the effectiveness of the approach, extensive experiments are
conducted using real-world SAS data sets containing various types of
underwater objects. The performance of the MoCo-SAS framework is compared with
a traditional supervised learning ResNet18 model for SAS data processing,
classification, and pattern recognition.

In extensive experiments, the proposed MoCo-SAS framework demonstrated
significant improvements over traditional supervised learning methods in terms
of F1-score. Figure 4 summarizes the results of the experiments, indicating
the F1-score for each method in the test data set.

Figure 4: Comparison of F1-score on the test data set using the proposed MoCo-
SAS, denoted as SSL, approach versus the typical ResNet18.

The MoCo-SAS approach outperforms the ResNet18 supervised learning method by
significant margins in terms of F1-score. This is primarily due to the
framework’s ability to leverage unlabeled data to learn useful
representations, thereby overcoming the scarcity of labeled data in SAS
applications. Moreover, the robustness of the learned features is evident from
the consistent performance of the method across various underwater
environments. These features capture the underlying patterns in the SAS data,
allowing the classification model to generalize well to new data. These
results confirm the effectiveness of the MoCo-SAS framework for SAS data
processing. By exploiting the structure of unlabeled data, the method is
capable of learning robust representations and achieving superior performance
in object detection tasks.

###  V-A Discussion on SVM Performance

The results indicate that the Support Vector Machine (SVM) classifier
outperforms other classifiers for downstream tasks in the MoCo-SAS framework.
One possible explanation for this superior performance is the effectiveness of
the SVM in high-dimensional spaces. Given that the data features extracted
from the pre-trained model are high-dimensional, SVM is likely to perform
well.

Additionally, SVM is robust against overfitting, particularly in situations
where the number of dimensions exceeds the number of samples. This
characteristic may have contributed to its success in the task. Furthermore,
the use of a subset of training points in the decision function makes SVM
memory efficient, which can be beneficial for large-scale data.

###  V-B Ablation Study Results

Table IV presents an ablation study on the impact of backbone depth (ResNet18,
ResNet34, ResNet50), label percentages, and the number of channels on the
performance of a SVM classifier. From the results, it is evident that the
depth of the backbone, the percentages of labels and the number of channels
all have a significant impact on the performance of the SVM classifier.

Rank | Backbone | Label % | Channels | Performance Tier
---|---|---|---|---
1 | ResNet50 | 100 | 2 | High
2 | ResNet18 | 100 | 2 | High
3 | ResNet18 | 100 | 1 | High
4 | ResNet50 | 100 | 1 | High
5 | ResNet50 | 1 | 1 | High
6 | ResNet34 | 100 | 2 | High
7 | ResNet34 | 100 | 1 | High
8 | ResNet50 | 10 | 2 | High
9 | ResNet50 | 5 | 2 | Medium
10 | ResNet34 | 1 | 1 | Medium
11 | ResNet50 | 5 | 1 | Medium
12 | ResNet50 | 10 | 1 | Medium
13 | ResNet18 | 10 | 1 | Medium
14 | ResNet18 | 5 | 1 | Medium
15 | ResNet34 | 10 | 1 | Medium
16 | ResNet34 | 5 | 1 | Medium
17 | ResNet18 | 10 | 2 | Low
18 | ResNet18 | 5 | 2 | Low
19 | ResNet18 | 1 | 1 | Low
20 | ResNet34 | 10 | 2 | Low
21 | ResNet34 | 5 | 2 | Low
22 | ResNet50 | 1 | 2 | Low
23 | ResNet18 | 1 | 2 | Low
24 | ResNet34 | 1 | 2 | Low
TABLE IV: Experiments performed to examine whether backbone model depth in
MoCo-SAS improved performance for single and multi-channel scenarios with
varying label percentages.

  * •

Backbone Depth: The deeper models (ResNet34 and ResNet50) generally outperform
the shallower model (ResNet18), especially when the label percentage is high
(100%). This suggests that deeper models are capable of extracting more
complex features from the data, which can lead to improved performance.
However, when the percentage of labels is low (1%), the performance of all
models decreases, with ResNet34 and ResNet50 performing only slightly better
than ResNet18. This could be because fewer labels provide the models with less
information, making it harder to extract useful features.

  * •

Label Percentages: The performance of all models improves as the percentage of
labels increases. This is expected, as more labels provide more information
from which models can learn. However, even at low label percentages (1% and
5%), the models are still able to achieve medium to high performance,
suggesting that they are able to effectively leverage unlabeled data.

  * •

Number of Channels: The number of channels also has a significant impact on
performance. In general, the models perform better with two channels than with
one. This could be because two channels provide more information than one,
allowing the models to extract more complex features. However, the performance
gain from using two channels is more pronounced for the deeper models
(ResNet34 and ResNet50) than for the shallower model (ResNet18), suggesting
that the deeper models are better able to leverage additional information.

These findings highlight the importance of considering the depth of the
backbone, the percentages of labels, and the number of channels when training
an SVM classifier for sonar imagery. Future work could explore these factors
in more depth, as well as investigate other potential influences on classifier
performance.

##  VI Conclusion and Future Work

This study introduced MoCo-SAS, demonstrating the potential of such techniques
in advancing SAS data processing, classification, and pattern recognition. The
approach leveraged contrastive learning and clustering-based methods to
exploit the structure of unlabeled data and extract high-level features,
contributing to the performance enhancement of the downstream tasks in image
classification and object recognition. Significant improvements in the
F1-score and robustness of the learned features, which highlights the
generalizability of the approach.

Future research directions include exploring advanced self-supervised learning
algorithms that can further enhance the learning of robust features.
Integrating the framework with other sonar modalities can lead to a more
comprehensive and robust system. Synthetic data augmentation and its effects
on classification performance also deserve further investigation. We also
intend to develop real-time processing capabilities for practical deployment
in Autonomous Underwater Vehicle (AUV) object detection applications. This
will involve optimizing the computational efficiency of the method to meet the
real-time processing requirements of such applications.

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