Approximation properties of modified (p, q)-Szász-Mirakyan-Kantorovich operators

1.   Introduction

The study of curves in classical differential geometry is one of the most fascinating topics. The concept of the involute-evolute curve pair was introduced by Huygens in 1968 in ^[1]. If the tangents of the first curve are normal to the second curve, we classify the second curve as an involute and the first curve as an evolute. In 1850, Bertrand first introduced the Bertrand curve ^[2]. When two curves share a common normal vector at corresponding points, they are referred to as Bertrand curves. We refer to the first curve as the Bertrand curve, and the second as the Bertrand mate. One significant characteristic of three-dimensional Euclidean space is that the distance between the corresponding points of the Bertrand curves remains constant. Bertrand curves were extended from Euclidean $3$ space to Riemannian $n$ space by L. R. Pears ^[3] who gave generalized results of the Bertrand curves. For $n \ge 4$, no special Frenet curve in $\mathbb{E}^{n}$ is a Bertrand curve. Matsuda and Yorozu ^[4] gave a new definition of the Bertrand curve called the (1, 3)-Bertrand curve and a characterization of the (1, 3)-Bertrand curves. After that, many researchers have studied (1, 3)-Bertrand curve ^[5,6,7]. Furthermore, some studies of characterizations of curves in different spaces and with different frames can be found in ^[8,9,10,11].

The study of involutes and evolutes has also been extended to Minkowski space, a pseudo-Euclidean space with an indefinite metric. In Minkowski space, the geometric properties of curves differ significantly from those in Euclidean space due to the presence of timelike, spacelike, and lightlike vectors. The concept of involute-evolute pairs in Minkowski space has been explored by several authors ^[12,13].

In 1843, Hamilton introduced the concept of a quaternion, a type of number system that exists in a four-dimensional vector space and can take on different forms, such as real, complex, dual, and split varieties. According to Clifford ^[14], in 1871, the quaternion was generalized to biquaternions. In $1987$, Bharathi and Nagaraj ^[15] studied the quaternionic curves ($Qu$-curves) in both $\mathbb{E}^{3}$ and $\mathbb{E}^{4}$ and provided the Frenet formula for $Qu$-curves. For other results of $Qu$-curves, we refer to ^{[16,17,18,19,20,21]}. Aksoyak in ^[22] defined the quaternionic frame $Qu$-frame for ($Qu$-curves) in $\mathbb{E}^{4}$, which is called type-$2$ $Qu$-frame. For more results see ^{[23,24,25,26,27,28]}. If a ($Qu-B$)-curve exists in $\mathbb{E}^{4}$, then its torsion or bi-torsion vanishes, so we can say that there is no ($Qu-B$)-curve whose torsion or bi-torsion is non-zero. Hence, we use the method given by Matsuda and Yorozu ^[4] to define the ($Qu-B$)-curve according to the type-$2$ $Qu$-frame. For other results regarding involutes and evolutes, we refer to the papers ^[29,30,31].

In this context, we determine the involute and evolute with the ($Qu-B$)-curve using a type-$2$ $Qu$-frame. We also deduce a relation between the Frenet frame and curvature functions.

2.   Preliminaries

A real quaternion is defined as:

where $\hat{i}, \hat{j}, \hat{k}$ are unit vectors in three-dimensional vector space and $a, b, c, d\in\mathbb{R}$. Any quaternion $u$ consists of two parts: one is the scalar part denoted as $S_u$ and the second is a vector part which is denoted as $V_u$, where

and

We also present any real quaternion as

Let us consider two quaternions, i.e.,

and

Then their addition and multiplication by a scalar $c$ and conjugate will be represented as

Let us denote the four-dimensional real vector space, in which addition and multiplication by a scalar $c$ are defined as described by $Q$, and refer to its elements as quaternions. The basis of this vector space is $\{1, \hat{i}, \hat{j}, \hat{k}\}$, and satisfies

Any quaternion $u$ can be considered as an element (a, b, c, d) of $\mathbb{R}^{4}$. When the scalar part is zero, we refer to the quaternion as spatial. In this case, it can be considered as an ordered triple (b, c, d) of $\mathbb{R}^{3}$ ^[15].

Multiplication of quaternions can be defined as

for every $u, u'\in Q$, where $\langle, \rangle$ and $\wedge$ denote the scalar product and cross product in $\mathbb{R}^{3}$. The quaternion multiplication is associative and distributive but non-commutative. Hence, $Q$ is a real algebra, referred to as a quaternion algebra. Moving forward, we define a symmetric and non-degenerate bilinear $h$ on $Q$ as

for $u, u'\in Q$ and the norm of $u$ is defined as

Therefore, the mapping denoted by $h$ is referred to as the quaternion scalar product. The $Qu$-curve in $\mathbb{R}^{4}$ is represented by $\gamma^4$ and the spatial $Qu$-curve in $\mathbb{R}^{3}$ associated by $\gamma^4\in \mathbb{R}^{4}$ is denoted by $\gamma$.

Theorem 2.1. (^[15]) Let

and $S$ be the set of spatial $Qu$-curves. Suppose that

is a curve parameterized by the arc length $s$. Then, the Frenet equations of $\gamma$ are given as

where $t$ represents the unit tangent vector, denoted by $\gamma'$, $n$ is the normal vector, and $b$ is the binormal vector, which is calculated as the cross product of $t$ and $n$. The principle curvature is denoted by $k$ and is equal to the norm of the derivative of the unit tangent vector, while the torsion of the curve $\gamma$ is represented by $-r$. Moreover, these Frenet vectors hold the following equations:

Theorem 2.2. Suppose that

and

is an arc-length parameterized curve in $\mathbb{R}^4$. Then Frenet equations of $\gamma^4$ are given as

where

$N_1$–$N_3$ represent the Frenet vectors of the curve $\gamma^{4}$,

is the principle curvature, $-r$ is the torsion and $(K-k)$ is the bitorsion of the curve $\gamma^{4}$. These Frenet vectors of the above theorems satisfy the following equations:

In four-dimensional Euclidean space, the product of two vectors is defined using the wedge product (or exterior product). For two vectors $u$ and $v$ in $\mathbb{E}^4$, the wedge product is given by

where $\otimes$ denotes the tensor product. The result is a bivector, which represents the plane spanned by $u$ and $v$.

In the context of the Frenet frame, the first normal vector $N_1$ is constructed as a vector orthogonal to the tangent vector $T(s)$ and the binormal vector $b(s)$. This is achieved by taking the wedge product of $b(s)$ and $T(s)$ and normalizing the result:

Here, $\|b(s) \wedge T(s)\|$ is the norm of the bivector $b(s) \wedge T(s)$, and $N_1$ is a unit vector orthogonal to both $b(s)$ and $T(s)$.

The vector product of $\vec{x}$, $\vec{y}$, $\vec{z}$ is given by the determinant as follows:

where^[32,33]

and

After establishing the fundamental properties of quaternionic curves and their Frenet frames in four-dimensional Euclidean space, we now focus on the geometric relationships between these curves and their associated involute and evolute pairs. The concepts of involutes and evolutes, which describe the relationship between two curves where the tangents of one curve are normal to the other, have been extensively studied in classical differential geometry. In the context of quaternionic Bertrand curves, these relationships take on a more intricate form due to the additional dimensions and the non-commutative nature of quaternion algebra. Using the type-2 quaternionic frame, we can derive explicit expressions for the (0, 2)-involute and (1, 3)-evolute curves, providing deeper insights into the geometric structure of these curves and their curvature properties.

This analysis not only extends classical results to higher dimensions, but also lays the groundwork for applications in fields such as robotics, computer graphics, and theoretical physics.

Definition 2.1. Let

and

be two $Qu$-curves. If each point of $\gamma^4(s)$ corresponds to points on $\beta^4(s)$ for all $s \in J$ via a regular $C^\infty$ function, and the normal plane spanned by the normal vectors at each point $\gamma^4(s)$ coincides with the normal plane spanned by the normal vectors of

then $\gamma^4(s)$ is called a $(Qu-B)$-curve in $\mathbb{E}^4$, and $\beta^4(s)$ is called the $(Qu-B)$-mate of $\gamma^4(s)$.

Definition 2.2. Let

and

be two $Qu$-curves. If the tangent vectors of $\gamma^4$ are normal to $\beta^4$, then $\gamma^4$ is called a (1, 3)-evolute when it is spanned by the first and third normal vectors, and $\beta^4$ is called a (0, 2)-involute when it is spanned by the first tangent and the third normal vector.

3.   The (0, 2)-involute of a $Qu-B$ curve via a type-2 quaternionic frame in $\mathbb{E}^{4}$

Suppose that

is a regular $Qu$-$B$ curve whose curvature functions are $K$, $-r$, and $K - k$. Let $\beta^4(\bar{s})$ be a $Qu-B$ (0, 2)-mate of $\gamma^4$ with the $Qu$-frame $\{T^*, N_1^*, N_2^*, N_3^*\}$ and curvature functions $K^*$, $-r^*$, and $(K - k)^*$. If the inner product condition

holds, then the pair $(\gamma^4, \beta^4(\bar{s}))$ is called a real quaternion Bertrand involute-evolute curve pair.

The relationship between these vector fields is given by

Also, $\beta^4(\bar{s})$ is called the (0, 2)-involute of $\gamma^4$. Thus, we can write $\beta^4(\bar{s})$ as

where $\xi(s)$ and $\eta(s)$ are $C^\infty$ functions on $[0, 1]$. By differentiating (3.2) with respect to $s$ using Eq (2.1), we obtain

Taking the scalar product by vectors $T$ and $N_2$ on both sides of Eq (3.3), we obtain

So, Eq (3.3) can be rewritten as

Taking the substitution for some differentiable functions $\Gamma$ and $\Theta$ as

Then Eq (3.4) becomes

where

because $N_1\perp N_2$.

Case 1. Suppose $\xi\not = 0$. In this case, $\eta\not = 0$. Denoting

then,

and

Differentiating Eq (3.6), and using Eq (2.1), we obtain

Taking the scalar product of Eq (3.8) with $N_1$ and $N_3$, we find that $\Gamma' = 0$ and $\Theta' = 0$ which implies that $\Gamma$ and $\Theta$ are constants.

Now, we rewrite Eq (3.8) as

Taking the substitution for some differentiable functions $\Phi$ and $\Psi$ as

then Eq (3.9) becomes

Denote

This implies that

and

and then

From Eqs (3.7) and (3.12), we have

Denote

This implies that

Using Eq (3.10), we obtain

Differentiating (3.11) with respect to $s$ using Eq (2.1), we obtain

Taking the scalar product on both sides of Eq (3.15) with $T$ and $N_2$, we find that

which implies that $\Phi$ and $\Psi$ are constants.

Thus, we can write

Substituting Eqs (3.6) and (3.14) into (3.16), we obtain

From Eq (3.17), we can choose

for some differentiable functions $\Theta$ and $\Gamma$. Differentiating (3.18) with respect to to $s$ using Eq (2.1), we obtain

From this, we obtain

Also, we can write

Equation (3.20) becomes

Theorem 3.1. Let

be a unit speed ($Qu-B$)-curve with non-zero curvatures $K$, $-r$, and $(K-k)$. Then $\gamma^4$ has the $(0, 2)$-involute of quaternionic Bertrand curve.

Proof. Let $\beta^4(\bar{s})$ be the (0, 2)-involute curve of $\gamma^4(s)$. Then the equation for $\beta^4(\bar{s})$ can be written as:

with $\eta\not = 0$.

Then, the curvatures satisfy

where $\xi_0, \eta, x_1$, and $x_2$ are constants.

Furthermore, the curvatures of $\beta(\bar{s})$ are given by

where $\Phi\not = 0$, the related frame is given by

This completes the proof. □

Corollary 3.1. If $\frac{-r}{R}$ or $\frac{(K-k)}{K}$ is constant, then the ($Qu-B$)-curve $\gamma^4$ does not have a $(0, 2)$-involute of $(Qu-B)$-curve in the form

Case 2. Suppose $\eta\not = 0$. Thus, Eq (3.21) reads as

Differentiating (3.23) with respect to $s$ using Eq (2.1), we obtain

taking the scalar product, we have

This implies that

By differentiating (3.25) with respect to $s$ using Eq (2.1), we obtain:

Let

Then

This implies that

Differentiating (3.27) with respect to $s$ using Eq (2.1), we deduce that $\upsilon$ and $\nu$ are constants. Hence,

We assume that

By differentiating the Eq (3.31) with respect to $s$ using Eq (2.1), we obtain

Thus, we have

Corollary 3.2. Let

be a unit speed ($Qu-B$)-curve with non-zero curvatures $K$, $-r$, and $(K-k)$. If $\gamma^4$ have a (0, 2)-involute of the ($Qu-B$)-curve

then curvature $K$ and $-r$ satisfy the equality

where $\xi_0$, $\upsilon$, and $\nu$ are constants and the curvatures of $\beta^4(\bar{s})$ are given as

The required $Qu$-frame is given by

4.   The (1, 3)-evolute of a ($Qu-B$)-curve via a type-2 quaternionic frame in $\mathbb{E}^{4}$

Definition 4.1. Let

be two unit-speed ($Qu-B$)-curves. Suppose that $\gamma^4$ has nonzero curvatures $K$, $-r$, and $(K - k)$. Additionally, assume that $\beta^4(\bar{s})$ has a Frenet frame $\{T^*, N_1^*, N_2^*, N_3^*\}$ with nonzero curvatures $K^*$, $-r^*$, and $(K - k)^*$.

Then, $\beta^4(\bar{s})$ is called a (1, 3)-evolute if the following conditions hold:

In other words, $\beta^4(\bar{s})$ has the parametric representation

where $\lambda$ and $\mu$ are smooth functions defined on the closed unit interval $J$.

Differentiating (4.2) with respect to $s$, we obtain

Also,

If we denote

then Eq (4.4) becomes

Differentiating (4.4) with respect to $s$ and using Eq (2.1), we obtain

From Eq (4.5), we obtain

Consequently, $c$ and $d$ are constants. Hence, Eq (4.7) becomes

Let

Then Eq (4.9) becomes

Differentiating Eq (4.11) with respect to $s$ using (2.1), we obtain

So, from Eq (4.1), we have $m' = 0$ and $n' = 0$, which implies that $m$ and $n$ are constants. Thus, Eq (4.12) takes the form

Let

Equation (4.13) becomes

Since $T^*\perp N_2^*$, from Eqs (4.6) and (4.15), we have

Then Eq (4.14) becomes

Differentiating Eq (4.16) with respect to $s$ using Eq (2.1), we obtain

from which we obtain

From Eq (4.17), we have

Theorem 4.1. If

is a unit speed $(Qu-B)$-curve with non-zero curvatures $K$, $-r$, and $(K-k)$. Then $\gamma^4(s)$ possesses the (1, 3)-evolute of the $Qu-B$-curve.

Proof. Suppose that

Then curvatures of $\beta^4(\bar{s})$ are obtained as

where

The $Qu$-frame of the curve $\beta^4(\bar{s})$ is given by

where $m$, $n$, $c$, and $d$ are constants.

In the following theorem, we provide both necessary and sufficient conditions for a ($Qu-B$)-curve to possess a (1, 3)-$(Qu-B)$-evolute curve.

This completes the proof. □

Theorem 4.2. Let

be two ($Qu-B$)-curves with non-zero curvatures. Then, $\beta^4(\bar{s})$ is a (1, 3)-evolute of ($Qu-B$)-curve of $\gamma^4(s)$, if and only if there exist $\Phi$ and $\Psi$ differentiable of $s$ and constants $\Delta\not = \pm 1$ and $\mu$ satisfying:

Proof. The curve $\beta^4(\bar{s}$) can be written as:

for all $\bar{s}, s\in I$, where $\Phi(s)$ and $\Psi(s)$ are $C^\infty$ functions on the unit interval $J$. Differentiating (4.23) with respect to $s$ using Eq (2.1), we have

Since

so

and

and from this, we obtain

Therefore, Eq (4.25) becomes

By squaring Eq (4.25), we obtain

If we denote

then from Eqs (4.25) and (4.27), we have

Differentiating (4.28), with respect to $s$ using Eq (2.1), we obtain

Since

we have

This means that $\epsilon$ and $\varepsilon$ are constants. Then,

Squaring Eq (4.31), we have

From Eq (4.27), we have

and

where

for $\varepsilon\not = 0$.

By Eqs (4.27) and (4.32), we obtain

Also,

From (4.34) and (4.35), we get

Denote

Thus, we can write

where

for $\delta_1\not = 0$. Using values of $\delta_1$, $\delta_2$ in Eq (4.31), we obtain

Taking the derivative of (4.39), with respect to $s$ using Eq (2.1), we obtain

Since

we get

From (4.28) and $(4.37)$–$(4.40)$, we obtain

where

Since

we get the result $(4.22)$:

Conversely, let $\gamma^4(s)$ be an evolute curve satisfying $(4.42)$–$(4.44)$. Then, we can write

Differentiating (4.46) with respect to $s$ using Eq (2.1), we get

Using Eqs (4.47) and (4.20), we obtain

From this,

since $\Psi' > 0$. Then Eq (4.48) becomes

Substituting from Eq (4.49) into (4.50), we obtain

Differentiating (4.51), with respect to $s$ using Eq (2.1), we obtain

By using Eq (4.52), we obtain

From Eqs (4.52) and (4.53), we get

Let

Then, we have

Taking the derivative of Eq (4.56), with respect to $s$ using Eq (2.1), we obtain

Differentiating (4.21), we have

From Eq (4.1), we deduce

because $N'^*\perp N^*$.

Using Eqs (4.54) and (4.59) in Eq (4.57), we get

From (4.51) and (4.53), we have

By Eqs (4.60) and (4.61), we obtain:

and from Eq (4.62),

Combining Eqs (4.62) and (4.63), we obtain

Also, $N_3^*$ can be stated as

that is

In the end, we find $(K-k)^*$

Hence, we find that $\beta^4(\bar{s})$ is the (1, 3)-evolute curve of the ($Qu-B$)-curve $\gamma^4(s)$. Therefore,

□

5.   Examples

Consider the following quaternionic Bertrand curve $\gamma^4(s)$ in $\mathbb{E}^4$ parameterized by arc length $s$

This curve lies on a Clifford torus in $\mathbb{E}^4$. The tangent vector $T(s)$ is given by

The principal curvature $K(s)$ is the norm of $T'(s)$

Thus, the first normal vector is

The torsion $-r(s)$ is

Thus, the second normal vector is

The bitorsion $K(s) - k(s)$ is the norm of $N_2'(s)$

Thus, the third normal vector is

5.1.   The $(0, 2)$-involute curve of the curve $\gamma^4(s)$

The $(0, 2)$-involute curve $\beta^4(\bar{s})$ is given by

where $\xi_0$ and $\eta$ are constants. Substituting the expressions for $\gamma^4(s)$, $T(s)$, and $N_2(s)$, we get:

Simplifying, the $(0, 2)$-involute curve is

5.2.   The $(1, 3)$-evolute curve of the curve $\gamma^4(s)$

The $(1, 3)$-evolute curve $\beta^4(\bar{s})$ is given by

Substituting the curvature functions

we get

Substituting the expressions for $\gamma^4(s)$, $N_1(s)$, and $N_3(s)$, we get

6.   Conclusions

This study has established a comprehensive framework for analyzing the properties of (0, 2)-involute and (1, 3)-evolute curves associated with quaternionic Bertrand curves in four-dimensional Euclidean space. By employing a type-2 quaternionic frame, we derived explicit mathematical expressions for these curves and elucidated their relationships with curvature functions and Frenet frames. These results deepen our understanding of the geometric structure of quaternionic curves and their involute-evolute pairs, contributing to the broader field of differential geometry.

The findings of this study have significant potential applications in both theoretical and applied mathematics. For instance, in robotics and motion planning, the geometric properties of quaternionic curves and their involute-evolute pairs can be utilized to design motion trajectories for robotic systems operating in higher-dimensional spaces. In computer graphics and animation, the explicit formulas derived in this work can be applied to model complex curves and surfaces, particularly in the rendering of four-dimensional objects.

Looking ahead, several promising directions for future research emerge. First, the results of this study could be generalized to $n$-dimensional Euclidean space ($\mathbb{E}^n$) for $n > 4$, providing a more comprehensive understanding of the geometric properties of curves in higher-dimensional spaces. Second, investigating the properties of quaternionic curves and their involute-evolute pairs in Minkowski space could yield insights into the behavior of curves in pseudo-Riemannian manifolds, with potential applications in relativity and cosmology. Finally, exploring the connection between quaternionic curves and physical systems, such as rigid body dynamics or quantum mechanics, could lead to new insights into the geometric structure of these systems.

Author contributions

Ayman Elsharkawy: created conceptualizations, supervised the research, reviewed and edited the manuscript, guided the theoretical framework, and prepared the manuscript draft; Ahmer Ali: collected data, supervised the study, provided critical insights, and contributed to refining the manuscript; Muhammad Hanif: created methodology, conducted the theoretical analysis, and developed the main results. Fatimah Alghamdi: reviewed and edited the manuscript, provided critical insights to refine interpretations, ensured adherence to publication standards, and contributed to improving the overall clarity and coherence of the work. All authors have read and agreed to the published version of the manuscript.

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

The authors declare that they have no conflicts of interest.