A new approach to Leonardo number sequences with the dual vector and dual angle representation

1.   Introduction

The concept of dual numbers ^[1], which is an extension of real numbers, was initially introduced by Clifford in 1873. It is defined as follows:

These numbers form a commutative and associative algebra over the real numbers. Kotelnikov was the first to explore the practical applications of dual numbers, laying the groundwork for their utilization in various fields ^[2]. Notably, Study incorporated dual numbers and associated vectors into line geometry and kinematics, pioneering their use in these domains ^[3]. In the context of vectors, the set of dual vectors can be represented as $\mathbb{D}^3$, indicating all possible combinations of dual number pairs. Additionally, the collection of all unit dual vectors forms what is known as the unit dual sphere, denoted by

which plays a significant role in geometric interpretations and calculations involving dual vectors. This rich framework of dual numbers and vectors offers a versatile toolset for mathematical and geometric analyses, with applications ranging from kinematics to computer graphics and beyond.

Study proved that there exists an isomorphism between the points of the dual unit sphere in $\mathbb{D}^3$ and the directed lines of Euclidean 3-space, which is known as Study mapping^[3]. Dual numbers have been applied in most fields including research on rigid body motion, displacement analysis of spatial mechanisms, surface shape analysis and computer graphics, kinematic synthesis, human body motion analysis, and others ^{[4,5,6,7,8,9,10]}.

The algebra of dual numbers is represented by a ring with the following addition and multiplication operations:

The multiplicative inverse of $\mathbb{A} = \gamma_{1}+\varepsilon \gamma_{2}$ is given by

and there is no inverse for pure dual numbers; hence, this algebra of numbers is not a field over real numbers. By using the inverse of dual numbers we can define the division operation of two dual numbers $\mathbb{A}$ and $\mathbb{B}$ as $\mathbb{A}\mathbb{B}^{-1}$ where $\mathbb{B}$ is not a pure dual number and $\mathbb{B}\neq0$. The set given by

is amodule over the ring of dual numbers $\mathbb{D}$, the elements of this module are called dual vectors.

Proposition 1. Let $\vec{\mathbb{A}}$ and $\vec{\mathbb{B}}$ be dual vectors; then, their scalar and cross products are respectively given by

If $\vec{\mathbb{A}}$ and $\vec{\mathbb{B}}$ are unit dual vectors then

where $\Phi = \phi+\varepsilon\phi^*$ is a dual angle between them^[1].

Proposition 2. The norm of the dual vector $\vec{\mathbb{A}}$ is given by

If $\vec{\mathbb{A}}$ is a unit dual vector, then $\|\vec{\gamma_{1}}\| = 1$ and $\langle\vec{\gamma_{1}}, \vec{\gamma}_{2}\rangle = 0$ ^[1].

Sequences of integers hold an important place in mathematical literature with many famous sequences playing crucial roles across various branches of mathematics. These sequences often arise naturally in mathematical investigations and have been extensively studied due to their properties and profound connections to different areas of mathematics.

The Fibonacci and Lucas number sequences ^{[11,12,13,14,15,16]} are defined by

The characteristic equation of recurrences $\mathrm{F}_n$ and $\mathrm{L}_n$ is given by $x^2-x-1 = 0$. The Binet formulas for the $\mathrm{F}_n$ and $\mathrm{L}_n$ sequences are respectively given by

where $\varphi = \frac{1+\sqrt{5}}{2}$ and $\psi = \frac{1-\sqrt{5}}{2}$ are the roots of the characteristic equation. The following identities are hold for Fibonacci and Lucas number sequences

The Leonardo number sequence ^{[17,18,19,20,21,22,23,24]} is defined as follows

with the initial condition the $\mathrm{Le}_0 = \mathrm{Le}_1 = 1$. The Leonardo numbers are related to the Fibonacci numbers through the below relation:

This Binet's formula for the Leonardo number sequence is given by

where $\varphi = \frac{1+\sqrt{5}}{2}$ and $\psi = \frac{1-\sqrt{5}}{2}$ are roots of the characteristic equation $x^3-2x^2+1 = 0$ and the other root of this equation is $\lambda = 1$.

Here, our goal was to study this a new number system that can by generated by using dual numbers and well known Leonardo numbers as dual Leonardo number sequences and we describe dual Leonardo vectors and angles to apply these dual vectors and angles in the geometry of dual space.

2.   Dual Leonardo number sequences

In this section, we demonstrate dual Leonardo number sequences, presenting their fundamental identities and properties. Some fundamental properties and identities of these number sequences are discussed, such as Binet's formula, Catalan's, Cassini's and D'ocagne's identities. These identities play crucial roles in obtaining on understanding of the properties of dual Leonardo number sequences, and we aim to establish and prove them by using Binet's formulas.

The dual Leonardo number is defined by the following relation:

where $\mathrm{Le}_n$ is the Leonardo number, $\varepsilon^2 = 0$ and $\varepsilon \neq 0$. The following recurrence relation can be obtained directly by using the definition of dual Leonardo numbers

where $\mathcal{L}_0 = 1+\varepsilon, \mathcal{L}_1 = 1+3\varepsilon$.

Theorem 1. The Binet formula for dual Leonardo numbers is given by

where $\underline\varphi = 1+\varepsilon\varphi$, $\underline\psi = 1+\varepsilon\psi$ and $\underline\varphi\; \underline\psi = 1+\varepsilon = \mathcal{L}_0$.

Proof. From (1.3) and (2.1), we find that

For the next proof we have taken into account the following equalities:

Theorem 2. (Catalan's identity) For any integers $n$ and $r$ such that $n\ge r$, we have

Proof. Using (1.2), (1.3), (2.3) and the Binet formula for dual Leonardo numbers in (2.2), we get

Corollary 1. (Cassini's identity) For any integer $n$ and $n\ge 1$, the following identity holds:

Proof. This identity can be obtained from Catalan's identity by taking $r = 1$.

Theorem 3. (D'ocagne's identity) For any integers $m$ and n, we have

Proof. Using (2.2) and (2.3), we will have

By taking (1.3), we get

The proof is completed.

3.   Dual Leonardo vector and dual angle

In this section we introduce the dual Leonardo vector $\vec{\mathcal{L}}_n$ and dual angle $\Phi$, respectively. Moreover, we will give the main identities for this vector.

Definition 1. The dual Leonardo vector $\vec{\mathcal{L}}_n$ in $\mathbb{D}^{3}$ is defined by

where $\vec{\mathrm{L}}e_n = (\mathrm{Le}_{n}, \mathrm{Le}_{n+1}, \mathrm{Le}_{n+2})$ and $\vec{\mathrm{L}}e_{n+1} = (\mathrm{Le}_{n+1}, \mathrm{Le}_{n+2}, \mathrm{Le}_{n+3})$ are real vectors.

Theorem 4. If $\vec{\mathcal{L}}_{n}$ is a unit dual vector, then

and

Proof. From (1.1), (1.3) and (3.1), we get

Using (1.2) and (1.3), we have

On the other hand

Using (2.3) and performing necessary calculations, we obtain

Then, by using (1.1), the proof is completed.

Theorem 5. For dual Leonardo vectors $\vec{\mathcal{L}}_n$ and $\vec{\mathcal{L}}_m$, we have

Proof. Using (1.3), (2.2) and (2.3), we obtain

and by using the properties of determinants, we get

where $\vec{e_1}, \; \vec{e_2}, \; \vec{e_3}$ are unit direction vectors. By Proposition (1) and the above calculations we have

and

Proposition 3. Let $\vec{\mathbb{A}} = \vec{\gamma_{1}}+\varepsilon\vec{\gamma}_{2}$ be a unit dual vector in $\mathbb{D}^{3}$; then the directed line that corresponds to $\vec{\mathbb{A}}$ has an equation of the form

where $0\le \mu \le 1$.

Proof. Let $\vec{\mathbb{A}} = \vec{\gamma}_{1}+\varepsilon\vec{\gamma}_{2}$ be the unit dual vector and if $T$ and $X$ are points on the corresponding line $d$ and $O$ is the origin (see Figure 1), then

where $\lambda$ is a real parameter. A point $X$ is on the line of vectors $\vec{\gamma}_{1}$ and $\vec{\gamma}_{2}$ if and only if

Then,

and

By taking $\mu = \langle \vec{\gamma}_{1}, \vec{t}\; \rangle$ +$\lambda$, we get the result as follows

Theorem 6. Suppose that $\vec{\mathcal{L}}_{0_{n}}$ is a dual Leonardo vector and let $\vec{\mathcal{L}}_{n} = \vec{\mathrm{L}}e_{n}+\mathfrak{ \varepsilon }\vec{\mathrm{Le}}_{n+1}$ be its unitized vector, that is $\vec{\mathcal{L}}_{n}\in \mathbb{S}^2$; then, the equation for the corresponding line is given by

Proof. By using (1.3) and Proposition (3), we obtain

The dual number $\Phi = \phi+\varepsilon\phi^*$ was defined by E. Study in 1903 as a dual angle, where $\phi$ is the angle between oriented lines $l_1$ and $l_2$ in $\mathbb{R}^3$ and $\phi^*$ is the vertical distance between these lines ^[3].

Corollary 2. For unit vectors $\vec{\mathcal{L}}_n$ and $\vec{\mathcal{L}}_m$, we have

The following cases can be given for a dual angle $\Phi$ satisfying that $cos{\Phi } = cos{\phi}-{\varepsilon\phi^*}sin{\phi}$, as wellas the conditions for the corresponding lines ${l}_{1}$ and ${l}_{2}$.

Case 1. Assume that $\phi = \frac{\pi}{2}$ and $\phi^{\ast }\neq 0$; then,

and

Additionally the corresponding lines ${l}_{1}$ and ${l}_{2}$ are perpendicular such that not intersect each other (see Figure 2).

Case 2. Assume that $\phi^{\ast } = 0$ and $\phi\neq0$; then, we obtain

and

in this case lines ${l}_{1}$ and ${l}_{2}$ intersect each other (see Figure 3).

Case 3. Assume that $\phi = \frac{\pi}{2}$ and $\phi^{\ast } = 0$; then,

and

in this case lines ${l}_{1}$ and ${l}_{2}$ intersect each other at a right angle (see Figure 4).

Case 4. Assume that $\phi = 0$ and $\phi^*\neq0$; then,

and

in this case corresponding lines ${l}_{1}$ and ${l}_{2}$ are parallel (see Figure 5).

4.   Conclusions

In this paper, we have introduced the concept of dual Leonardo number sequences and derived the fundamental identities associated with them. Then, we introduced the notion of dual Leonardo vectors and investigated their scalar and cross products. Additionally, we have provided the properties of dual Leonardo vectors, including Study mapping and relationship with dual angles. These findings provide valuable insights into the interference between dual Leonardo number sequences, vectors and geometric concepts within the framework of dual space. Moreover, our explanation of the fundamental identities and properties opens the door further research on the geometry of dual space.

Use of 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 no conflicts of interest.