On Fractional Geometry of Curves

Abstract

Fractional Differential Geometry of curves is discussed, with the help of a new fractional derivative, the Λ-fractional derivative, with the corresponding Λ-fractional space. Λ-Fractional derivative completely conforms with the demands of Differential Topology, for the existence of a differential. Therefore Fractional Differential Geometry is established in that Λ-space. The results are pulled back to the initial space.

1. Introduction

Fractional analysis has recently been considered an indispensable tool in describing real-life models. Although physicists consider fractional calculus as an indispensable tool in their work, the question of the existence of fractional geometry is always negative. Many scientists have wondered how fractional calculus may help in the betted description of various problems, without the existence of fractional geometry. The answer is discussed in the present article, establishing the main principles of fractional differential geometry.

The origins of fractional calculus go back to Leibnitz [1], Liouville [2], and Riemann [3]. Fractional calculus has been employed to describe more intricate real-world models ever since. While conventional mathematical analysis is almost restricted to a local description of a function and fractional analysis is inherently a global one, the latter is considered more suitable for describing the real world. Various viscoelastic responses have been described by fractional differential analysis [4], as well as other physical problems, dependent upon time derivatives [5]. Moreover, problems described by fractals are better expressed through fractional analysis [6]. Further, control problems have been analyzed through fractional calculus. Extensive information about fractional analysis and fractional differential equations are explicated in [7,8,9,10]. Lazopoulos [11] has also introduced an elastic uniaxial model based upon fractional derivatives. This model succeeded in lifting Noll’s axiom of local action. Hence fractional analysis from the solely time-dependent problems, extended to space-dependent ones, just for considering inhomogeneous space fields. Nevertheless, Carpinteri et al. [12] have also introduced a fractional approach considering non-local mechanics. Let us point out that many researchers suggested fractional calculus for solving problems in mechanics [13,14], Jumarie [15,16,17]. Drapaca and Sivaloganathan [18], Sumelka [19] have adopted fractional analysis in problems of continuum mechanics with microstructure, where non-local elasticity is necessary. Another favorite field of fractional continuum mechanics is hydrodynamics [13,20]. Yet, another application of fractional calculus was the description of peridynamic theory [21,22,23]. Nevertheless, viscoelasticity is the main area for fractional calculus applications in Mechanics [4,5]. In addition, fractional differential geometry describes successfully rigid body dynamics, in holonomic and non-holonomic systems [17,24,25,26]. Differential geometry which is revisited by fractional calculus might be found in Quantum Mechanics, Physics, and Relativity theory [27,28], along with applications in various physical areas, may also be found in various books [8,9,10,29]. Different aspects concerning the fractional geometry of manifolds [13,15,16,17,30] have also been presented. Further, researchers attempted to apply fractional differential geometry to various fields of mechanics, quantum mechanics, physics, relativity, finance, probabilities, etc. Indeed, fractal functions exhibiting self-similarity are non-differentiable functions but they exhibit fractional differentiability of order 0 < γ < 1. See reference [6,31,32,33]. However, mathematicians are doubting about the basis of fractional geometry, since the various and many fractional derivatives do not satisfy the requirements of Differential Topology for forming differentials and being able to formulate geometry. Adda [7,34] has proposed for the fractional differential $d^{a}f=g(x){(dx)}^a$ instead of the classical one $df(x)=f^{\prime }(x)dx$, where g(x) is the fractional derivative of a function f(x).

In the first attempt to establish fractional differential, Lazopoulos [35,36] introduced the L-Fractional derivative. Nevertheless, even that fractional derivative failed to satisfy the Differential Topology conditions for the existence of a fractional differential. Lazopoulos then [37] introduced the Λ-fractional derivative that conforms with the conditions required by Differential Topology in the Λ-fractional space conjugate to the original one. Then the various results are pulled back to the original space. No differential geometry is valid in the original space. Hence the establishment of fractional differential geometry is possible only in the Λ-space and the various results may be transferred to the original space. Lazopoulos [38] has already presented the Λ-fractional beam theory, using the proposed Λ-fractional curvature of the elastic curves of the beams. Further, the fractional deformation of a bar based upon the Λ- fractional derivative has also been presented [39]. Recently, the fractional plane elasticity theory with biharmonic functions has been presented [40], along with the discussion of Fractional Taylor’s Series and the Variational Euler–Lagrange Equations [41]. Further, a recent publication deals with the Λ-Fractional Elastic Solid Mechanics theory [42].

In the present work, fractional differential geometry is formulated in the Λ-fractional space for curves. Furthermore, the various results may be transferred back to the original space. Specifically, the fractional differential geometry of curves with their tangent spaces, their normals, the curvature vectors, and the curves of curvature centers is established. In addition, the fractional Serret–Frenet equations are discussed in the Λ-fractional space. Lazopoulos [38] has already presented the Λ-fractional beam theory, using the proposed Λ-fractional curvature of the elastic curves of the beams.

2. Basic Properties of Fractional Calculus

With many applications in engineering and physics, Fractional Calculus has been considered one of the most active fields in applied mathematics. Fractional Calculus has lately become a branch of pure mathematics with many applications in Physics and Engineering. There are many definitions of fractional derivatives. Fractional Calculus was stemmed by Leibniz, looking for the possibility of defining the derivative $\frac{d^{n}g}{d{x}^n}$ when $n=\frac{1}{2}$ the order of derivation is not an integer. The various types of fractional derivatives exhibit some advantages over the other derivatives. Nevertheless, they all are non-local. On the other hand, the conventional derivatives express strictly locality. Information about fractional analysis and its applications may be found in the classical books of Kilbas et al. [29], Podlubny [9], Samko et al. [8].

Recalling the n-fold integral of a function f(x)

$${}_{\mathrm{a}}I{}_x^{n}f\left(x\right)=\frac{1}{(n-1)!}{\displaystyle \underset{\mathrm{a}}{\overset{x}{\int }}{(x-s)}^{n-1}}f(s)ds,\hspace{1em}x>0,n\in N,$$

Leibniz defined the γ-multiple integral with 0 < γ < 1 by,

$${}_a{}^{}I{}_x^{\gamma }f\left(x\right)=\frac{1}{\Gamma \left(\gamma \right)}{{\displaystyle \int }}_a^x\frac{f\left(s\right)}{{\left(x-s\right)}^{1-\gamma }}ds,$$

(1)

With Γ(γ) Euler’s Gamma function.

Further, the left Riemann–Liouville (R-L) derivatives are defined by:

$${}_a{}^{RL}D{}_x^{\gamma }f\left(x\right)=\frac{d}{dx}\left({}_{\alpha }{}^{}I{}_x^{1-\gamma }f\left(x\right)\right)=\frac{1}{\Gamma \left(1-\gamma \right)}\frac{d}{dx}{{\displaystyle \int }}_a^x\frac{f\left(s\right)}{{\left(x-s\right)}^{\gamma }}ds,$$

(2)

With corresponding definitions for the right fractional integrals and derivatives (Poldubny [9]).

3. The Λ-Fractional Derivative

The L-Fractional derivative was introduced by the authors, in an attempt to devise a fractional derivative satisfying the properties of a derivative demanded by the Differential Topology, for the existence of the corresponding differential. Indeed, the Differential Topology requirements for the existence of a differential are (see [34,43,44]):

• Linearity: D(af + bg) (x) = aDf(x) + bDg(x);
• Composition (chain rule) $D\left(f\left(g\right)\right)\left(x\right)=Df\left(g\right)\cdot D\left(g\right)\left(x\right)$;
• Leibniz’s (product) rule: $D(f\cdot g)(x)=Df(x)\cdot g(x)+f(x)\cdot D(g(x)$.

Although the various fractional derivatives satisfy the linearity property, they fail to satisfy the composition and Leibniz’s rules. Lazopoulos [35] introduced the L-fractional derivative in an attempt for the fractional derivative to satisfy the Differential Topology requirements for the existence of differential and hence the existence of fractional differential geometry. Nevertheless, the initial definition of the L-fractional derivative failed to satisfy all the requirements for the existence of differential. The revision of the L-fractional derivative is targeting that purpose.

The Λ-fractional derivative (Λ-FD) has been introduced as:

$${}_a{}^{\mathsf{\Lambda }}D{}_x^{\gamma }f(x)=\frac{{}_{\alpha }{}^{RL}D{}_x^{\gamma }f(x)}{{}_a{}^{RL}D{}_x^{\gamma }x}.$$

(3)

Considering the definition of the fractional derivative, Equation (3), the Λ-FD is expressed by

$${}_a{}^{\mathsf{\Lambda }}D{}_x^{\gamma }f\left(x\right)=\frac{\frac{d{}_{a}I{}_x^{1-\gamma }f(x)}{dx}}{\frac{d{}_{a}I{}_x^{1-\gamma }x}{dx}}=\frac{d{}_{a}I{}_x^{1-\gamma }f(x)}{d{}_{a}I{}_x^{1-\gamma }}$$

(4)

Defining as $X={}_{a}I{}_x^{1-\gamma }x$ and $F(X)={}_a{}^{}I{}_x^{1-\gamma }f(x)$, the Λ-FD is defined as a conventional derivative in the fractional space (X, F(X)). The Fractional Differential Geometry is defined as a conventional differential geometry in the Λ-fractional space, (X, F(X)), with the derivative,

$${}_a{}^{\mathsf{\Lambda }}D{}_x^{\gamma }f\left(x\right)=\frac{dF(X)}{dX}.$$

(5)

Further the relation,

$${}_a{}^{RL}D{}_x^{1-\gamma }({}_{a}I{}_x^{1-\gamma }f(x))=f(x),$$

(6)

is quite important for the pulling back of the various functions from the fractional Λ-space to the original one.

It will be clarified in the application, how, from the initial space (x, f(x)), the fractional Λ-space (X, F(X)) is defined. Further, the pullback of the results in the initial space will also be demonstrated. For simplicity reasons, only the left fractional integrals and derivatives will be taken into consideration. Nevertheless, applications with symmetric space may be found in [21].

Just to clarify the ideas, let us work as an example on the function,

$$f\left(x\right)=x^2.$$

(7)

Then the Λ-fractional plane (X, F(X)) is defined (with a = 0) by

$$X=\frac{x^{2-\gamma }}{\Gamma (3-\gamma )},$$

(8)

and

$$F(X){=}_0{I}_x^{1-\gamma }f\left(x\left(X\right)\right)=\frac{1}{\Gamma \left(1-\gamma \right)}{\displaystyle \underset{0}{\overset{x}{\int }}\frac{s^2}{{\left(x-s\right)}^{\gamma }}}ds=\frac{2}{\Gamma (4-\gamma )}{x}^{\left(3-\gamma \right)}.$$

(9)

Further considering from Equation (8),

$$x={\left(\Gamma \left(3-\gamma \right)\hspace{0.17em}{\rm X}\right)}^{\frac{1}{2-\gamma }},$$

(10)

Equation (10) yields

$$F\left(X\right)={\frac{2\hspace{0.17em}\left({\left(\Gamma \left(3-\gamma \right)X\right)}^{\frac{1}{2-\gamma }}\right)}{\Gamma \left(4-\gamma \right)}}^{3-\gamma }.$$

(11)

Thus, the derivative in the Λ-fractional space is expressed by,

$$\frac{dF\left(X\right)}{dX}=\frac{24{\left(X\Gamma \left(3-\gamma \right)\right)}^{\frac{3}{2-\gamma }}}{\left(2-\gamma \right)\left(3-\gamma \right)\left(4-\gamma \right)}.$$

(12)

For X₀ = 0.6 and γ = 0.6, the derivative in the Λ-fractional plane is equal to D(F(X₀)) = 1.1580. Since the Λ-fractional derivative behaves in the Λ-fractional space, exactly in the conventional way, the tangent Y(X) of the curve at a point X₀ is defined by the line,

$$Y\left(X\right)=F\left(X_0\right)+\frac{d}{dX}\left(F\right(X_0\left)\right)(X-X_0).$$

(13)

Further, the corresponding tangent space, in the original plane (x, f(x), is defined by the curve that will be built as follows:

The x₀ = 0.81 in the initial plane, corresponding to X₀ = 0.60, is defined recalling Equation (10). Then substituting in the derivative $\frac{dF\left(X\right)}{dX}$, Equation (12), the variable X by Equation (8), the ${}_0{}^{\mathsf{\Lambda }}D{}_x^{\gamma }f\left(x\right)$ of the Λ-fractional space is defined as a function of x. Hence the corresponding function in the real space (x, f(x)) may be pulled back by the relation ${}_0{}^{RL}D{}_x^{1-\gamma }\left({}_0{}^{\mathsf{\Lambda }}D{}_x^{\gamma }f\left(x\right)\right).$ Indeed

$${}_0{}^{RL}D{}_x^{1-\gamma }\left({}_0{}^{\mathsf{\Lambda }}D{}_x^{\gamma }f\left(x\right)\right)=\frac{1}{\Gamma \left(\gamma \right)}\frac{d}{dx}{{\displaystyle \int }}_0^x{\left(x-s\right)}^{\gamma -1}{}_0{}^{\mathsf{\Lambda }}D{}_x^{\gamma }f\left(s\right)ds.$$

(14)

In the present case for the function f(x) = x²

$${}_0{}^{RL}D{}_x^{1-\gamma }{\left({}_0{}^{\mathsf{\Lambda }}D{}_x^{\gamma }{x}^2\right)}_{x=0.81}=1.41.$$

(15)

Thus, the fractional tangent space g(x) in the original space (x, f(x)) is defined by

$$g\left(x\right)=f{\left(x\right)}_{x_0}+{}_0{}^{RL}D{}_x^{1-\gamma }{\left({}_0{}^{L}D{}_x^{\gamma }f\left(x\right)\right)}_{x=x_0}\left(\frac{x^{2-\gamma }}{\Gamma (3-\gamma )}-X_0\right)$$

(16)

In the present case at X₀ = 0.6 for γ = 0.6, x₀ = 0.81 the tangent space is defined by

$$g\left(x\right)={\left(x^2\right)}_{x=0.81}+1.41(0.81x^{1.41}-0.6).$$

(17)

More details with illustrating diagrams may be found in [37], since the basic principles of the theory are exhibited there.

4. The Fractional Arc-Length

Let y = f(x) be a function, with fractional derivative of order 0 < γ < 1. The fractional differential in the Λ-fractional plane (X, Y(X)) is defined by:

$$dY(X)=\frac{dY(X)}{dX}\hspace{0.17em}dX,$$

(18)

where X and Y(X) are defined by $X={}_{a}I{}_x^{1-\gamma }x$ and $F(X)={}_a{}^{}I{}_x^{1-\gamma }f(x)$. Then the arc-length in the Λ-fractional plane is defined by:

$$S\left(X\right)={\left(\frac{{\left(dF(X)\right)}^2}{{\left(dX\right)}^2}+1\right)}^{\frac{1}{2}}dX.$$

(19)

Furthermore, the arc-length s(x) in the original plane is defined by,

$$s(x)={}_0{}^{RL}D{}_x^{1-\gamma }(S(X))={}_0{}^{RL}D{}_x^{1-\gamma }(S(\frac{x^{2-\gamma }}{\left(2-3\gamma +{\gamma }^2\right)\Gamma \left(1-\gamma \right)})).$$

(20)

Nevertheless, for the parametric curves of the type:

$$x=g\left(t\right),\hspace{1em}y=f\left(t\right),$$

(21)

The fractional differential of the arc-length in the Λ-fractional plane is expressed by:

$$dS\left(T\right)=\sqrt{{\left(\frac{dY(T)}{dT}\right)}^2+{\left(\frac{dX(T)}{dT}\right)}^2}\hspace{0.17em}dT,$$

(22)

and the arc-length

$$S\left(T\right)={\displaystyle \underset{0}{\overset{T}{\int }}dS({\rm T})}.$$

(23)

The arc-length s(t) in the original plane is defined by the integral equation,

$$s(t)={}_a{}^{RL}D{}_x^{1-\gamma }(S(T))={}_a{}^{RL}D{}_x^{1-\gamma }(S(\frac{t^{2-\gamma }}{\Gamma \left(3-\gamma \right)})).$$

(24)

5. The Fractional Tangent Space of a Space Curve

Let a representation of a space curve C be r = r(s) in the initial space, where s is the fractional length of the curve. Then the fractional tangent space of the curve in the Λ-space is defined by the first-order derivative:

$$R_1=\frac{d^{\gamma }r}{d^{\gamma }s}=\frac{d{I}^{1-\gamma }r}{d{I}^{1-\gamma }s}=\frac{dR(S)}{dS}.$$

(25)

Since,

$$d\left|R(S)\right|=\left|dS\right|,$$

(26)

The length $\left|R_1\right|$ of the tangent vector in the Λ-fractional space is unity. Further, the corresponding tangent vector expressed in variables of the original plane is defined through the equation,

$$R_1(s)=R_1(S)=R_1(\frac{s^{2-\gamma }}{\Gamma \left(3-\gamma \right)}).$$

(27)

The tangent space of the curve r^t = r(s) at the point r₀ = r(s₀) is defined through the Λ-fractional space with,

R^t = R(S₀) + kR₁(S₀) 0 < k,

(28)

and the corresponding tangent space in the original space may be defined with,

$$r^t\left(s\right)=r(s_0)+({}_a{}^{RL}D{}_s^{1-\gamma }{R}_1(\frac{s_0{}^{2-\gamma }}{\Gamma \left(3-\gamma \right)}))\left(\frac{s^{2-\gamma }}{\Gamma \left(3-\gamma \right)}-S_0\right).$$

(29)

The study may go on for the definition of the fractional curvature and fractional radius of curvature in the fractional space. Indeed, following conventional approaches in the Λ-fractional space, the results may be pulled back to the original space.

The plane through R₀ = R(S₀), orthogonal to the tangent line at R₀ defines the normal plane to the curve at S₀. That normal plane in the Λ-fractional space is defined by:

$$\left(Y-R\left(S_0\right)\right)\cdot T\left(S_0\right)=\left(Y-R\left(S_0\right)\right)\cdot R_1\left(S_0\right)=0.$$

(30)

The corresponding normal space y in the original space is defined by,

$$y\left(s\right)={}_a{}^{RL}D{}_s^{\gamma }Y\left(S\left(s\right)\right).$$

(31)

6. Fractional Curvature of Curves

Considering the fractional tangent vector in the Λ- fractional space,

$$T=R_1\left(S\right)=\frac{dR\left(S\right)}{dS}={}_a{}^{\mathsf{\Lambda }}D{}_s^{\gamma }\left(r\left(s\right)\right),$$

(32)

With its Fractional derivative,

$$R_2\left(S\right)=\frac{dT}{dS}={}_a{}^{\mathsf{\Lambda }}D{}_s^{\gamma }\left({}_a{}^{\mathsf{\Lambda }}D{}_s^{\gamma }T\right)=T_1\left(S\right),$$

(33)

The fractional curvature vector K on the curve C at the point R(S) is defined by

$$K=K\left(S\right)=T_1.$$

(34)

In fact, the fractional curvature vector $T_1$ on the curve, considered in the Λ-fractional space, is orthogonal to $T$ and parallel to the fractional normal plane. The fractional curvature of C at R(S), in the Λ-fractional space, is the magnitude of the fractional curvature vector:

$$K=⎣K\left(S\right)⎦.$$

(35)

Likewise, the fractional radius of curvature in the fractional space is defined as the reciprocal of the curvature K at R(S):

$${\rm P}=\frac{1}{{\rm K}}=⎣\frac{1}{{\rm K}\left(S\right)}⎦.$$

(36)

7. The Fractional Serret–Frenet Equations

Let r(s) be a curve with its conjugate in the Λ-fractional space R(S) with unit speed, where the fractional velocity vector,

$$T\left(S\right)=R_1\left(S\right)={}_a{}^{\mathsf{\Lambda }}D{}_s^{\gamma }r\left(s\right)=\frac{dR\left(S\right)}{dS},$$

(37)

is of unit length. Then the vector

$$T_1\left(S\right)=R_2\left(S\right)=\frac{d^{2}R\left(S\right)}{d{S}^2}={}_a{}^{\mathsf{\Lambda }}D{}_s^{\gamma }\left({}_a{}^{\mathsf{\Lambda }}D{}_s^{\gamma }r\left(s\right)\right),$$

(38)

is normal to the curve R = R(S), since $T\left(S\right)·T\left(S\right)=1$ and

$$T_1\left(S\right)·T\left(S\right)=0,$$

(39)

because ${}_{\alpha }{}^{\mathsf{\Lambda }}D{}_s^{\gamma }\beta =0$ for any constant, in the Λ-fractional space.

Further, ${{\rm T}}_1\left(S\right)=K\left(S\right)N\left(S\right)$, with N(S) be the unit principal normal to R at S, provided that K(S) $\ne$ 0, and K(S) denoting the curvature of R at S. Consequently, there is a possibility for the definition of the equations for the fractional focal curve C(S) by,

$$\left(C(S)-R(S)\right)\cdot R_1(S)=0,$$

(40)

$$\left(C\left(S\right)-R\left(S\right)\right)·K\left(S\right)N\left(S\right)=1.$$

(41)

Hence, the center of curvature C(S), in the Λ-fractional space, is defined by the point R(S) + P(S)N(S) with $P\left(S\right)=\frac{1}{K\left(S\right)}$. In addition, the principal normal vector N(S), which is orthogonal to the tangent line, is pointing towards the locus of the curvature centers that is called the focal line in the Λ-fractional space. In that fractional space, the binormal unit vector B(S) = T(S)xN(S) forms a right-oriented orthonormal basis T(S), N(S), B(S) for the tangent vector space of the Λ-fractional mapping R(S) of the initial curve $r(s)$. Additionally, the derivatives of the aforementioned orthonormal basis with respect to S, i.e., T₁(S), N₁(S), B₁(S) depend linearly upon the vectors, T(S), N(S), B(S). Yet, from the evident equations:

$$T_1·T=0\mathrm{with}{T}_1·N=0\mathrm{and}{T}_1·N+N_1·T=0,$$

(42)

The Serret–Frenet equations are formulated in the conjugate Λ-fractional space:

$$T_1=KN,$$

(43)

$$N_1=-KT+\tau {\rm B},$$

(44)

$${{\rm B}}_1=-\tau N.$$

(45)

The coefficient τ is the torsion of the curve R(S) in the Λ-fractional space conjugate of the curve r(s). These equations are the Fractional Equations for the fractional Serret–Frenet system in the fractional Λ-space.

8. The Fractional Radius of Curvature of a Plane Curve

The Λ-fractional curvature or the conjugate curve R(S) in the Λ-fractional space, for a plane curve r(s) in the initial space, is studied in the present chapter. In fact, according to Porteous [19], we study at each point of the fractional curve R(S), how closely the neighborhood of the curve approximates to a parameterized circle. In the Λ-fractional tangent space at a point R(T₀), the circle with center C and radius P is described by all R(T) in the differential space such that:

$$\left(R\left(T\right)-C\left(T\right)\right)·\left(R\left(T\right)-C\left(T\right)\right)=P^2.$$

(46)

Further Equation (41) yields:

$$C·R-\frac{1}{2}R·R=\frac{1}{2}\left(C·C-P^2\right),$$

(47)

with the right-hand side being constant.

Therefore, the differentiation of the function

$$V\left(C\right):T\to C·R\left(T\right)-\frac{1}{2}R\left(T\right)·R\left(T\right),$$

(48)

yields,

$$\frac{dV\left(C\right)}{dT}=\left(C-R\left(T\right)\right)·R_1\left(T\right)=0,$$

(49)

$$\frac{d^{2}V\left(C\right)}{d{T}^2}=\left(C-R\left(T\right)\right)·R_2\left(T\right)-R_1\left(T\right)·R_1\left(T\right)=0.$$

(50)

Suppose that R is a parametric curve with $R(t)$ in the virtual tangent space of the Λ-fractional space. Then Equation (49) indicates that:

$$\frac{dV}{dT}=0$$

(51)

when the tangent vector $R_1\left(T\right)$, in the Λ-fractional space, is orthogonal to the vector $-R\left(T\right)$, that is the normal line. When $R_2\left(T\right)$ is not linearly dependent upon $R_1\left(T\right)$, there will be a unique point $C-R\left(T\right)$ on the normal line such that also

$$\frac{d^{2}V}{d{T}^2}=0.$$

(52)

Specifically, for plane curves in the initial space,

$$r\left(x\right)=x{e}_1+y\left(x\right)e_2.$$

(53)

Hence, the corresponding Λ-fractional space with a = 0 is defined by,

$$X=\frac{1}{\Gamma \left(1-\gamma \right)}{{\displaystyle \int }}_0^x\frac{s}{{\left(x-s\right)}^{\gamma }}ds,Y\left(X\right)=\frac{1}{\Gamma \left(1-\gamma \right)}{{\displaystyle \int }}_0^x\frac{\left(y\left(s\right)-y\left(0\right)\right)}{{\left(x-s\right)}^{\gamma }}ds$$

then, the equations defining the fractional centers of curvature $C=C_1{e}_1+C_2{e}_2$, in the Λ-fractional space, become,

$$\left(C_1-X\right)+\left(C_2-Y(X\right))\frac{dY\left(X\right)}{dX}=0,$$

(54)

$$\left(C_2-Y\left(X\right)\right)\frac{d^{2}Y\left(X\right)}{d{X}^2}-\left(1+{\left(\frac{dY\left(X\right)}{dX}\right)}^2\right)=0.$$

(55)

Likewise, the fractional radius of curvature is defined, in the Λ-fractional space, by,

$${{\rm P}}^{\gamma }={{\rm P}}_1^{\gamma }{e}_1+{{\rm P}}_2^{\gamma }{e}_2=\left(C_1-X\right)e_1+\left(C_2-Y\left(X\right)\right)e_2$$

(56)

hence, the fractional curvature is defined, in the Λ-fractional space, by its components,

$${{\rm P}}_1^{\gamma }=-\frac{1+{\left(\frac{dY\left({\rm X}\right)}{dX}\right)}^2}{\frac{d^{2}Y\left(X\right)}{d{X}^2}}\frac{dY\left(X\right)}{dX},$$

(57)

$${{\rm P}}_2^{\gamma }=\frac{1+{\left(\frac{dY\left({\rm X}\right)}{dX}\right)}^2}{\frac{d^{2}Y\left(X\right)}{d{X}^2}}.$$

(58)

9. Application

The Fractional Geometry of a parabola.

Let $r$ be a parabola $t\to (t,t^2)$. Then its vector equation in the original space is defined by,

$$r(t)=t{e}_1+t^2{e}_2.$$

(59)

Hence the fractional Λ-space is defined by,

$$R\left(T\right)={}_0{}^{}{\rm I}{}_t^{1-\gamma }t{e}_1+{}_0{}^{}{\rm I}{}_t^{1-\gamma }{t}^2{e}_2=\frac{t^{2-\gamma }}{\left(2-3\gamma +{\gamma }^2\right)\Gamma \left(1-\gamma \right)}{e}_1-\frac{2t^{3-\gamma }}{\left(-6+11\gamma -6{\gamma }^2+{\gamma }^3\right)\Gamma \left(1-\gamma \right)}{e}_2.$$

(60)

Consequently,

$$T=\frac{t^{2-\gamma }}{\Gamma \left(3-\gamma \right)}\mathrm{and}\left(T\right)=\frac{2t^{3-\gamma }}{\Gamma \left(4-\gamma \right)}=-\frac{2{\left(\left(2-3\gamma +{\gamma }^2\right){\rm T}\right)}^{\frac{3-\gamma }{2-\gamma }}\Gamma {\left(1-\gamma \right)}^{\frac{1}{2-\gamma }}}{\left(-6+11\gamma -6{\gamma }^2+{\gamma }^3\right)}.$$

The functions y(t) = t² in the original space and (T, Y(T)) in the fractional Λ-space are represented in Figure 1 and Figure 2.

Therefore,

R(T) = Te₁ + Y(T)e₂,

(61)

and

$${\mathbf{R}}_1\left(T\right)={\mathbf{e}}_1+ \frac{dY\left(T\right)}{dT} e_2=e_1-\frac{2{\left(\left(2-3\gamma +{\gamma }^2\right)T\Gamma \left(1-\gamma \right)\right)}^{\frac{1}{2-\gamma }}}{-2+\gamma }{e}_2.$$

(62)

Hence, the tangent G(T) line at a point T₀, in the fractional Λ-space, is defined by:

$$G\left(T\right)=Y\left(T_0\right)+\frac{dY\left(T\right)}{dT}\left[{}_{T_0}{}^{}\left(T-T_0\right)\right].$$

(63)

Figure 3 shows the curve with its tangent, at a point in the Λ-space.

Let us remind that the relation between t and T is,

$$T=\frac{t^{2-\gamma }}{\Gamma \left(3-\gamma \right)}.$$

(64)

The corresponding tangent space in the original space (t, t²) is defined by the curve,

$$f\left(t\right)=t_0^2+{}_0{}^{c}D{}_{t_0}^{1-\gamma }\left(\frac{dY\left({\rm T}\right)}{dT}\right)(\frac{t^{2-\gamma }}{\Gamma \left(3-\gamma \right)}-{\mathrm{T}}_0).$$

(65)

For the specific case of γ = 0.6 and T₀ = 0.6 in the Λ- fractional space, the corresponding point in the original space is defined by t₀ = 0.81. Therefore, the fractional tangent space g(t) in the original plane is described by:

$$g\left(t\right)=0.6561+1.4083\left(0.8051 t^{1.4}-0.6\right).$$

(66)

Figure 4 shows the original curve and the fractional tangent space of the curve.

Furthermore, the curvature centers of the parabola in the Λ-space describe a curve,

$C=C_1\left(T\right)e_1+C_2\left(T\right)e_2$, where Equation (53a,b) define the coefficients C₁(T) and C₂(T).

Consider in the present case,

$$Y\left(T\right)=-\frac{2{\left(\left(2-3\gamma +{\gamma }^2\right){\rm T}\right)}^{\frac{3-\gamma }{2-\gamma }}\Gamma {\left(1-\gamma \right)}^{\frac{1}{2-\gamma }}}{\left(-6+11\gamma -6{\gamma }^2+{\gamma }^3\right)}\mathrm{and}\frac{dY\left(T\right)}{dT}=-\frac{2{\left(\left(2-3\gamma +{\gamma }^2\right)T\Gamma \left(1-\gamma \right)\right)}^{\frac{1}{2-\gamma }}}{-2+\gamma },$$

(67)

$$\frac{d^{2}Y\left(T\right)}{d{T}^2}=\frac{2{\left(\left(2-3\gamma +{\gamma }^2\right)T\Gamma \left(1-\gamma \right)\right)}^{\frac{1}{2-\gamma }}}{{\left(-2+\gamma \right)}^{2}T}.$$

(68)

Hence, solving the system, (Equations (67) and (68)), the curve of curvature centers is defined in the Λ-space by,

$$C_1=T+1.67T^{0.71}\left(-T+0.84 T^{0.29}\left(1+2.78 T^{1.43}\right)\right),$$

(69)

$$C_2=0.84T^{0.29}\left(1+2.78T^{1.43}\right).$$

(70)

Recalling the equation for the curve C(T), the curve of the centers of curvature, and the corresponding conjugate curve Y(T) in the Λ-space is shown in Figure 5.

Proceeding to the definition of the curve c(t) of the image of the curve C(T) in the initial plane (t, y(t)), where

$$c\left(t\right)=c_1{e}_1+c_2{e}_2.$$

(71)

Indeed,

$$c_i\left(t\right)={}_0{}^{c}D{}_{t_0}^{1-\gamma }\left(C_i\left(T\right)\right),$$

(72)

with $T=\frac{t^{2-\gamma }}{\Gamma \left(3-\gamma \right)}$.

Performing the algebra,

$$c_1\left(t\right)=2.4t-1.7t^2+3.9t^3,c_2\left(t\right)=0.7t+3.2t^3.$$

(73)

The image of the curve of curvature centers in the original space is shown in Figure 6.

10. Conclusions

Since the well-known fractional derivatives fail to satisfy the necessary conditions for corresponding to a fractional differential, direct fractional differential geometry is not possible. Adopting the new definition of fractional derivative, the Λ-fractional derivative, along with a new fractional space, the Λ-fractional space, where the Λ-fractional derivative behaves as a conventional one, Fractional differential geometry of the curves is formulated in the Λ-fractional space. Then, the results are transferred into the initial space. The fractional geometry of curves was discussed. The principles of the Fractional geometry presented here may be transferred in manifolds that are of major importance for various applications in mechanics and generally in physics.