Quaternionic Serret-Frenet Frames for Fuzzy Split Quaternion Numbers

TheSerret-Frenet formulas describe the kinematic properties of a particle moving along a continuous and differentiable curve in Euclidean space E3 or Minkowski space E3 1. These formulas are used in many areas such as mathematics, physics (especially in relative theory), medicine, and computer graphics. Quaternions were discovered by SirWilliam R. Hamilton in 1843. The most widely used and most important feature of quaternions is that each unit quaternion represents a transformation. This representation has a special and important role on turns in 3-dimensional vector spaces. This situation is detailed in the study [1]. Nowadays, quaternions are used in many areas such as physics, computer graphics, and animation. For example, visualizing and translating with computer graphics are much easier with quaternions. It is known by especially mathematicians and physicists that any unit (split) quaternion corresponds to a rotation in Euclidean and Minkowski spaces. The notion of a fuzzy subset was introduced by Zadeh [2] and later applied in various mathematical branches. According to the standard condition, a fuzzy number is a convex and a normalized fuzzy subset of real numbers. Basic operations on fuzzy quaternion numbers can be seen in study [3]. There are many applications of quaternions. In physics, we have highlighted applications in quantum mechanics [4] and theory of relativity [5]. In addition, there are applications in aviation projects and flight simulators [6]. On the other hand, the study [7] is a basic study for quaternionic fibonacci forms. All of references that we reviewed guided us to studying the geometry of quaternions. In this paper, we have described the basic operations of fuzzy split quaternions. With this number of structures we aimed to achieve the frenet frame equation. Previously, frenet frame has been created by split quaternions in [8]. In these studies, we obtained Frenet frame by the fuzzy split quaternion.


Introduction
The Serret-Frenet formulas describe the kinematic properties of a particle moving along a continuous and differentiable curve in Euclidean space  3 or Minkowski space  3  1 .These formulas are used in many areas such as mathematics, physics (especially in relative theory), medicine, and computer graphics.
Quaternions were discovered by Sir William R. Hamilton in 1843.The most widely used and most important feature of quaternions is that each unit quaternion represents a transformation.This representation has a special and important role on turns in 3-dimensional vector spaces.This situation is detailed in the study [1].Nowadays, quaternions are used in many areas such as physics, computer graphics, and animation.For example, visualizing and translating with computer graphics are much easier with quaternions.It is known by especially mathematicians and physicists that any unit (split) quaternion corresponds to a rotation in Euclidean and Minkowski spaces.
The notion of a fuzzy subset was introduced by Zadeh [2] and later applied in various mathematical branches.According to the standard condition, a fuzzy number is a convex and a normalized fuzzy subset of real numbers.Basic operations on fuzzy quaternion numbers can be seen in study [3].There are many applications of quaternions.In physics, we have highlighted applications in quantum mechanics [4] and theory of relativity [5].In addition, there are applications in aviation projects and flight simulators [6].On the other hand, the study [7] is a basic study for quaternionic fibonacci forms.All of references that we reviewed guided us to studying the geometry of quaternions.
In this paper, we have described the basic operations of fuzzy split quaternions.With this number of structures we aimed to achieve the frenet frame equation.Previously, frenet frame has been created by split quaternions in [8].
In these studies, we obtained Frenet frame by the fuzzy split quaternion.
Let  →  () be any second-order differentiable space curve with nonvanishing second derivative.We can choose this local coordinate system to be the Serret- The Serret-Frenet frame for the curve  →  () is given as the following differential equation.Writing this frame with matrices is easily for the mathematical calculations. [ The speed value of the curve  →  () is denoted by

Split Quaternion Frames
In this section, firstly we will give the split quaternions definition and their characteristics properties.
∈ } is a vector space over  having basis {1, , , } with the following properties: Every element of the set H  is called a split quaternion.[9].
Definition 2. Let two split quaternions be  =  0 1+ 1 + 2 +  3  and  =  0 1+ 1 + 2 + 3 .These two split quaternions multiplication is calculated as Definition 3. The conjugate of the split quaternion Definition 4. A unit-length split quaternion's norm is Definition 5.Because of H  ≃  4 2 , we can define the timelike, spacelike, and lightlike quaternions for  = ( 0 ,  1 ,  2 ,  3 ) as follows: (i) Spacelike quaternion for   < 0 (ii) Timelike quaternion for   > 0 (iii) Lightlike quaternion for   = 0 Here,   =  = .[1].We can add to Definition 5 following descriptions.Timelike, spacelike, and lightlike vectors are important for the Minkowski space  3  1 .The Minkowski space  3 1 is the accepted common space for the physical reality.We know that the general properties of the quaternions are similar to Minkowski space  4  2 .The Minkowski space  4 2 is a vector space with real dimension  4  and index  2  .Elements of Minkowski space  4  2 are called events or four vectors.On Minkowski space  4  2 , there is an inner product of signature two "plus" and two "minus".Also, we prefer to define the vector structure of Minkowski space with quaternions.
Every possible rotation R (a 3 × 3 special split orthogonal matrix) can be constructed from either one of the two related split quaternions  =  0 1 +  1  +  2  +  3  or − = − 0 1 −  1  −  2  −  3  using the transformation law [8]: where All columns of this matrix expressed in this form are orthogonal but not orthonormal.This matrix form is a special orthogonal group (1, 2).On the other hand, the matrix  can be obtained by the unit split quaternions  and −.
There are two unit timelike quaternions for every rotation in Minkowski 3-space.These timelike quaternions are  and −.
For this reason, a timelike quaternion   can be supposed as a 3 × 3 dimensional orthogonal rotation matrix.
The equations obtained as a result of this coincidence are quaternion valued linear equations.If we derive the column equation of ( 9), respectively, then we obtain the following results:

Serret-Frenet Frames of Split Quaternions
In this section, we give the Serret-Frenet Frame equations for split quaternions.If we calculate the differential equations corresponding to Serret-Frenet Frames with split quaternions, we can obtain the following differential equations.These equations are the formulas Serret-Frenet frames with split quaternions.

Serret-Frenet Frames of Fuzzy Split Quaternions
In this section, we study obtaining the Frenet frame equations with split quaternions in the fuzzy space.For this, firstly we define a fuzzy real set and fuzzy real numbers.(ii) For all  ∈ (0, 1], the set [] = { ∈  : () ≥ } is a limited set.
Similarly, a fuzzy split quaternion number is given by ℎ  : The fuzzy quaternion number's set is denoted by   and the set of all fuzzy split quaternion numbers is denoted by    and identified as  4   , where every element ℎ  is associated with (, , , ).
:  → ,  ∈ ; the function  is said to be fuzzy differentiable at the point  if there is a function  that is fuzzy continuous at the point  and have
Definition 10.Let ℎ  = ( 0 ,  1 ,  2 ,  3 ); the conjugate of ℎ  is defined as The norm of ℎ  is defined as Because of   ⊂    , the following equation can be written: where Here,   is the component of the matrix  and the matrix is calculated from (17) as follows: In this matrix (23), we calculate the derivative of the columns, respectively, to the elements  0 ,  1 ,  2 , and  3 .We will get the Fuzzy tangent vector  →   to the derivation from the first column to the elements  0 ,  1 ,  2 , and  3 : We will get the fuzzy normal vector  →   to the derivation from the second column to the elements  0 ,  1 ,  2 , and  3 : We will get the fuzzy binormal vector  →   to the derivation from the third column to the elements  0 ,  1 ,  2 , and  3 : If we write, respectively, these founded matrices in (11), (12), and (13), we can obtain the following equalities for Serret-Frenet frame equations: The differential of fuzzy split quaternion ℎ  is expressed with matrix form as follows: Here, ( 0 ,  1 ,  2 ,  3 ) is the real and imaginary elements of the fuzzy split quaternionic vector.Now, we must need to calculate the elements   ,   ,   ,   , (0 ≤  ≤ 3) of the coefficient matrix.We need solutions of ( 27), (28), and (29) to obtain the elements   ,   ,   ,   , (0 ≤  ≤ 3).For this reason, we put the differential of fuzzy split quaternion ℎ  , fuzzy tangent vector  →   , fuzzy normal vector  →   , and fuzzy binormal vector  →   in (27), (28), and (29) in its places.When we make the needed calculations, we can obtain the following results: Finally, we get results for the elements   ,   ,   ,   , (0 ≤  ≤ 3) as follows: Therefore, by using these values (35) we obtain the fuzzy split quaternionic Serret-Frenet frame equation as

Conclusion and Discussion
In this study, we redefined the algebraic operations for split quaternions on fuzzy split quaternions.The set of split quaternions is a subset of fuzzy split quaternions (  ⊂    ).This condition is important because the given definitions for fuzzy split quaternions are provided with it.As a result of this, given definitions are similar to definitions for split quaternions.We have seen that these definitions are similar to the split quaternion structures.We have obtained in this study fuzzy tangent vector  →   , fuzzy normal vector  →   , and fuzzy binormal vector  →   .These vector forms are a new description and calculation.Also, we have redefined these Serret-Frenet frames for fuzzy split quaternions on familiar Serret-Frenet frames.For fuzzy quaternionic forms the torsion and curvature functions are defined as  :  ⊂  → [0, 1]  :  ⊂  → [0, 1] For this reason, Serret-Frenet frame elements in (36) for fuzzy split quaternions get values in the range [−1, 1].In Definition 7, we can see that if we take equal fuzzy split quaternion to the split quaternion, the function ℎ  ∈   can take the value  1  and if we take not equal fuzzy split quaternion to the split quaternion, the function ℎ  can take the value  0  .Hence, for calculating (27), (28), and (29), the necessary rule is ℎ  ( 0 1 +  1  +  2  +  3 ) = 1 (38)