In this paper, we define new quaternionic associated curves called quaternionic principal-direction curves and quaternionic principal-donor curves. We give some properties and relationships between Frenet vectors and curvatures of these curves. For spatial quaternionic curves, we give characterizations for quaternionic helices and quaternionic slant helices by means of their associated curves.
Curves with a mathematical relationship between them are called associated curves. The study of associated curves is an interesting and important research area of the fundamental theory of curves. Some properties, such as Frenet vectors and curvatures of original curves, can be characterized by using their associated curves. Within this area of interest, various associated curves have been defined, such as Bertrand curve mates, Mannheim partner curves and involute-evolute curve couples, and they have been studied in different spaces, such as Euclidean spaces, Minkowski spaces, and dual spaces [1, 3, 7, 12, 14, 17].
Recently, Choi and Kim [4] introduced a new associated curve for a given curve as the integral curve of the vector field generated by the Frenet frame along it. This associated curve has been called the direction curve. Using these associated curves provides a canonical method to construct general helices and slant helices, which are widely used in various research areas in science and nature. These curves have attracted many authors to begin to study them. While non-null direction curves have been studied by Choi
In this paper, we give the definition of quaternionic direction curves as the quaternionic integral curve of a quaternion-valued function generated by Frenet vectors for a given quaternionic curve. Then, by taking this quaternion-valued function as the principal normal vector field of the curve, we define principal-direction and principal-donor curves for spatial quaternionic and quaternionic curves. We provide relationships between Frenet vectors and curvatures of a given quaternionic curve and its quaternionic principal-direction curve, and then obtain some properties of these curves. Moreover, for spatial quaternionic curves, we provide characterizations for quaternionic helices and quaternionic slant helices with the aid of their associated curves.
In this section, we give a brief summary of basic concepts concerning quaternionic curves and some definitions and theorems about these curves in Euclidean 3-space
A real quaternion,
where (
A real quaternion can also be given the form
where 〈, 〉 and ∧ denote the inner product and vector product of
The conjugate of
The norm of a quaternion
If ||
Following the basic concepts above, we can give some definitions and theorems concerning quaternionic curves.
([2]) The three-dimensional Euclidean space
([16]) Let
and
where the prime denotes the derivative with respect to
Moreover, the following relationship between the Frenet vectors holds [8]:
([2])
([2]) The four-dimensional Euclidean space
([16]) Let
and
where the prime denotes the derivative with respect to
([2])
([9]) A spatial quaternionic curve
([9])
([10]) A spatial quaternionic curve
([10])
In this section, we define spatial quaternionic principal-direction and principal-donor curves in Euclidean 3-space and obtain some relationships between these curves.
For a spatial quaternionic curve
where
By differentiating
The spatial quaternionic curve
Now, we can give the definitions for the spatial quaternionic
Let
Let
Since
and
From Definition 2.2, we get
and
respectively.
On the other hand, the curvatures of spatial quaternionic principal-donor curve
Let the ratio
This means that
By putting
and
From Theorem 3.3, we have the following corollary.
Thus, we can produce the following theorem that can be used to construct a spatial quaternionic slant helix from a spatial quaternionic helix by using the spatial quaternionic direction curves.
The proof is clear from Corollary 3.5, Theorem 2.10, and Theorem 2.12.
Now, we can discuss the condition where the spatial quaternionic principal-direction curve of
By differentiating (
On the other hand, since
Multiplying the first, second and third equations in (
From
In this section, we give definitions of the quaternionic
Let
where
By differentiating
The quaternionic curve
Now, we can give definitions for the quaternionic
Let
Let
By using Definition 4.2, we can produce the following theorem:
Since
By taking the third order derivative of
By using Definition 2.5, the torsion and bitorsion of the quaternionic principal-direction curve
and
Moreover, the first and second binormal vector fields of the quaternionic principal-direction curve
and
Now, we can discuss the condition where the quaternionic principal-direction curve of
By differentiating (
Since
Multiplying the first, second, third and fourth equations in (
From
Specifically, if the ratio
By differentiating the second equation and using the third equation of system (
where
By using the third equation of system (
Thus, we have a solution of system (
where