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 et al. [5], null curves have been studied by Qian and Kim [15] in three-dimensional Minkowski space E13. Körpınar et al. [11] used the Bishop frame to study these curves. Macit and Düldül [13] defined a W-direction curve by using a unit Darboux vector field W for a given curve and introduced a V-direction curve associated with a curve lying on the surface. They also studied direction curves in four-dimensional Euclidean space.

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 E³ and in Euclidean 4-space E⁴.

A real quaternion, q, is defined as q = a₀e₀ + a₁e₁ + a₂e₂ + a₃e₃ where a_i ( i = 0, 1, 2, 3) are real numbers, and the basis {e₀, e₁, e₂, e₃} has the following properties:

where (ijk) is an even permutation of (123). The algebra for quaternions is denoted by Q.

A real quaternion can also be given the form q = s_q + v_q where s_q = a₀ and v_q = a₁e₁ + a₂e₂ + a₃e₃ are the scalar part and vector part of q, respectively [6]. If q = s_q +v_q and p = s_p+v_p are two quaternions in Q. Then the quaternion product of q and p is given by [6]

where 〈, 〉 and ∧ denote the inner product and vector product of E³, respectively.

The conjugate of q = s_q + v_q is defined by q̄ = s_q − v_q. The term q is called a spatial quaternion whenever q + q̄ = 0 [2]. The quaternion inner product can be defined as follows:

The norm of a quaternion q = a₁e₁ + a₂e₂ + a₃e₃ + a₄e₄ is defined as [6]

If ||q|| = 1, then q is called unit quaternion.

Following the basic concepts above, we can give some definitions and theorems concerning quaternionic curves.

Definition 2.1

([2]) The three-dimensional Euclidean space E³ is identified by the space of the spatial quaternions {q ∈ Q: q + q̄ = 0}. Let I = [0, 1] be an interval in ℝ, s ∈ I be a parameter and Q a set of quaternions. A curve defined as α: I ⊂ ℝ → Q, α(s)=∑i=13αi(s)ei is called a spatial quaternionic curve.

Definition 2.2

([16]) Let α: I ⊂ ℝ → Q be a spatial quaternionic curve and s ∈ I be the arc-length parameter of α. The Frenet vectors and curvatures of the spatial quaternionic curve α(s) can be given, respectively, as follows:

and

where the prime denotes the derivative with respect to s, k(s) and r(s), which are called the curvature and torsion of the spatial quaternionic curve α(s), respectively.

Moreover, the following relationship between the Frenet vectors holds [8]:

Theorem 2.3

([2]) Let α: I ⊂ ℝ → Q, α(s)=∑i=13αi(s)ei be an arc-lengthed spatial quaternionic curve with Frenet frame {t, n, b} and curvatures {k, r}. Then the Frenet formulae of the quaternionic curve α(s) can be given in matrix form as follows:

Definition 2.4

([2]) The four-dimensional Euclidean space E⁴ is identified by the space of the quaternions. Let I = [0, 1] be an interval in ℝ, s ∈ I be a parameter and Q a set of quaternions. A curve defined by β: I ⊂ ℝ → Q, β(s)=∑i=03βi(s)ei is called a quaternionic curve.

Definition 2.5

([16]) Let β: I ⊂ ℝ → Q, β(s)=∑i=03βi(s)ei be a quaternionic curve and s ∈ I be the arc-length parameter of β. The Frenet vectors and curvatures of β(s) can be given respectively as follows:

and

where the prime denotes the derivative with respect to s, K(s), k(s) and r(s)−K(s), which are called the principal curvature, torsion and bitorsion of β, respectively, and B₂(s) × T(s) × N(s) is the ternary product of the vectors.

Theorem 2.6

([2]) Let β: I ⊂ ℝ → Q, β(s)=∑i=03βi(s)ei be a quaternionic curve in E⁴with Frenet frame {T, N, B₁, B₂} and curvatures {K, k, r − K}. Then the Frenet formulae for the quaternionic curve β(s) can be given in matrix form as follows:

Definition 2.7

([9]) A spatial quaternionic curve α is called a spatial quaternionic helix if its unit tangent vector t makes a constant angle with a fixed unit quaternion U.

Theorem 2.8

([9]) Let α be a spatial quaternionic curve with nonzero curvatures. Then α is a spatial quaternionic helix if and only if the following applies:

Definition 2.9

([10]) A spatial quaternionic curve α is called the spatial quaternionic slant helix if its unit normal vector n makes a constant angle with a fixed unit quaternion U.

Theorem 2.10

([10]) Let α be a spatial quaternionic curve with nonzero curvatures. Then α is a spatial quaternionic slant helix if and only if the following applies:

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 α: I → E³ with Frenet frame {t, n, b}, consider a quaternion valued function X given by

where x, y and z are differentiable functions of s which is the arc-length parameter for α. It is assumed that X is unit, i.e, the following equality holds:

By differentiating equation (3.2), we have the following:

The spatial quaternionic curve ᾱ(s̄) defined by dα¯ds¯=X(s) is called the spatial quaternionic integral curve of X(s). Since X is unit, it is clear that the arclength parameter s̄ of ᾱ is equal to s+c, where c is a constant. Without loss of generality, we can assume that s̄ = s. The spatial quaternionic curve ᾱ is unique up to translation.

Now, we can give the definitions for the spatial quaternionic X-direction curve, spatial quaternionic X-donor curve, spatial quaternionic principal-direction curve and the spatial quaternionic principal-donor curve in E³ in the following sections.

Definition 3.1

Let α be a spatial quaternionic curve in E³ and X be a quaternion-valued function satisfying equations (3.1) and (3.2). The spatial quaternionic integral curve ᾱ: I → E³ of X is called the spatial quaternionic X-direction curve of α. The curve α whose spatial quaternionic X-direction curve is ᾱ is called the spatial quaternionic X-donor curve of ᾱ.

Definition 3.2

Let α be a spatial quaternionic curve in E³. The spatial quaternionic integral curve ᾱ of n(s) in (3.1) is called the spatial quaternionic principal-direction curve. In other words, a spatial quaternionic principal-direction curve is a spatial quaternionic integral curve of X(s) with x(s) = z(s) = 0, y(s) = 1 in (3.1). Moreover, α is called the spatial quaternionic principal-donor curve of ᾱ. With the aid of Definition 3.2, we can produce the following theorem:

Theorem 3.3

Let α be a spatial quaternionic curve in E³with Frenet frame {t, n, b} and curvatures {k, r}, and let ᾱ be the spatial quaternionic principal-direction curve of α with Frenet frame {t̄, n̄, b̄} and curvatures {k̄, r̄}. The principal curvature k̄ and the torsion r̄ of the spatial quaternionic principal-direction curve ᾱ of α can be given, respectively, as follows:

Theorem 3.4

Let α be the spatial quaternionic curve in E³with Frenet frame {t, n, b} and curvatures {k, r} and let ᾱ be the spatial quaternionic principal-direction curve of α with Frenet frame {t̄, n̄,b̄} and curvatures {k̄, r̄}. The principal curvature k and the torsion r of spatial quaternionic principal-donor curve α of ᾱ can be given, respectively, as follows:

Corollary 3.5

Let α be a spatial quaternionic curve in E³with curvatures {k, r} and let ᾱ be a spatial quaternionic principal-direction curve of α with curvatures {k̄, r̄}. The following relationship is satisfied:

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.

Theorem 3.6

Let α be a spatial quaternionic curve with nonzero curvatures and let ᾱ be the spatial quaternionic principal-direction curve of α. The curve α is a spatial quaternionic helix if and only if ᾱ is a spatial quaternionic slant helix.

Proposition 3.7

Let α be a spatial quaternionic curve and ᾱ be a spatial quaternionic integral curve of (3.1). The spatial quaternionic principal-direction curve of ᾱ is equal to α if and only if the functions x(s), y(s), z(s) in (3.1) are as follows:

In this section, we give definitions of the quaternionic V–direction and quaternionic principal-direction curves and study the relationship between a given quaternionic curve and its quaternionic principal-direction curve.

Let β: I ⊂ ℝ → Q, β(s)=∑i=03βi(s)ei be a quaternionic curve in E⁴ with Frenet frame {T, N, B₁, B₂} and curvatures {K, k, r − K}, where e_i, (i = 0, 1, 2, 3) are quaternionic units as mentioned in the Preliminaries. Consider a quaternion valued function V in E⁴ given by

where u_i (i = 1, 2, 3, 4) are differentiable functions of s which is the arc-length parameter of β. If we take the quaternion valued function V as unit, the following equality holds

By differentiating equation (4.2), we have

The quaternionic curve β̄(s̄) defined by dβ¯ds¯=V(s) is called the quaternionic integral curve of V (s). Since V (s) is unit, it is clear that the arc-length parameter of β̄ is equal to s+c, where c is a constant. Without loss of generality, we can assume that s̄ = s. The quaternionic curve β̄ is unique up to translation in E⁴.

Now, we can give definitions for the quaternionic V-direction, quaternionic V-donor, quaternionic principal-direction, and quaternionic principal-donor curves.

Definition 4.1

Let β: I ⊂ ℝ → Q, β(s)=∑i=03βi(s)ei be a quaternionic curve and V (s) a quaternion-valued function in E⁴ satisfying equations (4.1) and (4.2). The quaternionic integral curve β̄: I ⊂ ℝ → Q of V (s) is called the quaternionic V-direction curve of β. The quaternionic curve β whose quaternionic V-direction curve is β̄ is called the quaternionic V-donor curve of β̄ in E⁴.

Definition 4.2

Let β: I ⊂ ℝ → Q, β(s)=∑i=03βi(s)ei be a quaternionic curve in E⁴. A quaternionic curve β̄ of N(s) in (4.1) is called the quaternionic principal-direction curve of β, and β is called the quaternionic principal-donor curve of β̄.

By using Definition 4.2, we can produce the following theorem:

Theorem 4.3

Let β be a quaternionic curve in E⁴with Frenet frame {T, N, B₁, B₂} and curvatures {K, k, r–K} and β̄ be the quaternionic principal-direction curve of β with Frenet frame {T̄, N̄, b̄₁, b̄₂} and curvatures { K̄, k̄, r̄− k̄ }. The principal curvature k̄, the torsion k̄, and the bitorsion r̄ − k̄ of the quaternionic principal-direction curve β̄ can be given respectively as:

Proposition 4.4

Let β be a quaternionic curve and β̄ be a quaternionic integral curve of (4.1). The quaternionic principal-direction curve of β̄ is equal to β if and only if u₁ = 0 and the following system of differential equations is satisfied:

where u_i, (i = 1, 2, 3, 4) in (4.1).