On some new frames along a space curve and integral curves with Darboux q-vector fields in $\mathbb{E}^3$

1.   Introduction

The Frenet frame is the best known and the most used frame for characterizing curves in differential geometry. However, there exist other frames for characterizing curves. Some of these alternative frames are the Bishop frame (rotation minimizing frame), the Flc (Frenet-like curve) frame, the $\{\mathsf{N}, \mathsf{C}, \mathsf{W}\}$ frame, etc. The Bishop frame ^[2] is also very suitable for engineering applications ^[10] because of being defined at points where even curvatures vanish. That's why many studies have been done by using this frame. On the other hand, the Bishop frame cannot be calculated analytically ^[6]. Therefore, Dede ^[5] recently introduced a new moving frame called the Flc frame. This frame has much easier calculations and a more analytical form that the Bishop frame does not have. The other alternative frame, $\{\mathsf{N}, \mathsf{C}, \mathsf{W}\}$, has been defined by Scofield ^[14], providing a different approach. Uzunoğlu et al. ^[15] showed that the $\{\mathsf{N}, \mathsf{C}, \mathsf{W}\}$ frame has an important advantage according to the Frenet frame since the expression of slant helix characterization is more short with the new curvatures. As an alternative to the Frenet frame, Dede et al. ^[4] defined a new adapted frame along a space curve. They called this new frame as q-frame and obtained the relations between the Frenet frame and the q-frame.

By using the Darboux frame $\{\mathsf{T}, \mathsf{V}, \mathsf{U}\}$ along a regular curve $\alpha$ lying on an oriented surface $M$ in $\mathbb{E}^3$, Hananoi et al. ^[9] defined the osculating Darboux vector field $\mathsf{D}_o$, the normal Darboux vector field $\mathsf{D}_n$, and the rectifying Darboux vector field $\mathsf{D}_r$, where $\mathsf{T}$ is the unit tangent vector field of $\alpha$, $\mathsf{U}$ is the unit normal vector field of $M$ restricted to $\alpha$, and $\mathsf{V} = \mathsf{U}\times\mathsf{T}$. Considering these vector fields, Önder ^[13] defined three special curves on a surface as $\mathsf{D}_i$-Darboux slant helices, where $i\in\{o, n, r\}$. In recent days, Alkan et al. ^[1] defined osculator Darboux frame, normal Darboux frame, and rectifying Darboux frame.

Recently, many researchers ^{[3,7,8,11,12,16]} have studied associated curves and have revealed the relationships between the main curve and the associated curves. The spherical indicators, the involute-evolute curve couple, the Bertrand curve couple, the Mannheim curve couple, and the integral curves are the most familiar ones among these associated curves. Integral curves are one of the interesting curves among these curves since they are tangent to the vector field at every point.

In this study, considering the Darboux vector field of the q-frame, we first define some new frames called the osculating q-frame, the normal q-frame, and the rectifying q-frame along a space curve in $\mathbb{E}^3$ and obtain their derivative equations. Second, by using some vector fields of these new q-frames, we define new integral curves called $\overline{\mathsf{d}}_o$-direction curve, $\overline{\mathsf{d}}_n$-direction curve, and $\overline{\mathsf{d}}_r$-direction curve of a space curve.

2.   Preliminaries

Let $\alpha = \alpha(t)$ be a regular space curve with nondegenerate condition $\alpha'\times\alpha''\not = 0$. Then, the Frenet frame vector fields are defined as

where $\mathsf{t}$, $\mathsf{n}$, and $\mathsf{b}$ denote the tangent, the principal normal, and the binormal vector fields of the curve $\alpha$ and the prime denotes the derivative with respect to $t$. Then, we have the following Frenet equations:

where $v = ||\alpha'(t)||$, and the curvature and the torsion functions are

respectively.

The q-frame $\{\mathsf{t}, \mathsf{n}_q, \mathsf{b}_q, \mathsf{k}\}$, which is an alternative to the Frenet frame along a space curve $\alpha = \alpha(t)$, is given by Dede et al. ^[4], and its vector fields are defined as

where the vector $\mathsf{k} = (0, 0, 1)$ is the projection vector and the vectors $\mathsf{n}_q$ and $\mathsf{b}_q$ are the quasi-normal and the quasi-binormal vector, respectively. The relation matrix between the Frenet frame and the q-frame is given by

where $\theta$ is the angle between the vectors $\mathsf{n}$ and $\mathsf{n}_q$.

Let $\alpha = \alpha(s)$ be a space curve with arc length parameter $s$. Then, the q-frame equations are given by

where $k_1, k_2, k_3$ denote the q-curvatures of $\alpha$ ^[4]. The relations between the curvatures are

where $\kappa$, $\tau$ are the Frenet curvatures. The Darboux vector $\mathsf d_{q}$ of the q-frame $\{\mathsf{t}, \mathsf{n}_q, \mathsf{b}_q, \mathsf{k}\}$ is given by

Thus, the instantaneous angular speed of the q-frame is calculated as

Let us consider the vector fields $\mathsf{d}_o$, $\mathsf{d}_n$, and $\mathsf{d}_r$ which are called the osculating q-vector field, the normal q-vector field, and the rectifying q-vector field along $\alpha$, respectively:

Then, the unit osculating q-vector field, the unit normal q-vector field, and the unit rectifying q-vector field along $\alpha$ are given by, respectively:

3.   Some new frames along a space curve

In this section, let us define some new frames along a space curve using the Darboux vector field of the q-frame and give some theorems related with these frames.

Definition 3.1. Let $\alpha$ be a space curve in $\mathbb{E}^3$, $\{\mathsf{t}, \mathsf{n}_q, \mathsf{b}_q, \mathsf{k}\}$ be its q-frame, and $\overline{\mathsf{d}}_o$ be the unit osculating q-vector field along the curve $\alpha$. Then, the orthonormal frame $\{\overline{\mathsf{d}}_o, \mathsf{b}_q, \mathsf{e}_o\}$ is called the osculating q-frame along $\alpha$, where $\mathsf{e}_o = \overline{\mathsf{d}}_o\times\mathsf{b}_q$.

Theorem 3.2. Let $\{\overline{\mathsf{d}}_o, \mathsf{b}_q, \mathsf{e}_o\}$ be the osculating q-frame along the curve $\alpha$. Then, the derivative equations according to this frame can be found as

where $\rho_o = \frac{(k'_3k_2-k'_2k_3)+k_1(k_2^2+k_3^2)}{k_2^2+k_3^2}$ and $\eta_o = \sqrt{k_2^2+k_3^2}$ are the curvatures of $\alpha$ according to the osculating q-frame.

Proof. Since $\overline{\mathsf{d}}_o$ is the unit osculating q-vector field along the curve $\alpha$, we get $\overline{\mathsf{d}}_o\in \text{Sp}\{\mathsf{t}, \mathsf{n}_q\}$. So, $\overline{\mathsf{d}}_o\perp\mathsf{b}_q$, and, thus, $\{\overline{\mathsf{d}}_o, \mathsf{b}_q, \mathsf{e}_o\}$ is an orthonormal frame along $\alpha$, where $\mathsf{e}_o = \overline{\mathsf{d}}_o\times\mathsf{b}_q$. Since $\overline{\mathsf{d}}'_o\in \text{Sp}\{\overline{\mathsf{d}}_o, \mathsf{b}_q, \mathsf{e}_o\}$, we can write

Taking the inner product of both sides of this equation with $\overline{\mathsf{d}}_o$ yields $a_1 = \langle\overline{\mathsf{d}}'_o, \overline{\mathsf{d}}_o\rangle = 0$ because of $||\overline{\mathsf{d}}_o|| = 1$. So, we get

If we take the inner product of both sides of Eq (3.2) with $\mathsf{b}_q$ and $\mathsf{e}_o$, respectively, we obtain $a_2 = \langle\overline{\mathsf{d}}'_o, \mathsf{b}_q\rangle$ and $a_3 = \langle\overline{\mathsf{d}}'_o, \mathsf{e}_o\rangle$. Substituting

into Eq (2.2) gives

If we differentiate this equation with respect to s and use Eq (2.1), we have

If we use Eq (3.3), we get $a_2 = 0$. So,

Moreover, using Eq (3.4), we obtain

If we use Eqs (3.5) and (3.6), we obtain $a_3 = -(\phi' +k_1)$. Also, Eq.(3.3) gives $\tan\phi = \frac{k_3}{k_2}$, and by doing some calculations, we find

So, from Eq (3.2), we obtain

In a similar way, we may write ${\mathsf{b}}'_q$ as a linear combination of the vectors $\overline{\mathsf{d}}_o$, $\mathsf{b}_q$, and $\mathsf{e}_o$, i.e.,

Taking the inner product of both sides of Eq (3.8) with $\overline{\mathsf{d}}_o$ yields $b_1 = \langle{\mathsf{b}}'_q, \overline{\mathsf{d}}_o\rangle$. If we use Eqs (2.1), (3.3), and (3.4), we find $b_1 = 0$. Also, since $||{\mathsf{b}}_q|| = 1$, we get $b_2 = 0$. Then, we have

If we take the inner product of both sides of Eq (3.9) with $\mathsf{e}_o$ and use Eqs (2.1), (3.3), and (3.6), we obtain

Moreover, since ${\mathsf{e}}'_o\in \text{Sp}\{\overline{\mathsf{d}}_o, \mathsf{b}_q, \mathsf{e}_o\}$, it can be written as

Taking the inner product of both sides of Eq (3.11) with $\overline{\mathsf{d}}_o$ gives $c_1 = \langle{\mathsf{e}}'_o, \overline{\mathsf{d}}_o\rangle$. Using Eqs (2.1) and (3.6) yields

If we use Eqs (3.3), (3.4), and (3.12), we obtain

From Eqs (3.3), (3.11), and (3.12), we get $c_2 = -\sqrt{k_2^2+k_3^2}$. Also, since $||\mathsf{e}_o|| = 1$, we have $c_3 = 0$. Then, from Eq (3.11), we find

If we denote $\rho_o = \frac{(k'_3k_2-k'_2k_3)+k_1(k_2^2+k_3^2)}{k_2^2+k_3^2}$ and $\eta_o = \sqrt{k_2^2+k_3^2},$ Eqs (3.7), (3.10), and (3.13) give the desired equations. Here, $\rho_o$ and $\eta_o$ are called the curvatures of $\alpha$ according to the osculating q-frame. □

Definition 3.3. Let $\{\overline{\mathsf{d}}_o, \mathsf{b}_q, \mathsf{e}_o\}$ be the osculating q-frame along the space curve $\alpha$. The curve $\alpha$ is called a $\mathsf{b}_q$-slant helix relative to the osculating q-frame if the vector field $\mathsf{b}_q$ makes a constant angle with a fixed direction, i.e., $\langle\mathsf{b}_q, \mathsf{u}\rangle = \cos\psi$, where $\mathsf{u}$ is a constant unit vector and $\psi$ is a constant angle.

Theorem 3.4. Let $\{\overline{\mathsf{d}}_o, \mathsf{b}_q, \mathsf{e}_o\}$ be the osculating q-frame along the space curve $\alpha$. The curve $\alpha$ is a $\mathsf{b}_q$-slant helix relative to the osculating q-frame if, and only if, the expression $\frac{\eta_o}{\rho_o}$ is constant (for $\eta_o\not = 0$ and $\rho_o\not = 0$).

Proof. Let $\alpha$ be a $\mathsf{b}_q$-slant helix relative to the osculating q-frame. Then, $\langle{\mathsf{b}}_q, \mathsf{u}\rangle = \cos\psi = c\not = 0$, where $\mathsf{u}$ is a unit constant direction. So, it can be written as

If we differentiate this equation, we obtain

If we use Eq (3.1), we find

Since $\eta_o\not = 0$ and $\rho_o\not = 0$, we have $\lambda_2 = 0$ and $\lambda_1 = constant$. Thus, we get $\frac{\eta_o}{\rho_o} = constant.$

Conversely, let $\frac{\eta_o}{\rho_o}$ be constant. Choosing $\frac{\eta_o}{\rho_o} = \frac{\cos\psi}{\sin\psi}$ and taking $\mathsf{u} = \cos\psi\overline{\mathsf{d}}_o+\sin\psi\mathsf{b}_q$ gives $\mathsf{u}' = 0$ by using Eq (3.1). So, the vector $\mathsf{u}$ is constant. Also, by taking the inner product of both sides of $\mathsf{u} = \cos\psi\overline{\mathsf{d}}_o+\sin\psi\mathsf{b}_q$ with $\mathsf{b}_q$ yields to $\langle\mathsf{u}, {\mathsf{b}}_q\rangle = \sin\psi$. Then, the constant vector $\mathsf{u}$ and the vector $\mathsf{b}_q$ make a constant angle, i.e., the curve $\alpha$ is a $\mathsf{b}_q$-slant helix. □

Corollary 3.5. A space curve $\alpha$ with $(k_2(s), k_3(s))\not = (0, 0)$ is a $\mathsf{b}_q$-slant helix if, and only if,

is a constant function.

Definition 3.6. Let $\alpha$ be a space curve in $\mathbb{E}^3$, $\{\mathsf{t}, \mathsf{n}_q, \mathsf{b}_q, \mathsf{k}\}$ be its q-frame, and $\overline{\mathsf{d}}_n$ be the unit normal q-vector field along the curve $\alpha$. Then, the orthonormal frame $\{\overline{\mathsf{d}}_n, \mathsf{t}, \mathsf{e}_n\}$ is called the normal q-frame along $\alpha$, where $\mathsf{e}_n = \overline{\mathsf{d}}_n\times\mathsf{t}$.

Theorem 3.7. Let $\{\overline{\mathsf{d}}_n, \mathsf{t}, \mathsf{e}_n\}$ be the normal q-frame along the curve $\alpha$. Then, the derivative equations according to this frame can be obtained as

where $\rho_n = \frac{(k'_2k_1-k'_1k_2)+k_3(k_1^2+k_2^2)}{k_1^2+k_2^2}$ and $\eta_n = \sqrt{k_1^2+k_2^2}$ are the curvatures of $\alpha$ according to the normal q-frame.

Proof. Since $\overline{\mathsf{d}}_n$ is the unit normal q-vector field along $\alpha$, from Eq (2.3), we have $\overline{\mathsf{d}}_n\in \text{Sp}\{\mathsf{n}_q, \mathsf{b}_q\}$. Hence, $\overline{\mathsf{d}}_n\perp\mathsf{t}$. Let $\mathsf{e}_n = \overline{\mathsf{d}}_n\times\mathsf{t}$. Then, we obtain the orthonormal frame $\{\overline{\mathsf{d}}_n, \mathsf{t}, \mathsf{e}_n\}$ along $\alpha$. Since $\overline{\mathsf{d}}'_n\in \text{Sp}\{\overline{\mathsf{d}}_n, \mathsf{t}, \mathsf{e}_n\}$, it can be written as

If we take the inner product of both sides of Eq (3.15) with $\overline{\mathsf{d}}_n$ and take into consideration $||\overline{\mathsf{d}}_n|| = 1$, we get $a_1 = 0$. So, we have

Taking the inner product of both sides of Eq (3.16) with $\mathsf{t}$ and $\mathsf{e}_n$, respectively, yields $a_2 = \langle\overline{\mathsf{d}}'_n, \mathsf{t}\rangle$ and $a_3 = \langle\overline{\mathsf{d}}'_n, \mathsf{e}_n\rangle$. In Eq (2.3), if we take

we obtain

Differentiating this equation with respect to s and using Eq (2.1) gives

If we use Eq (3.17), we get $a_2 = 0$. So, we have

Also, using Eq (3.18) yields

Moreover, from Eqs (3.19) and (3.20), we have $a_3 = -(\phi' +k_3)$. By doing some calculations, we find

If we use Eq (3.16), we have

Similarly, since $\mathsf{t}'\in \text{Sp}\{\overline{\mathsf{d}}_n, \mathsf{t}, \mathsf{e}_n\}$, we can write

If we take the inner product of both sides of Eq (3.22) with $\overline{\mathsf{d}}_n$, we obtain $b_1 = \langle\mathsf{t}', \overline{\mathsf{d}}_n\rangle$. Eqs (2.1), (3.17), and (3.18) give $b_1 = 0$. Also, since $||\mathsf{t}|| = 1$, we have $b_2 = 0$. So, we get

Taking the inner product of both sides of Eq (3.23) with $\mathsf{e}_n$ and using Eqs (2.1), (3.17), and (3.20) yields $b_3 = \sqrt{k_1^2+k_2^2}$ and

Additionally, since ${\mathsf{e}}'_n\in \text{Sp}\{\overline{\mathsf{d}}_n, \mathsf{t}, \mathsf{e}_n\}$, we can write

If we take the inner product of both sides of Eq (3.25) with $\overline{\mathsf{d}}_n$, we get $c_1 = \langle{\mathsf{e}}'_n, \overline{\mathsf{d}}_n\rangle$. Eqs (2.1) and (3.20) give us

If we use Eqs (3.18) and (3.26), we find

Also, using Eqs (3.17), (3.25), and (3.26) yields $c_2 = -\sqrt{k_1^2+k_2^2}$. In addition, since $||\mathsf{e}_n|| = 1$, we have $c_3 = 0$. Thus, using Eq (3.25) gives

If we denote $\rho_n = \frac{(k'_2k_1-k'_1k_2)+k_3(k_1^2+k_2^2)}{k_1^2+k_2^2}$ and $\eta_n = \sqrt{k_1^2+k_2^2}$ and use Eqs (3.21), (3.24), and (3.27), we obtain the desired equations. □

Theorem 3.8. Let $\{\overline{\mathsf{d}}_n, \mathsf{t}, \mathsf{e}_n\}$ be the normal q-frame along the space curve $\alpha$. The curve $\alpha$ is a helix relative to the normal q-frame if, and only if, the expression $\frac{\eta_n}{\rho_n}$ is constant, for $\eta_n\not = 0$ and $\rho_n\not = 0$.

Proof. Let $\alpha$ be a helix relative to the normal q-frame. Then, $\langle\mathsf{t}, \mathsf{u}\rangle = \cos\psi = c\not = 0$, where $\mathsf{u}$ is a unit constant direction. So, we can write

Differentiating this equation with respect to s and using Eq (3.14) gives

Since $\eta_n\not = 0$ and $\rho_n\not = 0$, we have $\lambda_2 = 0$ and $\lambda_1 = constant$. So, we obtain $\frac{\eta_n}{\rho_n} = constant.$

Conversely, let $\frac{\eta_n}{\rho_n}$ be constant. If we choose $\frac{\eta_n}{\rho_n} = \frac{\cos\psi}{\sin\psi}$ and take $\mathsf{u} = \cos\psi\, \overline{\mathsf{d}}_n+\sin\psi\, \mathsf{t}$, we find $\mathsf{u}' = 0$ with the help of Eq (3.14). Then, the vector $\mathsf{u}$ is constant. Additionally, if we take the inner product of both sides of $\mathsf{u} = \cos\psi\, \overline{\mathsf{d}}_n+\sin\psi\, \mathsf{t}$ with $\mathsf{t}$, we get $\langle\mathsf{u}, \mathsf{t}\rangle = \sin\psi$. Consequently, the constant vector $\mathsf{u}$ and the tangent vector $\mathsf{t}$ make a constant angle, i.e., the curve $\alpha$ is a helix. □

Example 3.9. Let us consider the curve $\alpha(t) = (3t-t^3, 3t^2, 3t+t^3)$ in $\mathbb{E}^3$. The q-frame apparatus of the curve $\alpha$ is

with the projection vector $\mathsf{k} = (0, 0, 1)$. Also, the normal q-frame apparatus of the curve $\alpha$ is

Since the expression $\frac{\eta_n}{\rho_n}$ is constant, the curve $\alpha$ is a helix relative to the normal q-frame.

Corollary 3.10. A space curve $\alpha$ with $(k_1(s), k_2(s))\not = (0, 0)$ is a helix if, and only if,

is a constant function.

Definition 3.11. Let $\alpha$ be a space curve in $\mathbb{E}^3$, $\{\mathsf{t}, \mathsf{n}_q, \mathsf{b}_q, \mathsf{k}\}$ be its q-frame, and $\overline{\mathsf{d}}_r$ be the unit rectifying q-vector field along the curve $\alpha$. Then, the orthonormal frame $\{\overline{\mathsf{d}}_r, \mathsf{n}_q, \mathsf{e}_r\}$ is called the rectifying q-frame along $\alpha$, where $\mathsf{e}_r = \overline{\mathsf{d}}_r\times\mathsf{n}_q$.

Theorem 3.12. Let $\{\overline{\mathsf{d}}_r, \mathsf{n}_q, \mathsf{e}_r\}$ be the rectifying q-frame along the curve $\alpha$. Then, the derivative equations according to this frame can be calculated as

where $\rho_r = \frac{(k'_3k_1-k'_1k_3)-k_2(k_1^2+k_3^2)}{k_1^2+k_3^2}$ and $\eta_r = \sqrt{k_1^2+k_3^2}$ are the curvatures of $\alpha$ according to the rectifying q-frame.

Proof. Since $\overline{\mathsf{d}}_r$ is the unit rectifying q-vector field along $\alpha$, we have $\overline{\mathsf{d}}_r\in \text{Sp}\{\mathsf{t}, \mathsf{b}_q\}$. So, $\overline{\mathsf{d}}_r\perp\mathsf{n}_q$ and $\{\overline{\mathsf{d}}_r, \mathsf{n}_q, \mathsf{e}_r\}$ is an orthonormal frame along $\alpha$, where $\mathsf{e}_r = \overline{\mathsf{d}}_r\times\mathsf{n}_q$. Since $\overline{\mathsf{d}}'_r\in \text{Sp}\{\overline{\mathsf{d}}_r, \mathsf{n}_q, \mathsf{e}_r\}$, we can write

If we take the inner product of both sides of Eq (3.28) with $\overline{\mathsf{d}}_r$ and consider $||\overline{\mathsf{d}}_r|| = 1$, we have $a_1 = 0$. So, it can be written as

Taking the inner product of both sides of Eq (3.29) with $\mathsf{n}_q$ and $\mathsf{e}_r$, respectively, yields $a_2 = \langle\overline{\mathsf{d}}'_r, \mathsf{n}_q\rangle$ and $a_3 = \langle\overline{\mathsf{d}}'_r, \mathsf{e}_r\rangle$. In Eq (2.4), denoting

gives

If we differentiate Eq (3.31) with respect to s and use Eq (2.1), we obtain

So, from Eq (3.30), we get $a_2 = 0$ and

Moreover, from Eq (3.31), we have

Also, if we use Eqs (3.32) and (3.33), we obtain $a_3 = -(\phi' -k_2)$. Doing some calculations yields

and from Eq (3.29), it follows that

Similarly, since ${\mathsf{n}}'_q\in \text{Sp}\{\overline{\mathsf{d}}_r, \mathsf{n}_q, \mathsf{e}_r\}$, we can write

If we take the inner product of both sides of Eq (3.35) with $\overline{\mathsf{d}}_r$, we get $b_1 = \langle{\mathsf{n}}'_q, \overline{\mathsf{d}}_r\rangle$. Using Eqs (2.1), (3.30), and (3.31) yields $b_1 = 0$. Also, since $||{\mathsf{n}}_q|| = 1$, we have $b_2 = 0$. Then, we get

Taking the inner product of both sides of Eq (3.36) with $\mathsf{e}_r$ gives $b_3 = \langle{\mathsf{n}}'_q, \mathsf{e}_r\rangle$. If we use Eqs (2.1), (3.30), and (3.33), we obtain $b_3 = \sqrt{k_1^2+k_3^2}$. So, we have

Moreover, since ${\mathsf{e}}'_r\in \text{Sp}\{\overline{\mathsf{d}}_r, \mathsf{n}_q, \mathsf{e}_r\}$, it can be written as

If we take the inner product of both sides of Eq (3.38) with $\overline{\mathsf{d}}_r$, we have $c_1 = \langle{\mathsf{e}}'_r, \overline{\mathsf{d}}_r\rangle$. From Eqs (2.1) and (3.33), we have

If we use Eqs (3.31) and (3.39), we obtain

From Eqs (3.30), (3.38), and (3.39), we find $c_2 = -\sqrt{k_1^2+k_3^2}$. Also, we get $c_3 = 0$, since $||\mathsf{e}_r|| = 1$. So, Eq (3.38) yields

Thus, if we denote

and

we obtain the desired equations from Eqs (3.34), (3.37), and (3.40). □

Definition 3.13. Let $\{\overline{\mathsf{d}}_r, \mathsf{n}_q, \mathsf{e}_r\}$ be the rectifying q-frame along the space curve $\alpha$. The curve $\alpha$ is called an $\mathsf{n}_q$-slant helix relative to the rectifying q-frame if the vector field $\mathsf{n}_q$ makes a constant angle with a fixed direction, i.e., $\langle\mathsf{n}_q, \mathsf{u}\rangle = \cos\psi$, where $\mathsf{u}$ is a constant unit vector and $\psi$ is a constant angle.

Theorem 3.14. Let $\{\overline{\mathsf{d}}_r, \mathsf{n}_q, \mathsf{e}_r\}$ be the rectifying q-frame along the space curve $\alpha$. The curve $\alpha$ is an $\mathsf{n}_q$-slant helix relative to the rectifying q-frame if, and only if, the expression $\frac{\eta_r}{\rho_r}$ is constant (for $\eta_r\not = 0$ and $\rho_r\not = 0$).

Proof. The proof of the theorem can be done in a similar way to the proof of Theorem 3.4. □

Corollary 3.15. A space curve $\alpha$ with $(k_1(s), k_3(s))\not = (0, 0)$ is an $\mathsf{n}_q$-slant helix if, and only if,

is a constant function.

4.   Some integral curves with Darboux q-vector fields

In this section, let us define some new integral curves associated with a space curve using the osculating, the normal, and the rectifying q-frame vector fields.

Definition 4.1. Let $\alpha$ be a space curve in $\mathbb{E}^3$, $\{\mathsf{t}, \mathsf{n}_q, \mathsf{b}_q, \mathsf{k}\}$ be the q-frame along $\alpha$, and $\overline{\mathsf{d}}_o$ be the unit osculating q-frame vector field of $\alpha$. The integral curve of the vector field $\overline{\mathsf{d}}_o$ is called the $\overline{\mathsf{d}}_o$-direction curve of $\alpha$. Namely, if $\gamma(s)$ is the $\overline{\mathsf{d}}_o$-direction curve of $\alpha$, then $\gamma(s) = \int\overline{\mathsf{d}}_o(s)$ or $\overline{\mathsf{d}}_o(s) = \gamma'(s)$.

Now, let us find the Frenet apparatus $\{\mathsf{t}_\gamma, \mathsf{n}_\gamma, \mathsf{b}_\gamma, \kappa_\gamma, \tau_\gamma\}$ of $\gamma$. Since $\gamma$ is the $\overline{\mathsf{d}}_o$-direction curve of $\alpha$, it can be written $\gamma' = \overline{\mathsf{d}}_o$. So, the tangent vector $\mathsf{t}_\gamma$ of $\gamma$ is

If we differentiate this equation with respect to $s$ and use Eq (2.1), we obtain

Since $\rho_o = \frac{(k'_3k_2-k'_2k_3)+k_1(k_2^2+k_3^2)}{k_2^2+k_3^2},$ Eq (4.1) can be rewritten as

So, we have $||\overline{\mathsf{d}}'_o|| = \varepsilon\rho_o$, where $\varepsilon = \pm1$. Since $\mathsf{n}_{\gamma} = \frac{\overline{\mathsf{d}}'_o}{||\overline{\mathsf{d}}'_o||}$, we get

Additionally, since $\mathsf{b}_\gamma = \mathsf{t}_\gamma\times\mathsf{n}_\gamma$, the binormal vector is obtained as $\mathsf{b}_\gamma = \varepsilon\mathsf{b}_q.$ Moreover, the curvature and the torsion of $\gamma$ can be found as

Then, the following corollary can be given.

Corollary 4.2. Let $\gamma$ be the $\overline{\mathsf{d}}_o$-direction curve of a space curve $\alpha$. Then, the Frenet apparatus $\{\mathsf{t}_\gamma, \mathsf{n}_\gamma, \mathsf{b}_\gamma, \kappa_\gamma, \tau_\gamma\}$ of $\gamma$ can be obtained as

where $\varepsilon = \pm1$.

Corollary 4.3. $\gamma$ is a general helix if, and only if, $\alpha$ is a $\mathsf{b}_q$-slant helix.

Definition 4.4. Let $\alpha$ be a space curve in $\mathbb{E}^3$, $\{\mathsf{t}, \mathsf{n}_q, \mathsf{b}_q, \mathsf{k}\}$ be the q-frame along $\alpha$, and $\overline{\mathsf{d}}_n$ be the unit normal q-frame vector field of $\alpha$. The integral curve of the vector field $\overline{\mathsf{d}}_n$ is called the $\overline{\mathsf{d}}_n$-direction curve of $\alpha$. Namely, if $\zeta(s)$ is the $\overline{\mathsf{d}}_n$-direction curve of $\alpha$, then $\zeta(s) = \int\overline{\mathsf{d}}_n(s)$ or $\overline{\mathsf{d}}_n(s) = \zeta'(s)$.

Similarly, let us obtain the Frenet apparatus $\{\mathsf{t}_\zeta, \mathsf{n}_\zeta, \mathsf{b}_\zeta, \kappa_\zeta, \tau_\zeta\}$ of the curve $\zeta$. Taking into account the definition of the $\overline{\mathsf{d}}_n$-direction curve gives

Differentiating this equation with respect to $s$ and applying Eq (2.1) yields

Since $\rho_n = \frac{(k'_2k_1-k'_1k_2)+k_3(k_1^2+k_2^2)}{k_1^2+k_2^2},$ Eq (4.2) can be rewritten as

Then, we get $||\overline{\mathsf{d}}'_n|| = \varepsilon\rho_n$, where $\varepsilon = \pm1$. Thus, since $\mathsf{n}_{\zeta} = \frac{\overline{\mathsf{d}}'_n}{||\overline{\mathsf{d}}'_n||}$, we obtain

Also, the binormal vector $\mathsf{b}_\zeta$ is given by $\mathsf{b}_\zeta = \varepsilon\mathsf{t}.$ Besides, the curvatures of the $\overline{\mathsf{d}}_n$-direction curve $\zeta$ can be obtained as

Then, we can give the following corollary.

Corollary 4.5. Let $\zeta$ be the $\overline{\mathsf{d}}_n$-direction curve of a space curve $\alpha$. Then, the Frenet apparatus $\{\mathsf{t}_\zeta, \mathsf{n}_\zeta, \mathsf{b}_\zeta, \kappa_\zeta, \tau_\zeta\}$ of $\zeta$ can be found as

where $\varepsilon = \pm1$.

Corollary 4.6. $\zeta$ is a general helix if, and only if, $\alpha$ is a general helix.

Definition 4.7. Let $\alpha$ be a space curve in $\mathbb{E}^3$, $\{\mathsf{t}, \mathsf{n}_q, \mathsf{b}_q, \mathsf{k}\}$ be the q-frame along $\alpha$, and $\overline{\mathsf{d}}_r$ be the unit rectifying q-frame vector field of $\alpha$. The integral curve of the vector field $\overline{\mathsf{d}}_r$ is called the $\overline{\mathsf{d}}_r$-direction curve of $\alpha$. Namely, if $\varphi(s)$ is the $\overline{\mathsf{d}}_r$-direction curve of $\alpha$, then $\varphi(s) = \int\overline{\mathsf{d}}_r(s)$ or $\overline{\mathsf{d}}_r(s) = \varphi'(s)$.

In a similar way, let us calculate the Frenet apparatus $\{\mathsf{t}_\varphi, \mathsf{n}_\varphi, \mathsf{b}_\varphi, \kappa_\varphi, \tau_\varphi\}$ of $\varphi$. From the definition of the $\overline{\mathsf{d}}_r$-direction curve, we have

If we differentiate this equation with respect to $s$, we find

Since $\rho_r = \frac{(k'_3k_1-k'_1k_3)-k_2(k_1^2+k_3^2)}{k_1^2+k_3^2},$ from Eq (4.3) we obtain

Then, we have $||\overline{\mathsf{d}}'_r|| = \varepsilon\rho_r$, where $\varepsilon = \pm1$. Since $\mathsf{n}_{\varphi} = \frac{\overline{\mathsf{d}}'_r}{||\overline{\mathsf{d}}'_r||}$, we get

Also, the binormal vector $\mathsf{b}_\varphi$ of the curve $\varphi$ is found as $\mathsf{b}_\varphi = \varepsilon\mathsf{n}_q.$ In addition, the curvature and the torsion of the $\overline{\mathsf{d}}_r$-direction curve $\varphi$ can be obtained as

Then, the following corollary can be given.

Corollary 4.8. Let $\varphi$ be the $\overline{\mathsf{d}}_r$-direction curve of a space curve $\alpha$. Then, the Frenet apparatus $\{\mathsf{t}_\varphi, \mathsf{n}_\varphi, \mathsf{b}_\varphi, \kappa_\varphi, \tau_\varphi\}$ of $\varphi$ can be obtained as

where $\varepsilon = \pm1$.

Corollary 4.9. $\varphi$ is a general helix if, and only if, $\alpha$ is an $\mathsf{n}_q$-slant helix.

5.   Conclusions

In this study, we obtained the derivative equations of the osculating q-frame, the normal q-frame, and the rectifying q-frame which have been defined along a space curve by using the Darboux vector field of the q-frame in Euclidean 3-space. Then, we defined some new slant helices and new integral curves and gave their characterizations.

Use of AI tools declaration

The author declares she has not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

The author declares no conflicts of interest.