In this section, first we define the spherical indicatrices of a framed curve and we investigate the relations between framed slant helices and their spherical indicatrices.

### 4.1
*v*-Indicatrices of framed slant helices

### Definition 4.1

Let \((\gamma ,\overline{\mu }_{1},\overline{\mu }_{2}):I\rightarrow \mathbb{R}^{3}\times \Delta _{2}\) be a framed curve. Its *v*-indicatrix is the framed curve \((\beta ,\overline{\mu }_{1_{\beta }},\overline{\mu }_{2_{\beta }} ):I\rightarrow S^{2}\times \Delta _{2}\) defined by

$$ \beta (s)=v(s) \quad \text{for all $s\in I$}. $$

The adapted frame apparatus of *β* is given by the notation \(\lbrace v_{\beta },\overline{\mu }_{1_{\beta }},\overline{\mu }_{2_{\beta }} \rbrace \). Clearly, there exists a smooth mapping \(\alpha _{\beta }:I\rightarrow \mathbb{R}\) such that:

$$ \beta ^{\prime }(s)=\alpha _{\beta }(s)v_{\beta }(s). $$

### Theorem 4.1

*Let* \((\gamma ,\overline{\mu }_{1},\overline{\mu }_{2})\) *be a framed curve in* \(\mathbb{R}^{3}\) *and* \((\beta ,\overline{\mu }_{1_{\beta }},\overline{\mu }_{2_{\beta }} )\) *be the framed* *v*-*indicatrix of* *γ*. *Then*, *γ* *is a framed slant helix if and only if* *β* *is a framed helix*.

### Proof

We assume that *γ* is a framed slant helix in \(\mathbb{R}^{3}\) and *β* is the framed *v*-indicatrix of *γ*. From Definition 4.1, we have

then, differentiating the last equation according to parameter *s* and using equation (1), we obtain

$$ \beta ^{\prime }(s)=v^{\prime }(s), $$

that is

$$ \alpha _{\beta }(s)v_{\beta }(s)=p(s)\overline{\mu }_{1}(s). $$

From the norm of the last equation, we obtain

$$ \alpha _{\beta }(s)=p(s). $$

Hence, we obtain the following equation:

$$ v_{\beta }(s)=\overline{\mu }_{1}(s). $$

(6)

If we differentiate the last equation and use equation (1), we obtain

$$ p_{\beta }(s)\overline{\mu }_{1_{\beta }}(s)=-p(s)v(s)+q(s) \overline{\mu }_{2}(s). $$

From the norm of the above equation, we obtain

$$ p_{\beta }(s)=p(s)\sqrt{1+H^{2}(s)}. $$

(7)

Hence, we obtain the following equation

$$ \overline{\mu }_{1_{\beta }}(s)=-\frac{1}{\sqrt{1+H^{2}(s)}}v(s)+ \frac{H(s)}{\sqrt{1+H^{2}(s)}}\overline{\mu }_{2}(s). $$

(8)

Then, using equations (6) and (8), we obtain

$$ \overline{\mu }_{2_{\beta }}(s)=v_{\beta }(s)\times \overline{\mu }_{1_{\beta }}(s)= \frac{H(s)}{\sqrt{1+H^{2}(s)}}v(s)+\frac{1}{\sqrt{1+H^{2}(s)}} \overline{ \mu }_{2}(s). $$

(9)

From the norm of the derivative of the last equation, we obtain the following equation:

$$ q_{\beta }(s)=\frac{H^{\prime }(s)}{1+H^{2}(s)}. $$

(10)

Then, we can readily see that \(\frac{q_{\beta }(s)}{p_{\beta }(s)}= \frac{H^{\prime }(s)}{p(s)(1+H^{2}(s))^{\frac{3}{2}}}=\sigma \) is a constant function since *γ* is a framed slant helix. In other words, using Theorem 2.3 we can readily see that *β* is a framed helix.

Conversely, if we assume that *β* is a framed helix then it is clear that *γ* is a framed slant helix. This completes the proof. □

### Corollary 4.1

*The* *v*-*indicatrix curve* *β* *of a framed curve* *γ* *is a regular framed curve*.

### Proof

It is obvious from the equation \(\alpha _{\beta }(s)=p(s)>0\) in the proof of Theorem 4.1. □

### Corollary 4.2

*Let* \((\gamma ,\overline{\mu }_{1},\overline{\mu }_{2})\) *be a framed curve with the adapted frame* \(\lbrace v(s),\overline{\mu }_{1}(s),\overline{\mu }_{2}(s) \rbrace \) *in* \(\mathbb{R}^{3}\) *and* \((\beta ,\overline{\mu }_{1_{\beta }},\overline{\mu }_{2_{\beta }} )\) *be the framed* *v*-*indicatrix of* *γ* *with the adapted frame* \(\lbrace v_{\beta }(s),\overline{\mu }_{1_{\beta }}(s), \overline{\mu }_{2_{\beta }}(s) \rbrace \). *Then*, *we have the following relations between the adapted frames of* *γ* *and* *β*

$$\begin{aligned}& v_{\beta }(s) =\overline{\mu }_{1}(s), \\& \overline{\mu }_{1_{\beta }}(s) =-\frac{1}{\sqrt{1+H^{2}(s)}}v(s)+ \frac{H(s)}{\sqrt{1+H^{2}(s)}}\overline{\mu }_{2}(s), \\& \overline{\mu }_{2_{\beta }}(s) =\frac{H(s)}{\sqrt{1+H^{2}(s)}}v(s)+ \frac{1}{\sqrt{1+H^{2}(s)}}\overline{\mu }_{2}(s), \end{aligned}$$

*where* *H* *is the framed harmonic curvature function of the curve* *γ*.

### Proof

It is obvious from equations (6), (8) and (9). □

### Corollary 4.3

*Let* \((\gamma ,\overline{\mu }_{1},\overline{\mu }_{2})\) *be a framed curve with the framed curvatures* \(p(s)\), \(q(s)\) *in* \(\mathbb{R}^{3}\) *and* \((\beta ,\overline{\mu }_{1_{\beta }},\overline{\mu }_{2_{\beta }} )\) *be the framed* *v*-*indicatrix of* *γ* *with the framed curvatures* \(p_{\beta }(s)\), \(q_{\beta }(s)\). *Then*, *the relations between these framed curvatures functions are*

$$\begin{aligned}& p_{\beta }(s) =p(s)\sqrt{1+H^{2}(s)}, \\& q_{\beta }(s) =\frac{H^{\prime }(s)}{1+H^{2}(s)}, \end{aligned}$$

*where* *H* *is the framed harmonic curvature function of the curve* *γ*.

### Proof

It is obvious from equations (7) and (10). □

### 4.2
\(\overline{\mu }_{1}\)-Indicatrices of framed slant helices

### Definition 4.2

Let \((\gamma ,\overline{\mu }_{1},\overline{\mu }_{2}):I\rightarrow \mathbb{R}^{3}\times \Delta _{2}\) be a framed curve. Its \(\overline{\mu }_{1}\)-indicatrix is the framed curve \((\eta ,\overline{\mu }_{1_{\eta }},\overline{\mu }_{2_{\eta }} ):I \rightarrow S^{2}\times \Delta _{2}\) defined by

$$ \eta (s)=\overline{\mu }_{1}(s) \quad \text{for all $s\in I$}. $$

The adapted frame apparatus of *η* is given by the notation \(\lbrace v_{\eta },\overline{\mu }_{1_{\eta }},\overline{\mu }_{2_{\eta }} \rbrace \). Clearly, there exists a smooth mapping \(\alpha _{\eta }:I\rightarrow \mathbb{R}\) such that:

$$ \eta ^{\prime }(s)=\alpha _{\eta }(s)v_{\eta }(s). $$

### Theorem 4.2

*Let* \((\gamma ,\overline{\mu }_{1},\overline{\mu }_{2})\) *be a framed slant helix in* \(\mathbb{R}^{3}\) *and* \((\eta ,\overline{\mu }_{1_{\eta }},\overline{\mu }_{2_{\eta }} )\) *be the framed* \(\overline{\mu }_{1}\)-*indicatrix of* *γ*. *Then*, *the curve* *η* *is a plane curve on* \(S^{2}\).

### Proof

Let *γ* be a framed slant helix in \(\mathbb{R}^{3}\) and *η* a framed \(\overline{\mu }_{1}\)-indicatrix of *γ*. From Definition 4.2, we have

$$ \eta (s)=\overline{\mu }_{1}(s). $$

(11)

Then, differentiating equation (11) according to the parameter *s* and using equation (1), we obtain

$$ \eta ^{\prime }(s)=\overline{\mu }^{\prime }_{1}(s) $$

or

$$ \alpha _{\eta }(s)v_{\eta }(s)=-p(s)v(s)+q(s)\overline{\mu }_{2}(s). $$

From the norm of the last equality, we obtain

$$ \alpha _{\eta }(s)=p(s)\sqrt{1+H^{2}(s)}. $$

Hence, we obtain the following equation:

$$ v_{\eta }(s)=-\frac{1}{\sqrt{1+H^{2}(s)}}v(s)+ \frac{H(s)}{\sqrt{1+H^{2}(s)}}\overline{\mu }_{2}(s). $$

(12)

If we differentiate the last equation and use equation (1), we obtain

$$ p_{\eta }(s)\overline{\mu }_{1_{\eta }}(s)= \frac{H(s)H^{\prime }(s)}{ ( 1+H^{2}(s) )^{\frac{3}{2}}}v(s)- p(s) \sqrt{1+H^{2}(s)}\overline{\mu }_{1}(s)+ \frac{H^{\prime }(s)}{ ( 1+H^{2}(s) )^{\frac{3}{2}}} \overline{\mu }_{2}(s). $$

Since *γ* is a framed slant helix \(\sigma (s)\) is a constant function. Hence, we can obtain

$$ p_{\eta }(s)\overline{\mu }_{1}(s)=p(s) \sigma H(s)v(s)-p(s)\sqrt{1+H^{2}(s)} \overline{\mu }_{1}(s)+p(s)\sigma \overline{\mu }_{2}(s). $$

(13)

Then, the norm of the last equation gives us

$$ p_{\eta }(s)=p(s)\sqrt{\bigl(1+H^{2}(s)\bigr) \bigl(1+ \sigma ^{2}\bigr)} $$

and so

$$ \overline{\mu }_{1_{\eta }}(s)= \frac{\sigma H(s)}{\sqrt{(1+H^{2}(s))(1+\sigma ^{2})}}v(s)- \frac{1}{\sqrt{1+\sigma ^{2}}}\overline{\mu }_{1}(s)+ \frac{\sigma }{\sqrt{(1+H^{2}(s))(1+\sigma ^{2})}} \overline{\mu }_{2}(s). $$

(14)

Then, using equations (12) and (14), we obtain

$$\begin{aligned} \overline{\mu }_{2_{\eta }}(s)&=v_{\eta }(s) \times \overline{\mu }_{1_{\eta }}(s) \\ &=\frac{H(s)}{\sqrt{(1+H^{2}(s))(1+\sigma ^{2})}}v(s)+ \frac{\sigma }{\sqrt{1+\sigma ^{2}}}\overline{\mu _{1}}(s) \\ &\quad {}+\frac{1}{\sqrt{(1+H^{2}(s))(1+\sigma ^{2})}}\overline{\mu _{2}}(s). \end{aligned}$$

(15)

From the norm of the derivative of \(\overline{\mu }_{2}(s_{\eta })\), we obtain

Hence, *γ* is a plane curve. This completes the proof. □

### Corollary 4.4

*The* \(\overline{\mu }_{1}\)-*indicatrix curve* *η* *of a framed curve* *γ* *is a regular framed curve*.

### Proof

It is obvious from the equation \(\alpha _{\eta }(s)=p(s)\sqrt{1+H^{2}}>0\) in the proof of Theorem 4.2. □

### Corollary 4.5

*Let* \((\gamma ,\overline{\mu }_{1},\overline{\mu }_{2})\) *be a framed curve with the adapted frame* \(\lbrace v(s),\overline{\mu }_{1}(s),\overline{\mu }_{2}(s) \rbrace \) *in* \(\mathbb{R}^{3}\) *and* \((\eta ,\overline{\mu }_{1_{\eta }},\overline{\mu }_{2_{\eta }} )\) *be the framed* \(\overline{\mu }_{1}\)-*indicatrix of* *γ* *with the adapted frame* \(\lbrace v_{\eta }(s), \overline{\mu }_{1_{\eta }}(s),\overline{\mu }_{2_{\eta }}(s) \rbrace \). *Then*, *we have the following relations between the adapted frames of* *γ* *and η*

$$\begin{aligned}& v_{\eta }(s) =-\frac{1}{\sqrt{1+H^{2}(s)}}v(s)+ \frac{H(s)}{\sqrt{1+H^{2}(s)}} \overline{\mu }_{2}(s), \\& \overline{\mu }_{1_{\eta }}(s) =\frac{1}{\sqrt{1+\sigma ^{2}}} \biggl( \frac{\sigma H(s)}{\sqrt{1+H^{2}(s)}}v(s)-\overline{\mu }_{1}(s)+ \frac{\sigma }{\sqrt{1+H^{2}(s)}}\overline{\mu }_{2}(s) \biggr), \\& \overline{\mu }_{2_{\eta }}(s) =\frac{1}{\sqrt{1+\sigma ^{2}}} \biggl( \frac{H(s)}{\sqrt{1+H^{2}(s)}}v(s)+\sigma \overline{\mu _{1}}(s)+ \frac{1}{\sqrt{1+H^{2}(s)}}\overline{\mu _{2}}(s) \biggr), \end{aligned}$$

*where* *H* *is the framed harmonic curvature function of the curve* *γ*.

### Proof

It is obvious from equations (12), (14) and (15). □

### 4.3
\(\overline{\mu }_{2}\)-Indicatrices of framed slant helices

### Definition 4.3

Let \((\gamma ,\overline{\mu }_{1},\overline{\mu }_{2}):I\rightarrow \mathbb{R}^{3}\times \Delta _{2}\) be a framed curve. Its \(\overline{\mu }_{2}\)-indicatrix is the framed curve \((\delta ,\overline{\mu }_{1_{\delta }},\overline{\mu }_{2_{\delta }} ):I\rightarrow S^{2}\times \Delta _{2}\) defined by

$$ \delta (s)=\overline{\mu }_{2}(s) \quad \text{for all $s\in I$}. $$

The adapted frame apparatus of *δ* is given by the notation \(\lbrace v_{\delta },\overline{\mu }_{1_{\delta }},\overline{\mu }_{2_{\delta }} \rbrace \). Clearly, there exists a smooth mapping \(\alpha _{\delta }:I\rightarrow \mathbb{R}\) such that:

$$ \delta ^{\prime }(s)=\alpha _{\delta }(s)v_{\delta }(s_{\delta }). $$

### Theorem 4.3

*Let* \((\gamma ,\overline{\mu }_{1},\overline{\mu }_{2})\) *be a framed curve in* \(\mathbb{R}^{3}\) *and* \((\delta ,\overline{\mu }_{1_{\delta }},\overline{\mu }_{2_{\delta }} )\) *be the framed* \(\mu _{2}\)-*indicatrix of* *γ*. *Then*, *γ* *is a framed slant helix if and only if* *δ* *is a framed helix*.

### Proof

We assume that *γ* is a framed slant helix in \(\mathbb{R}^{3}\) and *δ* is the framed \(\overline{\mu }_{2}\)-indicatrix of *γ*. From Definition 4.3, we have

$$ \delta (s)=\overline{\mu }_{2}(s). $$

Then, differentiating the last equation according to the parameter *s* and using equation (1), we obtain

$$ \delta ^{\prime }(s)=\overline{\mu }^{\prime }_{2}(s) $$

that is

$$ \alpha _{\delta }(s)v_{\delta }(s)=-p(s)H(s)\overline{\mu }_{1}(s), $$

where \(H(s)\) is the framed harmonic curvature function of *γ*. From the norm of the above equation, assuming that \(\epsilon =1\) if \(H>0\) or \(\epsilon =-1\) if \(H<0\), we obtain

$$ \alpha _{\delta }(s)=\epsilon p(s)H(s). $$

Hence, we obtain the following equation:

$$ v_{\delta }(s)=-\epsilon \overline{\mu }_{1}(s). $$

(16)

If we differentiate the last equation and use equation (1), we obtain

$$ p_{\delta }(s)\overline{\mu }_{1_{\delta }}(s)=-\epsilon \bigl(-p(s)v(s)+q(s) \overline{\mu }_{2}(s)\bigr). $$

From the norm of the last equation, we obtain

$$ p_{\delta }(s)=p(s)\sqrt{1+H^{2}(s)}. $$

(17)

Hence, we have

$$ \overline{\mu }_{1_{\delta }}(s)=\frac{\epsilon }{\sqrt{1+H^{2}(s)}}v(s)- \frac{\epsilon H(s)}{\sqrt{1+H^{2}(s)}}\overline{\mu }_{2}(s). $$

(18)

Then, using equations (16) and (18), we have

$$ \overline{\mu }_{2_{\delta }}(s)=v_{\delta }(s)\times \overline{\mu }_{1_{\delta }}(s)=\frac{H(s)}{\sqrt{1+H^{2}(s)}}v(s)+ \frac{1}{\sqrt{1+H^{2}(s)}}\overline{ \mu }_{2}(s). $$

(19)

From the norm of the derivative of the last equation, we obtain the following equation:

$$ q_{\delta }(s)=\frac{H^{\prime }(s)}{1+H^{2}(s)}. $$

(20)

Then, we can readily see that \(\frac{q_{\delta }(s)}{p_{\delta }(s)}= \frac{H^{\prime }(s)}{p(s)(1+H^{2}(s))^{\frac{3}{2}}}=\sigma \) is a constant function since *γ* is a framed slant helix. In other words, using Theorem 2.3 we can see readily that *δ* is a framed helix.

Conversely, we assume that *δ* is a framed helix then it is clear that *γ* is a framed slant helix. This completes the proof. □

### Corollary 4.6

*Let* \((\gamma ,\overline{\mu }_{1},\overline{\mu }_{2})\) *be a framed curve with the adapted frame* \(\lbrace v(s),\overline{\mu }_{1}(s),\overline{\mu }_{2}(s) \rbrace \) *in* \(\mathbb{R}^{3}\) *and* \((\delta ,\overline{\mu }_{1_{\delta }},\overline{\mu }_{2_{\delta }} )\) *be the framed* \(\overline{\mu }_{2}\)-*indicatrix of* *γ* *with the adapted frame* \(\lbrace v_{\delta }(s), \overline{\mu }_{1_{\delta }}(s), \overline{\mu }_{2_{\delta }}(s) \rbrace \). *Then*, *we have the following relations between the adapted frames of* *γ* *and δ*

$$\begin{aligned}& v_{\delta }(s) =-\epsilon \overline{\mu }_{1}(s), \\& \overline{\mu }_{1_{\delta }}(s) =\frac{\epsilon }{\sqrt{1+H^{2}(s)}}v(s)- \frac{\epsilon H(s)}{\sqrt{1+H^{2}(s)}}\overline{\mu }_{2}(s), \\& \overline{\mu }_{2_{\delta }}(s) =\frac{H(s)}{\sqrt{1+H^{2}(s)}}v(s)+ \frac{1}{\sqrt{1+H^{2}(s)}}\overline{\mu }_{2}(s), \end{aligned}$$

*where* \(\epsilon =1\) *if* \(H>0\) *or* \(\epsilon =-1\) *if* \(H<0\), *H* *is the framed harmonic curvature function of the curve* *γ*.

### Proof

It is obvious from equations (16), (18) and (19). □

### Corollary 4.7

*Let* \((\gamma ,\overline{\mu }_{1},\overline{\mu }_{2})\) *be a framed curve with the framed curvatures* \(p(s)\), \(q(s)\) *in* \(\mathbb{R}^{3}\) *and* \((\delta ,\overline{\mu }_{1_{\delta }},\overline{\mu }_{2_{\delta }} )\) *be the framed* \(\overline{\mu }_{2}\)-*indicatrix of* *γ* *with the framed curvatures* \(p_{\delta }(s)\), \(q_{\delta }(s)\). *Then*, *the relations between these framed curvatures functions are*

$$\begin{aligned}& p_{\delta }(s) =p(s)\sqrt{1+H^{2}(s)}, \\& q_{\delta }(s) =\frac{H^{\prime }(s)}{1+H^{2}(s)}, \end{aligned}$$

*where* *H* *is the framed harmonic curvature function of the curve* *γ*.

### Proof

It is obvious from equations (17) and (20). □

### Example 4.1

Let \(\gamma : (-2\pi ,2\pi )\subset \mathbb{R}\rightarrow \mathbb{R}^{3}\) be a curve defined by

$$\begin{aligned} \gamma (t) =& \frac{\sqrt{6}}{5} \biggl(\sin \biggl(\frac{3 t}{5} \biggr)- \frac{2}{7} \sin \biggl(\frac{7 t}{5} \biggr)- \frac{\sin (t)}{5},- \cos \biggl(\frac{3 t}{5} \biggr)+ \frac{2}{7} \cos \biggl( \frac{7 t}{5} \biggr)+ \frac{\cos (t)}{5}, \\ &\frac{2 \sqrt{6} t}{5}-\sqrt{6} \sin \biggl( \frac{2 t}{5} \biggr) \biggr). \end{aligned}$$

The curve *γ* has a singular point at \(t=0\), so that it is not a Frenet curve. On the other hand, the curve *γ* is a framed curve with the mapping \((\gamma ,\overline{\mu }_{1},\overline{\mu }_{2}): (-2\pi ,2\pi )\subset \mathbb{R}\rightarrow \mathbb{R}^{3}\times \Delta _{2}\). The adapted frame vectors of the framed curve \((\gamma ,\overline{\mu }_{1},\overline{\mu }_{2})\) are given by

$$\begin{aligned}& v(t)= \frac{1}{5} \biggl(3 \sin \biggl(\frac{4 t}{5} \biggr)+2 \sin \biggl(\frac{6 t}{5} \biggr),-3 \cos \biggl(\frac{4 t}{5} \biggr)-2 \cos \biggl(\frac{6 t}{5} \biggr),\frac{2\sqrt{6}}{5} \sin \biggl( \frac{t}{5} \biggr) \biggr), \\& \overline{\mu }_{1}= \biggl(\frac{2}{5} \sqrt{6} \cos (t),\frac{2}{5} \sqrt{6} \sin (t),\frac{1}{5} \biggr), \\& \overline{\mu }_{2}= \frac{1}{5} \biggl(2 \cos \biggl( \frac{6 t}{5} \biggr)-3 \cos \biggl(\frac{4 t}{5} \biggr),2 \sin \biggl( \frac{6 t}{5} \biggr)-3 \sin \biggl(\frac{4 t}{5} \biggr), \frac{2 \sqrt{6}}{5} \cos \biggl(\frac{t}{5} \biggr) \biggr). \end{aligned}$$

Also, the framed curvatures of the framed curve \((\gamma ,\overline{\mu }_{1},\overline{\mu }_{2})\) are as follows:

$$ p(t)=\frac{2}{5} \sqrt{6} \cos \biggl(\frac{t}{5} \biggr) \quad \text{and}\quad q(t)=\frac{2}{5} \sqrt{6} \sin \biggl( \frac{t}{5} \biggr). $$

Moreover, we can readily see that the \(\sigma =\frac{1}{2\sqrt{6}}\) for the framed curve \((\gamma ,\overline{\mu }_{1},\overline{\mu }_{2})\) with the help of Theorem 3.1, so it is a framed slant helix. Finally, we show Figs. 1–4, which are the framed slant helix, the *v*-indicatrix of *γ*, the \(\overline{ \mu }_{1}\)-indicatrix of *γ*, and the \(\overline{\mu }_{2}\)-indicatrix of *γ*, respectively.