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.