In this section, we introduce a special spacelike equiform-Bishop Smarandache curves according to the equiform-Bishop frame in Minkowski 3-space \(\mathbb {R}_{1}^{3}\). Furthermore, we obtain the natural curvature functions of these curves and studying some properties on it when the spacelike base curve r=r(s) specially is contained in a plane. Let r=r(σ) be a regular unit speed spacelike curve with spacelike equiform-Bishop principal normal and timelike equiform-Bishop binormal.
Definition 2
Let \(r:I\subset \mathbb {R}\rightarrow S_{1}^{2}\) be a regular unit speed spacelike curve lying fully on \(S^{2}_{1}\). The spacelike equiform-Bishop TB1-Smarandache curve \({\varphi }:I\subset \mathbb {R}\rightarrow S_{1}^{2}\) of r defined by
$$ \varphi=\varphi(\sigma^{\ast})=\frac{1}{\sqrt{2}\,\rho}\left(a\,T(\sigma)+b\,B_{1}(\sigma)\right),{\quad} a^{2}+ b^{2}=2. $$
(6)
Theorem 1
Let \(r:I\subset \mathbb {R}\rightarrow S_{1}^{2}\) be a regular unit speed spacelike curve lying fully on \(S^{2}_{1}\) with the moving equiform-Bishop frame {T,B1,B2}. If \({\varphi }:I\subset \mathbb {R}\rightarrow S_{1}^{2}\) is the spacelike equiform-Bishop TB1-Smarandache curve of r=r(σ) with non-zero natural curvature functions, then its Frenet frame {Tφ,Nφ,Bφ} is given by
$$ \begin{aligned} \left(\begin{array}{c} T_{\varphi} \\ N_{\varphi} \\ B_{\varphi} \end{array} \right)=\left(\begin{array}{ccc} \frac{-b}{\rho\sqrt{b^{2}+a^{2}(1-K^{2}_{2})}} & \frac{a}{\rho\sqrt{b^{2}+a^{2}(1-K^{2}_{2})}} & \frac{-{aK}_{2}}{\rho\sqrt{b^{2}+a^{2}(1-K^{2}_{2})}} \\ \frac{\omega_{1}}{\rho\sqrt{\omega_{1}^{2}+\omega_{2}^{2}-\omega_{3}^{2}}} & \frac{\omega_{2}}{\rho\sqrt{\omega_{1}^{2}+\omega_{2}^{2}-\omega_{3}^{2}}}& \frac{{\omega}_{3}}{\rho\sqrt{{\omega}_{1}^{2}+{\omega}_{2}^{2}-{\omega}_{3}^{2}}}\\ \frac{-a(\omega_{3}+\omega_{2}K_{2})}{\Delta_{1}}& \frac{a\omega_{1}K_{2}-b\omega_{3}}{\Delta_{1}} & \frac{-(a\omega_{1}+b\omega_{2})}{\Delta_{1}} \end{array} \right)\left(\begin{array}{c} T \\ B_{1} \\ B_{2} \end{array} \right), \end{aligned} $$
(7)
where
$$ \begin{aligned} &\omega_{1}=a\left(K^{2}_{2}-1\right)\left[b^{2}+a^{2}\left(1-K^{2}_{2}\right)\right]-2a^{2}{bK}_{2}K'_{2}, \\&\omega_{2}=2a^{3}K_{2}K'_{2}-b\left[b^{2}+a^{2}\left(1-K^{2}_{2}\right)\right],\\ &\omega_{3}=\left({bK}_{2}-aK'_{2}\right)\left[b^{2}+a^{2}\left(1-K^{2}_{2}\right)\right]-2a^{3}K^{2}_{2}K'_{2},\\ &\Delta_{1}=\rho^{2}\sqrt{\omega_{1}^{2}+\omega_{2}^{2}-\omega_{3}^{2}}\sqrt{b^{2}+a^{2}\left(1-K^{2}_{2}\right)}. \end{aligned} $$
(8)
Proof
Differentiationg Eq. (6) with respect to σ and using Eq. (5), we get
$$ \varphi'(\sigma^{\ast})=\frac{d\varphi}{d\sigma^{\ast}}\frac{d\sigma^{\ast}}{d\sigma}=\frac{1}{\sqrt{2}\,\rho}\left(b\,T(\sigma)+a\,B_{1}(\sigma)-{aK}_{2}B_{2}(\sigma)\right), $$
(9)
hence
$$ {}T_{\varphi}(\sigma^{\ast})=\frac{1}{\rho\sqrt{b^{2}+a^{2}\left(1-K^{2}_{2}\right)}}\left(b\,T(\sigma)+a\,B_{1}(\sigma)-{aK}_{2}B_{2}(\sigma)\right), $$
(10)
with the parameterization
$$ \frac{d\sigma^{\ast}}{d\sigma}=\frac{\rho\sqrt{b^{2}+a^{2}\left(1-K^{2}_{2}\right)}}{\sqrt{2}}\,. $$
(11)
Again differentiating Eq. (10) with respect to σ, we have
$${\kern-14.5pt}T'_{\varphi}(\sigma^{\ast})\,=\,\frac{\sqrt{2}}{\rho\left[b^{2}+a^{2}\left(1-K^{2}_{2}\right)\right]^{2}}\left(\omega_{1}T(\sigma)+\omega_{2}B_{1}(\sigma)+\omega_{3} B_{2}(\sigma)\right). $$
where
$$\begin{aligned} &\omega_{1}=a\left(K^{2}_{2}-1\right)\left[b^{2}+a^{2}\left(1-K^{2}_{2}\right)\right]-2a^{2}{bK}_{2}K'_{2}, \\&\omega_{2}=2a^{3}K_{2}K'_{2}-b\left[b^{2}+a^{2}\left(1-K^{2}_{2}\right)\right],\\ &\omega_{3}=\left({bK}_{2}-aK'_{2}\right)\left[b^{2}+a^{2}\left(1-K^{2}_{2}\right)\right]-2a^{3}K^{2}_{2}K'_{2}. \end{aligned} $$
The curvature and the principal normal of φ are given as follows
$${\kappa}_{\varphi}(\sigma^{\ast})=\left\|T'_{\varphi}(\sigma^{\ast})\right\|=\frac{\sqrt{2}\sqrt{\omega_{1}^{2}+\omega_{2}^{2}-\omega_{3}^{2}}}{\left[b^{2}+a^{2}\left(1-K^{2}_{2}\right)\right]^{2}}, $$
and
$$N_{\varphi}(\sigma^{\ast})=\frac{\omega_{1}T(\sigma)+\omega_{2}B_{1}(\sigma)+\omega_{3}B_{2}(\sigma)}{\rho\sqrt{\omega_{1}^{2}+\omega_{2}^{2}-\omega_{3}^{2}}}. $$
On the other hand, we can express
$$\begin{aligned} B_{\varphi}(\sigma^{\ast})&=\frac{1}{\Delta_{1}}\left\{-a(\omega_{3}+\omega_{2}K_{2})T(\sigma)+a\omega_{1}K_{2}-b\omega_{3}B_{1}(\sigma)\right.\\ & \left.\quad-(a\omega_{1}+b\omega_{2})B_{2}(\sigma)\right\}, \end{aligned} $$
where
$$\Delta_{1}=\rho^{2}\sqrt{\omega_{1}^{2}+\omega_{2}^{2}-\omega_{3}^{2}}\sqrt{b^{2}+a^{2}\left(1-K^{2}_{2}\right)}. $$
Now, from Eq. (9), we have
$${\begin{aligned} \varphi^{\prime\prime}(\sigma^{\ast})\,=\, \frac{1}{\sqrt{2}\,\rho}&\left\{a\left(K^{2}_{2}-1\right)T(\sigma)\,-\,b\,B_{1}(\sigma) +\left({bK}_{2}-aK'_{2}\right)B_{2}(\sigma) \right\}, \end{aligned}} $$
similarly
$$\varphi^{\prime\prime\prime}(\sigma^{\ast})= \frac{1}{\sqrt{2}\,\rho}\left(\mu_{1}T(\sigma)+\mu_{2}B_{1}(\sigma)+\mu_{3}B_{2}(\sigma)\right), $$
where
$$\begin{aligned} &\mu_{1}=aK'_{2}(1+k_{2})+K_{1}({aK}_{2}+{bK}_{1}-a)+2bK'_{1}, \\ &\mu_{2}=({aK}_{2}+{bK}_{1}-a)-b\left(K_{1}K'_{1}+ K^{\prime\prime}_{1}\right),\\ &\mu_{3}=- K_{2}({aK}_{2}+{bK}_{1}-a)-a\left(K_{1}K^{\prime}_{1}+ K^{\prime\prime}_{2}\right). \end{aligned} $$
As a consequence with the above computation, the torsion of φ is obtained as
$${\begin{aligned} {\tau}_{\varphi} \,=\,\frac{\!\sqrt{2}}{\rho}\!\left\{\!\frac{\left[{aK}_{2}+{bK}_{1}-a\right]\left[\mu_{3}(a-{bK}_{1})-a\mu_{2}K_{2}\right]-a\mu_{1}\left[K'_{2}({bK}_{1}-a)-bK'_{1}K_{2}\right]} {\left[a^{2}K'_{2}-2{abK}_{2}\right]^{2}+\left[b\left(aK'_{2}-{bK}_{2}\right)-a^{2}K_{2}\left(K^{2}_{2}-1\right)\right]^{2}-\left[b^{2}-a^{2}\left(K^{2}_{2}-1\right)\right]^{2}} \right\}\,. \end{aligned}} $$
□
Corollary 1
Let \(r:I\subset \mathbb {R}\rightarrow S_{1}^{2}\) be a regular unit speed spacelike curve lying fully on \(S^{2}_{1}\) with the moving equiform-Bishop frame {T,B1,B2}. If the base curve r=r(s) is contained in a plane, then the spacelike equiform-Bishop TB1-Smarandache curve is a circular helix if \(\,K_{2}\neq \pm \frac {\sqrt {2}}{a}\) and K2≠±1. Moreover, its natural curvature functions are dependent only on the second equiform-Bishop curvature and given by
$$ {\begin{aligned} &{\kappa}_{\varphi}(\sigma^{\ast})=\frac{\sqrt{2}\sqrt{a^{2}\left(1-K^{2}_{2}\right)^{2}+b^{2}\left(1-K^{2}_{2}\right)}} {b^{2}+a^{2}(1-K^{2}_{2})},\\& {\tau}_{\varphi}(\sigma^{\ast})=\left\{\frac{\sqrt{2}}{\rho}\right\}\left\{\frac{\sqrt{2}(1+a)(1-K_{2})} {4a^{2}b^{2}K^{2}_{2}+\left(K^{2}_{2}-1\right)\left[b^{2}-a^{2}\left(K^{2}_{2}-1\right)\right]^{2}}\right\}. \end{aligned}} $$
(12)
Definition 3
Let \(r:I\subset \mathbb {R}\rightarrow S_{1}^{2}\) be a regular unit speed spacelike curve lying fully on \(S^{2}_{1}\). The spacelike equiform-Bishop TB2-Smarandache curve \({\varphi }:I\subset \mathbb {R}\rightarrow S_{1}^{2}\) of r defined by
$$ \varphi=\varphi(\sigma^{\ast})=\frac{1}{\sqrt{2}\,\rho}\left(a\,T(\sigma)+b\,B_{2}(\sigma)\right),{\quad} a^{2}- b^{2}=2. $$
(13)
Theorem 2
Let \(r:I\subset \mathbb {R}\rightarrow S_{1}^{2}\) be a regular unit speed spacelike curve lying fully on \(S^{2}_{1}\) with the moving equiform-Bishop frame {T,B1,B2}. If \({\varphi }:I\subset \mathbb {R}\rightarrow S_{1}^{2}\) is the spacelike equiform-Bishop TB2-Smarandache curve of r=r(σ) with non-zero natural curvature functions, then its Frenet frame {Tφ,Nφ,Bφ} is given by
$$ {\left(\begin{array}{c} T_{\varphi} \\ N_{\varphi} \\ B_{\varphi} \end{array} \right)}\,=\,\left(\!\begin{array}{ccc} \frac{- {bK}_{2}}{\rho\sqrt{b^{2}K^{2}_{2}+ a^{2}(1-K^{2}_{2})}} & \frac{a}{\rho\sqrt{b^{2}K^{2}_{2}+ a^{2}(1-K^{2}_{2})}} & \frac{aK_{2}}{\rho\sqrt{b^{2}K^{2}_{2}+ a^{2}(1-K^{2}_{2})}} \\ \frac{\xi_{1}}{\rho\sqrt{\xi_{1}^{2}+\xi_{2}^{2}-\xi_{3}^{2}}} & \frac{\xi_{2}}{\rho\sqrt{\xi_{1}^{2}+\xi_{2}^{2}-\xi_{3}^{2}}}& \frac{\xi_{3}}{\rho\sqrt{\xi_{1}^{2}+\xi_{2}^{2}-\xi_{3}^{2}}} \\ \frac{a(\xi_{2}K_{2}-\xi_{3})}{\Delta_{2}}& \frac{a\xi_{1}K_{2}- b\xi_{3}K_{2}}{\Delta_{2}} & \frac{-(a\xi_{1}+ b\xi_{2}K_{2})}{\Delta_{2}} \end{array} \!\right)\!\!\left(\begin{array}{c}T \\ B_{1} \\ B_{2} \end{array} \right)\!, $$
(14)
where
$$ \begin{aligned} &\xi_{1}=\left[{bK}_{1}+a (K_{2}-1)\right]\left[(a-{bK}_{2})^{2}- a^{2}K^{2}_{2}\right], \\&\xi_{2}=-bK'_{1}\left[(a-{bK}_{2})^{2}- a^{2}K^{2}_{2}\right]+(a-{bK}_{1})\left[bK'_{2}(a-{bK}_{2})+ a^{2}K_{2}K'_{2}\right],\\&\xi_{3}=-2({aK}_{1}K_{2}+K'_{2})\left[(a-{bK}_{2})^{2}- a^{2}K^{2}_{2}\right]+{aK}_{2}\left[bK'_{2}(a-{bK}_{2})- a^{2}K_{2}K'_{2}\right],\\&\Delta_{2}=\rho^{2}\sqrt{\xi_{1}^{2}+\xi_{2}^{2}-\xi_{3}^{2}}\sqrt{b^{2}K^{2}_{2}+ a^{2}(1-K^{2}_{2})}:{\quad} K_{2}\neq\frac{\pm a}{\sqrt{a^{2}- b^{2}}}. \end{aligned} $$
(15)
Proof
Differentiationg Eq. (13) with respect to σ and using Eq. (5), we have
$$ {\begin{aligned}{ \varphi'(\sigma^{\ast})=\frac{d\varphi}{d\sigma^{\ast}}\frac{d\sigma^{\ast}}{d\sigma}=\frac{1}{\sqrt{2}\,\rho}\left({bK}_{2}\,T(\sigma)+a\, B_{1}(\sigma)+{aK}_{2}\,B_{2}(\sigma)\right),} \end{aligned}} $$
(16)
then, we have
$$ {\begin{aligned} T_{\varphi}(\sigma^{\ast})=\frac{1}{\rho\sqrt{b^{2}K^{2}_{2}+ a^{2}\left(1-K^{2}_{2}\right)}}\left({bK}_{2}\,T(\sigma)+a\, B_{1}(\sigma)+{aK}_{2}\,B_{2}(\sigma)\right), \end{aligned}} $$
(17)
where
$$ \frac{d\sigma^{\ast}}{d\sigma}=\frac{\sqrt{b^{2}K^{2}_{2}+ a^{2}\left(1-K^{2}_{2}\right)}}{\sqrt{2}}\,. $$
(18)
Then
$${}{\begin{aligned} T'_{\varphi}(\sigma^{\ast})&=\frac{\sqrt{2}}{\rho\left[b^{2}K^{2}_{2}+ a^{2}\left(1-K^{2}_{2}\right)\right]^{2}}\left(\xi_{1}T(\sigma)+\xi_{2}B_{1}(\sigma)\right.\\ & \left. \quad+\xi_{3} B_{2}(\sigma)\right), \end{aligned}} $$
where
$${\begin{aligned} &\xi_{1}=\left[{bK}_{1}+a (K_{2}-1)\right]\left[(a-{bK}_{2})^{2}- a^{2}K^{2}_{2}\right], \\&\xi_{2}=-bK'_{1}\left[(a-{bK}_{2})^{2}- a^{2}K^{2}_{2}\right]+(a-{bK}_{1})\left[bK'_{2}(a-{bK}_{2})+ a^{2}K_{2}K'_{2}\right],\\ &\xi_{3}=-2\left({aK}_{1}K_{2}+K'_{2}\right)\left[(a-{bK}_{2})^{2}-a^{2}K^{2}_{2}\right]\!+{aK}_{2}\left[bK'_{2}(a-{bK}_{2})- a^{2}K_{2}K'_{2}\right]. \end{aligned}} $$
Therefore, the natural curvature functions κφ,τφ can be expressed as follows:
$${\kappa}_{\varphi}(\sigma^{\ast})=\frac{\sqrt{2}\sqrt{\xi_{1}^{2}+\xi_{2}^{2}-\xi_{3}^{2}}}{\left[b^{2}K^{2}_{2}+ a^{2}\left(1-K^{2}_{2}\right)\right]^{2}}, $$
and
$$ N_{\varphi}(\sigma^{\ast})=\frac{\xi_{1}T(\sigma)+\xi_{2}B_{1}(\sigma)+\xi_{3} B_{2}(\sigma)}{\rho\sqrt{\xi_{1}^{2}+\xi_{2}^{2}-\xi_{3}^{2}}}. $$
Also, the binormal vector of φ is
$$\begin{aligned} {}B_{\varphi}(\sigma^{\ast})&\!=\frac{1}{\Delta_{2}}\left\{a(\xi_{2}K_{2}-\xi_{3})T(\sigma)\,+\,(a\xi_{1}K_{2}- b\xi_{3}K_{2})B_{1}(\sigma)\right. \\ & \left. \quad-(a\xi_{1}+ b\xi_{2}K_{2})B_{2}(\sigma)\right\}, \end{aligned} $$
where
$$\Delta_{2}=\rho^{2}\sqrt{\xi_{1}^{2}+\xi_{2}^{2}-\xi_{3}^{2}}\sqrt{b^{2}K^{2}_{2}+ a^{2}\left(1-K^{2}_{2}\right)}. $$
Differentiating Eq. (16) with respect to σ, we get
$$\begin{aligned} \varphi^{\prime\prime}(\sigma^{\ast})&=\frac{1}{\sqrt{2}\,\rho}\left\{-{\varepsilon}[a+b(K_{1}K_{2}+K'_{1})]T(\sigma)\right.\\ & \left.\quad-{\varepsilon} {bK}_{2}\,B_{1}(\sigma)+\left[{\varepsilon} b K^{2}_{2}-aK'_{1}\right]B_{2}(\sigma) \right\},\end{aligned} $$
and
$$\varphi^{\prime\prime\prime}(\sigma^{\ast})= \frac{1}{\sqrt{2}\,\rho}\left(\alpha_{1}T(\sigma)+\alpha_{2}B_{1}(\sigma)+\alpha_{3}B_{2}(\sigma)\right), $$
where
$$\begin{aligned} &\alpha_{1}=b K_{2}+\left[a K'_{1}- b K^{2}_{2}-K^{\prime\prime}_{1}-(K_{1}K_{2})'\right], \\&\alpha_{2}= b (2K_{2}K'_{2}-2K'_{1}-K_{1}K_{2})-aK^{\prime\prime}_{1},\\&\alpha_{3}=- K_{2}\left[a+b(K_{1}K_{2}+K'_{1})\right]. \end{aligned} $$
Then
$${\begin{aligned} {\tau}_{\varphi}=\frac{\sqrt{2}}{\rho}\left\{\frac{\begin{aligned}&[ {bK}_{2}^{2}-aK'_{1}][ b \alpha_{2}K_{2}-a \alpha_{1}]+[b^{2}\alpha_{3}- a b \alpha_{1}]K^{2}_{2}\\&+ a(\alpha_{3}+\alpha_{2}K_{2})[a+b(K_{1}K_{2}+K'_{1})] \end{aligned}} {\begin{aligned}&a^{4}{K'_{1}}^{4}+\left[ a K_{2}\left(a+b(K_{1}K_{2}+K'_{1})\right)\right]^{2}\\&- \left[b^{2}K^{2}_{2}+ a\left(a+b(K_{1}K_{2}+K'_{1})\right) \right]^{2}\end{aligned}} \right\}\,. \end{aligned}} $$
□
Corollary 2
Let \(r:I\subset \mathbb {R}\rightarrow S_{1}^{2}\) be a regular unit speed spacelike curve lying fully on \(S^{2}_{1}\) with the moving equiform-Bishop frame {T,B1,B2}. If the base curve r=r(s) is contained in a plane, then the spacelike equiform-Bishop TB2-Smarandache curve is a circular helix if \(\,K_{2}\neq \pm \frac {a}{\sqrt {2}}\) and K2≠±1 and its natural curvature functions are dependent only on the second equiform-Bishop curvature and given by
$$ \begin{aligned} &{\kappa}_{\varphi}(\sigma^{\ast})=\frac{\sqrt{2}\,{bK}_{2}\sqrt{1-K^{2}_{2}}} {b^{2}K^{2}_{2}+ a^{2}\left(1-K^{2}_{2}\right)},\\ & {\tau}_{\varphi}(\sigma^{\ast})=\left\{\frac{\sqrt{2}}{\rho}\right\}\left\{\frac{K_{2}\left(3 b^{2}K^{2}_{2}-a^{2}\right)} { a^{3}\left(1-K^{2}_{2}\right)}\right\}. \end{aligned} $$
(19)
Definition 4
Let \(r:I\subset \mathbb {R}\rightarrow S_{1}^{2}\) be a regular unit speed spacelike curve lying fully on \(S^{2}_{1}\). The spacelike equiform-Bishop B1B2-Smarandache curve \({\varphi }:I\subset \mathbb {R}\rightarrow S_{1}^{2}\) of r defined by
$$ \varphi=\varphi(\sigma^{\ast})=\frac{1}{\sqrt{2}\,\rho}\left(a\,B_{1}(\sigma)+b\,B_{2}(\sigma)\right),{\quad} a^{2}-b^{2}=2. $$
(20)
Theorem 3
Let \(r:I\subset \mathbb {R}\rightarrow S_{1}^{2}\) be a regular unit speed spacelike curve lying fully on \(S^{2}_{1}\) with the moving equiform-Bishop frame {T,B1,B2}. If \({\varphi }:I\subset \mathbb {R}\rightarrow S_{1}^{2}\) is the spacelike equiform-Bishop B1B2-Smarandache curve of r=r(σ) with non-zero natural curvature functions, then its Frenet frame {Tφ,Nφ,Bφ} is given by
$$ \left(\begin{array}{c} T_{\varphi} \\ N_{\varphi} \\ B_{\varphi} \end{array} \right)=\left(\begin{array}{ccc} \frac{-1 }{\rho}\,\,\, & 0\,\,\, & 0 \\ 0 \,\,\,& \frac{-1}{\rho\sqrt{1- K^{2}_{2}}}\,\,\,& \frac{K_{2}}{\rho\sqrt{1- K^{2}_{2}}} \\ 0\,\,\,& \frac{-K_{2}}{\rho^{2}\sqrt{1- K^{2}_{2}}}\,\,\, & \frac{1}{\rho^{2}\sqrt{1- K^{2}_{2}}} \end{array} \right)\left(\begin{array}{c} T \\ B_{1} \\ B_{2} \end{array} \right),{\quad} K_{2}\neq\pm\,1. $$
(21)
Proof
Differentiationg Eq. (20) with respect to σ and using Eq. (5), we get
$$ \varphi'(\sigma^{\ast})=\frac{d\varphi}{d\sigma^{\ast}}\frac{d\sigma^{\ast}}{d\sigma}=\frac{-(a+{bK}_{2})T(\sigma)}{\sqrt{2}\,\rho}, $$
(22)
hence
$$ T_{\varphi}(\sigma^{\ast})=\frac{- \,T(\sigma)}{\rho}, $$
(23)
with the parameterization
$$ \frac{d\sigma^{\ast}}{d\sigma}=\frac{a+{bK}_{2}}{\sqrt{2}}\,. $$
(24)
Differentiating Eq. (23) with respect to σ, we have
$$T'_{\varphi}(\sigma^{\ast})=\frac{-\sqrt{2}\,}{\rho(a+{bK}_{2})}\left(B_{1}(\sigma)-K_{2}\, B_{2}(\sigma)\right). $$
The curvature of φ is given by
$${\kappa}_{\varphi}(\sigma^{\ast})=\frac{\sqrt{2}\sqrt{1- K^{2}_{2}}} {a+{bK}_{2}},{\quad} K_{2}\neq \frac{-a}{b}. $$
Furthermore, the principal normal and binormal vectors of φ are defined as follows:
$$N_{\varphi}(\sigma^{\ast})=\frac{-1}{\rho\sqrt{1- K^{2}_{2}}}\left(B_{1}(\sigma)-K_{2}\, B_{2}(\sigma)\right), $$
$$ B_{\varphi}(\sigma^{\ast})=\frac{1}{\rho^{2}\sqrt{1-{\varepsilon} K^{2}_{2}}}\left(-K_{2}\, B_{1}(\sigma)+ B_{2}(\sigma)\right). $$
From Eq. (22), we get
$$\begin{aligned} \varphi^{\prime\prime}(\sigma^{\ast})&= \frac{-1}{\sqrt{2}\,\rho}\left\{bK'_{2}\,T(\sigma)+(a+{bK}_{2})B_{1}(\sigma)\right. \\& \left.\quad-K'_{2}(a+{bK}_{2})B_{2}(\sigma) \right\}, \end{aligned} $$
similarly
$${\begin{aligned} \varphi^{\prime\prime\prime}(\sigma^{\ast})=&\frac{-1}{\sqrt{2}\,\rho}\left(\left[bK^{\prime\prime}_{2}+(a+{bK}_{2})(K_{2}K'_{2}-1)\right]T(\sigma)+2bK'_{2}\,B_{1}(\sigma)\right.\\&-\left.\left[(a+{bK}_{2})K^{\prime\prime}_{2}+K'_{2}({aK}_{2}+bK'_{2})\right]B_{2}(\sigma)\right). \end{aligned}} $$
Then, we obtain the torsion of φ as follows. Then
$${\tau}_{\varphi}=\frac{\sqrt{2}}{\rho}\left\{\frac{(a+{bK}_{2})K^{\prime\prime}_{2}+K'_{2}({aK}_{2}+3bK'_{2})} {({K'_{2}}^{2}-1)(a+{bK}_{2})^{2}} \right\}\,. $$
□
Corollary 3
Let \(r:I\subset \mathbb {R}\rightarrow S_{1}^{2}\) be a regular unit speed spacelike curve lying fully on \(S^{2}_{1}\) with the moving equiform-Bishop frame {T,B1,B2}. If the base curve r=r(s) is contained in a plane, then the spacelike equiform-Bishop B1B2-Smarandache curve is also contained in a plane.
Definition 5
Let \(r:I\subset \mathbb {R}\rightarrow S_{1}^{2}\) be a regular unit speed spacelike curve lying fully on \(S^{2}_{1}\). The spacelike equiform-Bishop TB1B2-Smarandache curve \({\varphi }:I\subset \mathbb {R}\rightarrow S_{1}^{2}\) of r defined by
$$ \begin{aligned} \varphi=\varphi(\sigma^{\ast})&=\frac{1}{\sqrt{3}\,\rho}\left(a\,T(\sigma)+b\,B_{1}(\sigma)++c\,B_{2}(\sigma)\right), \\&\quad a^{2}+ b^{2}-c^{2}=3. \end{aligned} $$
(25)
Theorem 4
Let \(r:I\subset \mathbb {R}\rightarrow S_{1}^{2}\) be a regular unit speed spacelike curve lying fully on \(S^{2}_{1}\) with the moving equiform-Bishop frame {T,B1,B2}. If \({\varphi }:I\subset \mathbb {R}\rightarrow S_{1}^{2}\) is the spacelike equiform-Bishop TB1B2-Smarandache curve of r=r(σ) with non-zero natural curvature functions, then its Frenet frame {Tφ,Nφ,Bφ} is given by
$$ \left(\begin{array}{c} T_{\varphi} \\ N_{\varphi} \\ B_{\varphi} \end{array} \right)=\left(\begin{array}{ccc} \frac{-(b+{cK}_{2})}{\rho\sqrt{(b+{cK}_{2})^{2}+(a^{2}-c^{2}K^{2}_{2})}} & \frac{a}{\rho\sqrt{(b+{cK}_{2})^{2}+(a^{2}-c^{2}K^{2}_{2})}} & \frac{-{cK}_{1}}{\rho\sqrt{(b+{cK}_{2})^{2}+(a^{2}-c^{2}K^{2}_{2})}} \\ \frac{\ell_{1}}{\rho\sqrt{\ell_{1}^{2}+\ell_{2}^{2}-\ell_{3}^{2}}} & \frac{\ell_{2}}{\rho\sqrt{\ell_{1}^{2}+\ell_{2}^{2}-\ell_{3}^{2}}}& \frac{\ell_{3}}{\rho\sqrt{\ell_{1}^{2}+\ell_{2}^{2}-\ell_{3}^{2}}} \\ \frac{-(a\,\ell_{3}+c\,\ell_{2}K_{1})}{\Delta_{3}}& \frac{c\,\ell_{1}K_{1}-\ell_{3}(b+{cK}_{2})}{\Delta_{3}} & \frac{-[a\,\ell_{1}+\ell_{2}(b+{cK}_{2})]}{\Delta_{3}} \end{array} \right)\left(\begin{array}{c} T \\ B_{1} \\ B_{2} \end{array} \right), $$
(26)
where
$$ {\begin{aligned} &\ell_{1}=(b+{cK}_{2})\left[a^{2}K_{2}K'_{2}-c(b+{cK}_{2})K'_{2}\right]-\left[(b+{cK}_{2})^{2}+a^{2}(1-K^{2}_{2})\right]\left[cK'_{2}+a(1-K_{2}^{2})\right], \\&\ell_{2}=(b+{cK}_{2})\left[(b+{cK}_{2})^{2}+a^{2}(1-K^{2}_{2})\right]-a\left[a^{2}K_{2}K'_{2}-c(b+{cK}_{2})K'_{2}\right],\\&\ell_{3}=\left[b+{cK}_{2}-aK'_{2}\right]\left[(b+{cK}_{2})^{2}+a^{2}(1-K^{2}_{2})\right]+{aK}_{1}K_{2}\left[a^{2}K_{2}K'_{2}-c(b+{cK}_{2})K'_{2}\right],\\&\Delta_{3}=\rho^{2}\sqrt{\ell_{1}^{2}+\ell_{2}^{2}-\ell_{3}^{2}}\sqrt{(b+{cK}_{2})^{2}+(a^{2}-c^{2}K^{2}_{2})}. \end{aligned}} $$
(27)
Proof
Differentiationg Eq. (25) with respect to σ and using Eq. (5), this leads to
$$ {\begin{aligned} \varphi'(\sigma^{\ast})=\frac{d\varphi}{d\sigma^{\ast}}\frac{d\sigma^{\ast}}{d\sigma}=\frac{1}{\sqrt{3}\,\rho}\left(-(b+{cK}_{2})\,T(\sigma)+a\, B_{1}(\sigma)-{cK}_{1}\,B_{2}(\sigma)\right), \end{aligned}} $$
(28)
then
$$ T_{\varphi}(\sigma^{\ast})=\frac{-(b+{cK}_{2})\,T(\sigma)+a\, B_{1}(\sigma)-{cK}_{1}\,B_{2}(\sigma)}{\rho\sqrt{(b+{cK}_{2})^{2}+\left(a^{2}-c^{2}K^{2}_{2}\right)}}, $$
(29)
where
$$ \frac{d\sigma^{\ast}}{d\sigma}=\frac{\sqrt{(b+{cK}_{2})^{2}+\left(a^{2}-c^{2}K^{2}_{2}\right)}}{\sqrt{3}}\,. $$
(30)
Then, from Eq. (29), we get
$$T'_{\varphi}(\sigma^{\ast})=\frac{\sqrt{3}\left(\ell_{1}T(\sigma)+\ell_{2}B_{1}(\sigma)+\ell_{3} B_{2}(\sigma)\right)}{\rho\left[(b+{cK}_{2})^{2}+\left(a^{2}-c^{2}K^{2}_{2}\right)\right]^{2}}, $$
where
$${\begin{aligned} &\ell_{1}=(b+{cK}_{2})\left[a^{2}K_{2}K'_{2}-c(b+{cK}_{2})K'_{2}\right]-\left[(b+{cK}_{2})^{2}+a^{2}(1-K^{2}_{2})\right]\left[cK'_{2}+a(1-K_{2}^{2})\right], \\&\ell_{2}=(b+{cK}_{2})\left[(b+{cK}_{2})^{2}+a^{2}(1-K^{2}_{2})\right]-a\left[a^{2}K_{2}K'_{2}-c(b+{cK}_{2})K'_{2}\right],\\&\ell_{3}=\left[b+{cK}_{2}-aK'_{2}\right]\left[(b+{cK}_{2})^{2}+a^{2}(1-K^{2}_{2})\right]+{aK}_{1}K_{2}\left[a^{2}K_{2}K'_{2}-c(b+{cK}_{2})K'_{2}\right]. \end{aligned}} $$
Then, the curvature and the principal normal vector of φ are respectively
$${\kappa}_{\varphi}(\sigma^{\ast})=\frac{\sqrt{3}\sqrt{\ell_{1}^{2}+\ell_{2}^{2}-\ell_{3}^{2}}}{\left[(b+{cK}_{2})^{2}+\left(a^{2}-c^{2}K^{2}_{2}\right)\right]^{2}}, $$
and
$$ N_{\varphi}(\sigma^{\ast})=\frac{\ell_{1}T(\sigma)+\ell_{2}B_{1}(\sigma)+\ell_{3} B_{2}(\sigma)}{\rho\sqrt{\ell_{1}^{2}+\ell_{2}^{2}-\ell_{3}^{2}}}. $$
Besides, the binormal vector of φ is given by
$${\begin{aligned}B_{\varphi}(\sigma^{\ast})=&\frac{1}{\Delta_{3}}\left\{-(a\,\ell_{3}+c\,\ell_{2}K_{1})T(\sigma)+\left[c\,\ell_{1}K_{1}-\ell_{3}(b+{cK}_{2})\right]B_{1}(\sigma)\right. \\ & \left.-\left[a\,\ell_{1}+\ell_{2}(b+{cK}_{2})\right]B_{2}(\sigma)\right\},\end{aligned}} $$
where
$$\Delta_{3}=\rho^{2}\sqrt{\ell_{1}^{2}+\ell_{2}^{2}-\ell_{3}^{2}}\sqrt{(b+{cK}_{2})^{2}+\left(a^{2}-c^{2}K^{2}_{2}\right)}\,. $$
The derivatives φ′′ and φ′′′ of φ are
$${\begin{aligned} \varphi^{\prime\prime}(\sigma^{\ast})=&\frac{1}{\sqrt{3}\,\rho}\left\{-\left[a+c\left(K'_{2}-K_{1}K_{2}\right)\right]T(\sigma)-\left[b+{cK}_{2}\right]B_{1}(\sigma)\right. \\&\left.+\left[(b+{cK}_{2})K_{2}-cK'_{1}\right]B_{2}(\sigma) \right\}, \end{aligned}} $$
and
$$\varphi^{\prime\prime\prime}(\sigma^{\ast})=\frac{1}{\sqrt{3}\,\rho}\left(\gamma_{1}T(\sigma)+\gamma_{2}B_{1}(\sigma)+\gamma_{3}B_{2}(\sigma)\right), $$
where
$$\begin{aligned} &\gamma_{1}=c\left(K_{2}{-K}^{\prime\prime}_{2}\right)-K_{2}\left[b+{cK}_{2}-3aK'_{2}\right], \\&\gamma_{2}=-\left[2cK'_{2}+a\left(1-K_{2}^{2}\right)\right],\\&\gamma_{3}=cK'_{2}-aK^{\prime\prime}_{2}+K_{2}\left[2cK'_{2}+a\left(1-K_{2}^{2}\right)\right]. \end{aligned} $$
Then
$${\begin{aligned} \tau_{\varphi}=\frac{\sqrt{3}}{\rho}\left\{ \begin{array}{c} \frac{ \begin{aligned} &a^2K'_2+\left[b+cK_{2}\right]\left[(\gamma_2-\gamma_3)(b+cK_2)+a\,\gamma_{1}(1-K_1)-a\,\gamma_3K'_{2}\right]\\ &+a\left(\gamma_3+\gamma_2K_{1}\right)\left[cK'_2-a\left(1-K_{2}^{2}\right)\right] \end{aligned}} {\begin{aligned} &\left[a^2K'_2-a(b+cK_2)(1-K_1)\right]^2+\left[aK_{1}\left[cK'_2+a\left(1-K^{2}_{2}\right)\right]\right.\\ &\left.+(b+cK_2)\left[b+cK_2+aK'_{2}\right]\right]^2 -\left[(b+cK_2)^2+a\left[cK'_2+a\left(1-K_{2}^{2}\right)\right]\right]^2 \end{aligned}} \end{array}\right\}\,. \end{aligned}} $$
□
Corollary 4
Let \(r:I\subset \mathbb {R}\rightarrow S_{1}^{2}\) be a regular unit speed spacelike curve lying fully on \(S^{2}_{1}\) with the moving equiform-Bishop frame {T,B1,B2}. If the base curve r=r(s) is contained in a plane, then the spacelike equiform-Bishop TB1B2-Smarandache curve is a circular helix if \(\,K_{2}\neq \pm \frac {a^{2}+b^{2}}{2}\) and K2≠±1. Also, its natural curvature functions are dependent only on the second equiform-Bishop curvature and given by
$$ \begin{aligned} &{\kappa}_{\varphi}(\sigma^{\ast})=\frac{a\,\sqrt{3\left(1-K^{2}_{2}\right)}} {(b+{cK}_{2})^{2}+\left(a^{2}-c^{2}K^{2}_{2}\right)},\\ & {\tau}_{\varphi}(\sigma^{\ast})\,=\,\left\{\frac{\sqrt{3}}{\rho}\right\} \left\{\frac{\begin{aligned}&{abK}_{2}(b+{cK}_{2})(K_{1}+K_{2})-a^{3}K_{1}\left(1-K_{2}^{2}\right)^{2}\\ &+a(b+{cK}_{2})\left(1-K_{2}^{2}\right)\left[a+c(1+K_{1}K_{2})+2(b+{cK}_{2})\right]\end{aligned}} {\begin{aligned}&a^{4}(1-K_{1})^{2}\left(1-K_{2}^{2}\right)^{2}+a(1-K_{1})(b+{cK}_{2})\left[b+{cK}_{2}\right.\\&\left.-2(1-K_{2}^{2})\right]\end{aligned}}\right\}. \end{aligned} $$
(31)