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“…Then the Frenet formulas are $$\left(\begin{array}{c}array\dot{t}\\ array\dot{n}\\ array\dot{b}\end{array}\right)=\left(\begin{array}{ccc}array0& array\phantom{\rule{2pt}{0ex}}\kappa & array\phantom{\rule{2pt}{0ex}}0\\ array-\u03f5\kappa & array\phantom{\rule{2pt}{0ex}}0& array\phantom{\rule{2pt}{0ex}}\tau \\ array0& array\phantom{\rule{2pt}{0ex}}\tau & array\phantom{\rule{2pt}{0ex}}0\end{array}\right)\left(\begin{array}{c}arrayt\\ arrayn\\ arrayb\end{array}\right),$$ where g ( t , t )=1, g ( n , n )= ϵ =±1, g ( b , b )=− ϵ , g ( t , n )= g ( t , b )= g ( n , b )=0, κ and τ are the curvature and the torsion of α , respectively, and ${}^{.}=\frac{d}{ds}$. () - Let α be a spacelike curve with a lightlike (null) principal normal n . Then the Frenet formulas are
$$\left(\begin{array}{c}array\dot{t}\\ array\dot{n}\\ array\dot{b}\end{array}\right)=\left(\begin{array}{ccc}array0& array\phantom{\rule{2pt}{0ex}}1& array\phantom{\rule{2pt}{0ex}}0\\ array0& array\phantom{\rule{2pt}{0ex}}\tau & array\phantom{\rule{2pt}{0ex}}0\\ array-1& array\phantom{\rule{2pt}{0ex}}0& array\phantom{\rule{2pt}{0ex}}-\tau \end{array}\right)\left(\begin{array}{c}arrayt\\ arrayn\\ arrayb\end{array}\right),$$ where g ( t , t )= g ( n , b )=1 and g ( b , b )= g ( t , n )= g ( t , b )= g ( n , b )=0. …”

confidence: 99%

“…Then the Frenet formulas are $$\left(\begin{array}{c}array\dot{t}\\ array\dot{n}\\ array\dot{b}\end{array}\right)=\left(\begin{array}{ccc}array0& array\phantom{\rule{2pt}{0ex}}\kappa & array\phantom{\rule{2pt}{0ex}}0\\ array-\u03f5\kappa & array\phantom{\rule{2pt}{0ex}}0& array\phantom{\rule{2pt}{0ex}}\tau \\ array0& array\phantom{\rule{2pt}{0ex}}\tau & array\phantom{\rule{2pt}{0ex}}0\end{array}\right)\left(\begin{array}{c}arrayt\\ arrayn\\ arrayb\end{array}\right),$$ where g ( t , t )=1, g ( n , n )= ϵ =±1, g ( b , b )=− ϵ , g ( t , n )= g ( t , b )= g ( n , b )=0, κ and τ are the curvature and the torsion of α , respectively, and ${}^{.}=\frac{d}{ds}$. () - Let α be a spacelike curve with a lightlike (null) principal normal n . Then the Frenet formulas are
$$\left(\begin{array}{c}array\dot{t}\\ array\dot{n}\\ array\dot{b}\end{array}\right)=\left(\begin{array}{ccc}array0& array\phantom{\rule{2pt}{0ex}}1& array\phantom{\rule{2pt}{0ex}}0\\ array0& array\phantom{\rule{2pt}{0ex}}\tau & array\phantom{\rule{2pt}{0ex}}0\\ array-1& array\phantom{\rule{2pt}{0ex}}0& array\phantom{\rule{2pt}{0ex}}-\tau \end{array}\right)\left(\begin{array}{c}arrayt\\ arrayn\\ arrayb\end{array}\right),$$ where g ( t , t )= g ( n , b )=1 and g ( b , b )= g ( t , n )= g ( t , b )= g ( n , b )=0. …”

confidence: 99%

“…In the last case, κ has only 2 values:

- κ =0, when α is a straight line and
- κ =1, in all other cases. ()

confidence: 99%

“…1 .For an arbitrary curve α(s) in the space E 3 1 the following Frenet formulae are given in [2][3][4][5][6][7]. If α is timelike curve, then the Frenet formulae read If α is pseudo null curve, the Frenet formulas have the form…”

confidence: 99%

“…Meanwhile, as is known, the Mannheim curves in nonflat space forms have two cases: one case is the nonnull Mannheim curves; the other case is the null Mannheim curves which are mainly considered in 3-dimensional de Sitter space S 3 1 . Grbović et al discussed the null Mannheim curves in 3dimensional Minkowski space [12]. We know that the case of null curve that is immersed in a three-dimensional de Sitter space is more sophisticated and interesting than nonnull curve in de Sitter space.…”

confidence: 99%