v = ( 2 t , e t , e − t ) v ′ = ( 2 , e t , − e − t ) v ′ ′ = ( 0 , e t , e − t ) v ′ ′ ′ = ( 0 , e t , − e − t ) ∣ v ′ ∣ = 2 + e 2 t + e − 2 t = e t + e − t v ′ × v ′ ′ = ( e t ⋅ e − t − ( − e − t ) ⋅ e t , ( − e − t ) ⋅ 0 − 2 ⋅ e − t , 2 ⋅ e t − e t ⋅ 0 ) = 2 ( 2 , − e − t , e t ) ∣ v ′ × v ′ ′ ∣ = 2 ( e t + e − t ) curvature k = ∣ v ′ × v ′ ′ ∣ ∣ v ′ ∣ 3 = 2 ( e t + e − t ) 2 ( v ′ × v ′ ′ ) ⋅ v ′ ′ ′ = 2 ( 2 ⋅ 0 + ( − e − t ) ⋅ e t + e t ⋅ ( − e − t ) ) = − 2 2 torsion ( v ′ × v ′ ′ ) ⋅ v ′ ′ ′ ∣ v ′ × v ′ ′ ∣ 2 = − 2 2 2 ( e t + e − t ) 2 = − 2 ( e t + e − t ) 2 \begin{array}{l}
v = (\sqrt{2} t, e^{t}, e^{-t}) \\
v' = (\sqrt{2}, e^{t}, -e^{-t}) \\
v'' = (0, e^{t}, e^{-t}) \\
v''' = (0, e^{t}, -e^{-t}) \\
|v'| = \sqrt{2 + e^{2t} + e^{-2t}} = e^{t} + e^{-t} \\
v' \times v'' = (e^{t} \cdot e^{-t} - (-e^{-t}) \cdot e^{t}, (-e^{-t}) \cdot 0 - \sqrt{2} \cdot e^{-t}, \sqrt{2} \cdot e^{t} - e^{t} \cdot 0) = \sqrt{2} (\sqrt{2}, -e^{-t}, e^{t}) \\
|v' \times v''| = \sqrt{2} (e^{t} + e^{-t}) \\
\text{curvature} \quad k = \frac{|v' \times v''|}{|v'|^{3}} = \frac{\sqrt{2}}{(e^{t} + e^{-t})^{2}} \\
(v' \times v'') \cdot v''' = \sqrt{2} (\sqrt{2} \cdot 0 + (-e^{-t}) \cdot e^{t} + e^{t} \cdot (-e^{-t})) = -2\sqrt{2} \\
\text{torsion} \quad \frac{(v' \times v'') \cdot v'''}{|v' \times v''|^{2}} = \frac{-2\sqrt{2}}{2 (e^{t} + e^{-t})^{2}} = \frac{-\sqrt{2}}{(e^{t} + e^{-t})^{2}} \\
\end{array} v = ( 2 t , e t , e − t ) v ′ = ( 2 , e t , − e − t ) v ′′ = ( 0 , e t , e − t ) v ′′′ = ( 0 , e t , − e − t ) ∣ v ′ ∣ = 2 + e 2 t + e − 2 t = e t + e − t v ′ × v ′′ = ( e t ⋅ e − t − ( − e − t ) ⋅ e t , ( − e − t ) ⋅ 0 − 2 ⋅ e − t , 2 ⋅ e t − e t ⋅ 0 ) = 2 ( 2 , − e − t , e t ) ∣ v ′ × v ′′ ∣ = 2 ( e t + e − t ) curvature k = ∣ v ′ ∣ 3 ∣ v ′ × v ′′ ∣ = ( e t + e − t ) 2 2 ( v ′ × v ′′ ) ⋅ v ′′′ = 2 ( 2 ⋅ 0 + ( − e − t ) ⋅ e t + e t ⋅ ( − e − t )) = − 2 2 torsion ∣ v ′ × v ′′ ∣ 2 ( v ′ × v ′′ ) ⋅ v ′′′ = 2 ( e t + e − t ) 2 − 2 2 = ( e t + e − t ) 2 − 2
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#340153 on Dec 2023