Given the space curve r=r(t), show that its torsion τ is given by τ=((r^. •r^(..) ×r^(...) )/(∣r^. ×r^(..) ∣^2 )). It may help to know that its curvature is numerically given by κ=((∣r^. ×r^(..) ∣)/(∣r^. ∣^3 )). r^. is differentiation of r once with respect to t.


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Question Number 5115 by Yozzii last updated on 14/Apr/16

$G i v e n t h e s p a c e c u r v e r = r (t), s h o w$ $t h a t i t s t o r s i o n τ i s g i v e n b y$ $τ = \frac{\dot{r} ∙ \overset{. .}{r} \times \overset{. . .}{r}}{∣ \dot{r} \times \overset{. .}{r} ∣^{2}} . I t m a y h e l p t o k n o w t h a t i t s$ $c u r v a t u r e i s n u m e r i c a l l y g i v e n b y κ = \frac{∣ \dot{r} \times \overset{. .}{r} ∣}{∣ \dot{r} ∣^{3}} .$ $\dot{r} i s d i f f e r e n t i a t i o n o f r o n c e w i t h$ $r e s p e c t t o t .$
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