Curvature and Torsion

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Curvature

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Expression of Curvature

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Torsion

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Expression of Torsion

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Expression of Screw-Curvature:

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Theorems

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1. A necessary and sufficient condition for a curve to be a straight line is that curvature \( \kappa=0\) at all points on the curve.
2. A necessary and sufficient condition for a curve to be a plane curve is that\( \tau =0\) at all points on the plane curve.
   
3. Prove that \( \kappa =| \vec{r}'\times \vec{r}'' |, \tau =\frac{[ \vec{r}',\vec{r}'',\vec{r}''' ]}{{{| \vec{r}'\times \vec{r}'' |}^2}}\) and \( [ \vec{r}',\vec{r}'',\vec{r}''' ]={\kappa^2}\tau \).
4. Show that \( \kappa =\frac{| \dot{\vec{r}}\times \ddot{\vec{r}} |}{{{| {\dot{\vec{r}}} |}^3}},\tau =\frac{[ \dot{\vec{r}},\ddot{\vec{r}},\dddot{\vec{r}} ]}{{{| \dot{\vec{r}}\times \ddot{\vec{r}} |}^2}}\) and \([ \dot{\vec{r}},\ddot{\vec{r}},\dddot{\vec{r}} ]=\kappa^2 \tau \dot{s}^6\)
5. A necessary and sufficient condition for a curve to be a plane curve is that \( [ \vec{r}',\vec{r}'',\vec{r}''' ]=0\) at all points.
6. A necessary and sufficient condition for a curve to be a plane curve is that\( [ \dot{\vec{r}},\ddot{\vec{r}},\dddot{\vec{r}} ]=0\) at all points.

Exercise

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1. Find the curvature and torsion of a following curve
   1. \( x=a( 3t-t^3 ),y=3at^2,z=a( 3t+t^3 )\)
      

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   2. \( x=t,y=t^2,z=t^3\)
      

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   3. \( x=a\cos t,y=a\sin t,z=at\cot \alpha \)
      

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   4. \( x=a \cos t,y=a \sin t,z=ct\)
      

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   5. \( \vec{r}=( a-a \sin t,a-a \cos t,b t )\)
      

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   6. \( x=acos\theta ,y=asin\theta ,z=a\theta tan\alpha \)
      

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   7. \( x=a\sqrt{6}t^3,y=a( 1+3t^2 ),z=a\sqrt{6}t\)
      

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   8. \( x=3t,y=3t^2,z=2t^3\)
      

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2. For a space curve \( \vec{r}=\vec{r}( s )\) show that
   1. \( [ \vec{t}',\vec{t}'',\vec{t}''' ]=\kappa^5 \frac{d}{ds} ( \frac{\tau }{\kappa } ) \)
      

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   2. \( [ \vec{b}',\vec{b}'',\vec{b}''' ]=\tau^5 \frac{d}{ds}( \frac{\kappa }{\tau } ) \)