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Curvature and Torsion
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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- A necessary and sufficient condition for a curve to be a straight line is that curvature \( \kappa=0\) at all points on the curve.
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A necessary and sufficient condition for a curve to be a plane curve is that\( \tau =0\) at all points on the plane curve.
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Prove that \( \kappa =| \vec{r}'\times \vec{r}'' |, \tau =\frac{[ \vec{r}',\vec{r}'',\vec{r}''' ]}{{{| \vec{r}'\times \vec{r}'' |}^2}}\) and \( [ \vec{r}',\vec{r}'',\vec{r}''' ]={\kappa^2}\tau \).
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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 \([ \dot{\vec{r}},\ddot{\vec{r}},\dddot{\vec{r}} ]=\kappa^2 \tau \dot{s}^6\)
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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.
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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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Find the curvature and torsion of a following curve
- \( x=a( 3t-t^3 ),y=3at^2,z=a( 3t+t^3 )\)
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- \( x=t,y=t^2,z=t^3\)
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- \( x=a\cos t,y=a\sin t,z=at\cot \alpha \)
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- \( x=a \cos t,y=a \sin t,z=ct\)
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- \( \vec{r}=( a-a \sin t,a-a \cos t,b t )\)
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- \( x=acos\theta ,y=asin\theta ,z=a\theta tan\alpha \)
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- \( x=a\sqrt{6}t^3,y=a( 1+3t^2 ),z=a\sqrt{6}t\)
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- \( x=3t,y=3t^2,z=2t^3\)
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- For a space curve \( \vec{r}=\vec{r}( s )\) show that
- \( [ \vec{t}',\vec{t}'',\vec{t}''' ]=\kappa^5 \frac{d}{ds} ( \frac{\tau }{\kappa } ) \)
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- \( [ \vec{b}',\vec{b}'',\vec{b}''' ]=\tau^5 \frac{d}{ds}( \frac{\kappa }{\tau } ) \)
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