example of curvature (space curve)

Example space curves and calculating their curvatures using the formula

\kappa(t)=\frac{\|{\bf r}^{\prime}(t)\times{\bf r}^{\prime\prime}(t)\|}{\|{\bf r% }^{\prime}(t)\|^{3}}

1. ${\bf r}(t)=3t\hat{i}+t^{2}\hat{j}-4t^{2}\hat{k}$

the first derivative

${\bf r}^{\prime}(t)=3\hat{i}+2t\hat{j}-8t\hat{k}$

vector magnitude of the derivative

$\|{\bf r}^{\prime}(t)\|=\sqrt{3^{2}+(2t)^{2}+(-8t)^{2}}$

$\|{\bf r}^{\prime}(t)\|=\sqrt{9+4t^{2}+16t^{2}}=\sqrt{9+20t^{2}}$

the second derivative

${\bf r}^{\prime\prime}(t)=2\hat{j}-8\hat{k}$

the cross product

${\bf r}^{\prime}(t)\times{\bf r}^{\prime\prime}(t)=\left|\begin{array}[]{ccc}% \hat{i}&\hat{j}&\hat{k}\\ 3&2t&-8t\\ 0&2&-8\\ \end{array}\right|=(-16t+16t)\hat{i}-(-24)\hat{j}+6\hat{k}$

${\bf r}^{\prime}(t)\times{\bf r}^{\prime\prime}(t)=24\hat{j}+6\hat{k}$

$\|{\bf r}^{\prime}(t)\times{\bf r}^{\prime\prime}(t)\|=\sqrt{576+36}=\sqrt{612% }=2\sqrt{153}$

$\|{\bf r}^{\prime}(t)\|^{3}=(9+20t^{2})^{3/2}$

$\kappa(t)=\frac{2\sqrt{153}}{(9+20t^{2})^{3/2}}$

2. Calculate the curvature of the right circular helix as given in the plot below and defined as

${\bf r}(t)=\cos t\hat{i}+\sin t\hat{j}+t\hat{k}$

${\bf r}^{\prime}(t)=-\sin t\hat{i}+\cos t\hat{j}+\hat{k}$

$\|{\bf r}^{\prime}(t)\|=\sqrt{\sin^{2}t+\cos^{2}t+1^{2}}=\sqrt{2}$

${\bf r}^{\prime\prime}(t)=-\cos t\hat{i}-\sin t\hat{j}$

${\bf r}^{\prime}(t)\times{\bf r}^{\prime\prime}(t)=\left|\begin{array}[]{ccc}% \hat{i}&\hat{j}&\hat{k}\\ -\sin t&\cos t&1\\ -\cos t&-\sin t&0\\ \end{array}\right|=\sin t\hat{i}-\cos t\hat{j}+(\sin^{2}t+\cos^{2}t)\hat{k}$

${\bf r}^{\prime}(t)\times{\bf r}^{\prime\prime}(t)=\sin t\hat{i}-\cos t\hat{j}% +\hat{k}$

$\|{\bf r}^{\prime}(t)\times{\bf r}^{\prime\prime}(t)\|=\sqrt{\sin^{2}t+\cos^{2% }t+1^{2}}=\sqrt{2}$

$\|{\bf r}^{\prime}(t)\|^{3}=2^{3/2}$

$\kappa(t)=\frac{\sqrt{2}}{2^{3/2}}=\frac{1}{2}$

Title	example of curvature (space curve)
Canonical name	ExampleOfCurvaturespaceCurve
Date of creation	2013-03-22 15:40:58
Last modified on	2013-03-22 15:40:58
Owner	bloftin (6104)
Last modified by	bloftin (6104)
Numerical id	8
Author	bloftin (6104)
Entry type	Example
Classification	msc 53A04
Related topic	PositionVector