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[parent] example of curvature (space curve) (Example)

Example space curves and calculating their curvatures using the formula

$\displaystyle \kappa(t) =\frac{\Vert {\bf r}'(t)\times {\bf r}''(t)\Vert}{ \Vert {\bf r}'(t)\Vert^3}$

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

the first derivative

$ {\bf r}'(t) = 3 \hat{i} + 2t\hat{j} - 8t\hat{k}$

vector magnitude of the derivative

$ \Vert {\bf r}'(t)\Vert = \sqrt{ 3^2 + (2t)^2 + (-8t)^2}$

$ \Vert {\bf r}'(t)\Vert = \sqrt{ 9 + 4t^2 + 16t^2} = \sqrt{9 + 20 t^2}$

the second derivative

$ {\bf r}''(t) = 2 \hat{j} - 8 \hat{k}$

the cross product

\begin{displaymath}{\bf r}'(t)\times {\bf r}''(t) = \left\vert \begin{array}{ccc... ...ray}\right\vert = (-16t + 16t)\hat{i} - (-24)\hat{j} + 6\hat{k}\end{displaymath}

$ {\bf r}'(t)\times {\bf r}''(t) = 24\hat{j} + 6\hat{k}$

$ \Vert {\bf r}'(t)\times {\bf r}''(t) \Vert = \sqrt{576 + 36} = \sqrt{612} = 2\sqrt{153}$

$ \Vert {\bf r}'(t)\Vert^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}$

\includegraphics[scale=.6]{helix.eps}

$ {\bf r}'(t) = -\sin t \hat{i} + \cos t\hat{j} + \hat{k}$

$ \Vert {\bf r}'(t)\Vert = \sqrt{ \sin^2 t + \cos^2 t + 1^2} = \sqrt{2}$

$ {\bf r}''(t) = -\cos t \hat{i} -\sin t \hat{j}$

\begin{displaymath}{\bf r}'(t)\times {\bf r}''(t) = \left\vert \begin{array}{ccc... ... = \sin t\hat{i} - \cos t\hat{j} + (\sin^2 t + \cos^2 t)\hat{k}\end{displaymath}

$ {\bf r}'(t)\times {\bf r}''(t) = \sin t\hat{i} - \cos t\hat{j} + \hat{k}$

$ \Vert {\bf r}'(t)\times {\bf r}''(t) \Vert = \sqrt{\sin^2 t + \cos^2 t + 1^2} = \sqrt{2}$

$ \Vert {\bf r}'(t)\Vert^3 = 2^{3/2}$

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



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Cross-references: circular helix, right, calculate, cross product, second derivative, derivative, vector, first derivative, formula, curvatures, space curves

This is version 5 of example of curvature (space curve), born on 2006-02-16, modified 2006-03-20.
Object id is 7624, canonical name is ExampleOfCurvatureSpaceCurve.
Accessed 1639 times total.

Classification:
AMS MSC: 53A04 (Differential geometry :: Classical differential geometry :: Curves in Euclidean space)
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