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[parent] curvature of Nielsen's spiral (Example)

Nielsen's spiral is the plane curve defined in the parametric form

$\displaystyle x = a{\mathrm{ci}}{t},\;\; y = a{\mathrm{si}}{t}$ (1)

where $ a$ is a non-zero constant, “ $ {\mathrm{ci}}$” and “ $ {\mathrm{si}}$” are the cosine integral and the sine integral and $ t$ is the parameter ($ t > 0$). We determine the curvature $ \kappa$ of this curve using the expression
$\displaystyle \kappa = \frac{x'y”-y'x”}{{[}(x')^2+(y')^2{]}^{3/2}}.$ (2)

The first derivatives of (1) are

$\displaystyle x' = \frac{d}{dt}\left(a\int_\infty^t\frac{\cos{u}}{u} du\!\right) = \frac{a\cos{t}}{t},$

$\displaystyle y' = \frac{d}{dt}\left(a\int_\infty^t\frac{\sin{u}}{u} du\!\right) = \frac{a\sin{t}}{t},$
and hence the second derivatives
$\displaystyle x” = -a\cdot\frac{t\sin{t}+\cos{t}}{t^2},\;\;\; y” = a\cdot\frac{t\cos{t}-\sin{t}}{t^2}.$
Substituting the derivatives in (2) gives
$\displaystyle \kappa = a^2\!\cdot\!\frac{(\cos{t})(t\cos{t}-\sin{t})+(\sin{t})(... ...dot t^2} :\!\left(\frac{a^2\cos^2{t}+a^2\sin^2{t}}{t^2}\right)^{\frac{3}{2}}\!,$
which is easily simplified to
$\displaystyle \kappa = \frac{t}{a}.$ (3)

Note. As consequence of the expressions for $ x'$ and $ y'$ we get

$\displaystyle \frac{dy}{dx} = \frac{y'}{x'} = \frac{\sin{t}}{\cos{t}} = \tan{t},$
which says that the sense of the parameter $ t$ is the slope angle of the tangent line of the Nielsen's spiral.
Figure: Plot of Nielsen's spiral for $ 2 \leq t \leq 50$. Axis scaling is in units of $ a$. (Octave / MATLAB source program for plot; in PDF format)
\includegraphics{nielsen}



"curvature of Nielsen's spiral" is owned by pahio. [ full author list (2) ]
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See Also: cosine integral, sine integral, famous curves, derivative for parametric form

Also defines:  Nielsen's spiral

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Cross-references: tangent line, slope angle, consequence, derivatives, second derivatives, first derivatives, expression, curve, parameter, sine integral, cosine integral, parametric form, plane curve
There are 2 references to this entry.

This is version 13 of curvature of Nielsen's spiral, born on 2007-05-09, modified 2007-08-29.
Object id is 9350, canonical name is CurvatureOfNielsensSpiral.
Accessed 1197 times total.

Classification:
AMS MSC: 53A04 (Differential geometry :: Classical differential geometry :: Curves in Euclidean space)
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\parametricplot in pstricks by pahio on 2007-07-07 16:31:11
I have tried to use \parametricplot in pstricks for making the Nielsen's spiral (http://planetmath.org/encyclopedia/CurvatureOfNielsensSpiral.html), but not succeeded. Are there some masters of pstricks who knows what is the cause? Please feel free to correct the code (the equations are seen in "version 7").
Jussi
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