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

$\displaystyle x=a\,\mbox{ci}\,{t},\quad y=a\,\mbox{si}\,{t}$ | (1) |

where $a$ is a non-zero constant, “ci” and “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

$\displaystyle\kappa=\frac{x^{{\prime}}y^{{\prime\prime}}-y^{{\prime}}x^{{% \prime\prime}}}{{[}(x^{{\prime}})^{2}+(y^{{\prime}})^{2}{]}^{{3/2}}}.$ | (2) |

The first derivatives of (1) are

$\displaystyle x^{{\prime}}=\frac{d}{dt}\left(a\int_{\infty}^{t}\frac{\cos{u}}{% u}\,du\!\right)\;=\;\frac{a\cos{t}}{t},$ | (3) |

$\displaystyle y^{{\prime}}\;=\;\frac{d}{dt}\left(a\int_{\infty}^{t}\frac{\sin{% u}}{u}\,du\!\right)\;=\;\frac{a\sin{t}}{t},$ | (4) |

and hence the second derivatives

$x^{{\prime\prime}}=-a\cdot\frac{t\sin{t}+\cos{t}}{t^{2}},\quad y^{{\prime% \prime}}=a\cdot\frac{t\cos{t}-\sin{t}}{t^{2}}.$ |

Substituting the derivatives in (2) yields

$\kappa\;=\;a^{2}\!\cdot\!\frac{(\cos{t})(t\cos{t}-\sin{t})+(\sin{t})(t\sin{t}+% \cos{t})}{t\cdot t^{2}}\!:\!\left(\frac{a^{2}\cos^{2}{t}+a^{2}\sin^{2}{t}}{t^{% 2}}\right)^{{\frac{3}{2}}}\!,$ |

which is easily simplified to

$\displaystyle\kappa\;=\;\frac{t}{a}.$ | (5) |

The arc length of Nielsen’s spiral can also be obtained in a simple closed form; using (3) and (4) we get:

$s\;=\;\int_{1}^{t}\sqrt{x^{{\prime 2}}\!+\!y^{{\prime 2}}}\,dt\;=\;\int_{1}^{t% }\sqrt{\frac{a^{2}\cos^{2}t}{t^{2}}+\frac{a^{2}\sin^{2}t}{t^{2}}}\,dt\;=\;\int% _{1}^{t}\frac{a}{t}\,dt,$ |

i.e.

$\displaystyle s\;=\;a\ln{t}.$ | (6) |

Note. The expressions for $x^{{\prime}}$ and $y^{{\prime}}$ allow us determine as well

$\frac{dy}{dx}\;=\;\frac{y^{{\prime}}}{x^{{\prime}}}\;=\;\frac{\sin{t}}{\cos{t}% }\;=\;\tan{t},$ |

which says that the sense of the parameter $t$ is the slope angle of the tangent line of the Nielsen’s spiral.

Defines:

Nielsen's spiral

Keywords:

arc length

Related:

CosineIntegral, SineIntegral, FamousCurvesInThePlane, DerivativeForParametricForm

Synonym:

arc length of Nielsen's spiral

Type of Math Object:

Example

Major Section:

Reference

Parent:

Groups audience:

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## Comments

## \parametricplot in pstricks

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

## Re: the Nielsen's spiral plot

I think you will find it easier to plot analytically complicated curves like Nielsen's spiral with numerical software rather than with Postscript directly. (This is what I have done in the entry.)

In particular, I'm not sure those approximating equations that you have work for "large" $t$, say $0< t < 1/2$, as they are based on Taylor series expansions of $sin$ and $cos$ about $0$. The interesting part, of course, is for large $t$, say $t > 6.0$. For this part, I evaluate the functions with their asymptotic series as displayed on their Wikipedia entry (without proof :). For $t < 6.0$, I use direct numerical integration to compute the function values.

I can post the Octave code (should also run in MatLab) once I clean it up a bit.

// Steve

## Re: the Nielsen's spiral plot

Steve,

Thousands of thanks to you for the beautiful picture of the Nielsen's spiral! I am very happy about it.

I believe now that pstricks isn't rather good for making such graphs, especially when they require uncomfortable functions and when the author is so inexperienced as I am :) Before I have made only one graph (cissoid of Diocles).

Jussi