# curvature of Nielsen’s spiral

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    (http://planetmath.org/sineintegral) and the sine integral (http://planetmath.org/sineintegral) and $t$ is the parameter (http://planetmath.org/Parametre) ($t>0$).

We determine the curvature (http://planetmath.org/CurvaturePlaneCurve) $\kappa$ 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)
 $\displaystyle x^{\prime}=\frac{d}{dt}\left(a\int_{\infty}^{t}\frac{\cos{u}}{u}% \,du\!\right)\;=\;\frac{a\cos{t}}{t},$ (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  (http://planetmath.org/ArcLength) of Nielsen’s spiral can also be obtained in a closed form  (http://planetmath.org/ClosedForm4); 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. Figure 1: Plot of Nielsen’s spiral for 2≤t≤50. Axis scaling is in units of a. (Octave / MATLAB http://aux.planetmath.org/files/objects/9350/nielsen.msource program for plot; in http://aux.planetmath.org/files/objects/9350/nielsen.pdfPDF format)
 Title curvature of Nielsen’s spiral Canonical name CurvatureOfNielsensSpiral Date of creation 2015-02-06 12:53:54 Last modified on 2015-02-06 12:53:54 Owner pahio (2872) Last modified by pahio (2872) Numerical id 22 Author pahio (2872) Entry type Example Classification msc 53A04 Synonym arc length of Nielsen’s spiral Related topic CosineIntegral Related topic SineIntegral Related topic FamousCurvesInThePlane Related topic DerivativeForParametricForm Defines Nielsen’s spiral