curvature of Nielsen’s spiral

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

$x=a\text{ci}t,y=a\text{si}t$ (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

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

The first derivatives of (1) are

$x^{\prime }={\displaystyle \frac{d}{dt}}\left(a{\displaystyle {\int }_{\mathrm{\infty }}^t}{\displaystyle \frac{\cos u}{u}}𝑑u\right)={\displaystyle \frac{a\cos t}{t}},$ (3)

$y^{\prime }={\displaystyle \frac{d}{dt}}\left(a{\displaystyle {\int }_{\mathrm{\infty }}^t}{\displaystyle \frac{\sin u}{u}}𝑑u\right)={\displaystyle \frac{a\sin t}{t}},$ (4)

and hence the second derivatives

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

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

$\kappa ={\displaystyle \frac{t}{a}}.$ (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.

$s=a\ln t.$ (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 $2\le t\le 50$. 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