curvature of Nielsen’s spiral

Nielsen’s spiral is the plane curve defined in theparametric 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)