PlanetMath: curvature of Nielsen's spiral

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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

is a non-zero constant, `` ${\mathrm{ci}}$ '' and `` ${\mathrm{si}}$ '' are the cosine integral and the sine integral and

is the parameter (

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

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

(3)

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

(4)

and hence the second derivatives

$\displaystyle x'' = -a\cdot\frac{t\sin{t}+\cos{t}}{t^2},\;\;\; y'' = a\cdot\frac{t\cos{t}-\sin{t}}{t^2}.$

Substituting the derivatives in (2) yields

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

which is easily simplified to

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

(5)

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

$\displaystyle s \;=\; \int_1^t\sqrt{x'^2\!+\!y'^2}\,dt \;=\; \int_1^t\sqrt{ \frac{a^2\cos^2t}{t^2}+\frac{a^2\sin^2t}{t^2} }\,dt \;=\; \int_1^t\frac{a}{t}\,dt,$