# logarithmic spiral

The equation of the logarithmic spiral in polar coordinates $r,\,\varphi$ is

 $\displaystyle r\;=\;Ce^{k\varphi}$ (1)

where $C$ and $k$ are constants ($C>0$).  Thus the position vector of the point of this curve as the coordinate vector is written as

 $\vec{r}\;=\;(Ce^{k\varphi}\cos\varphi,\;Ce^{k\varphi}\sin\varphi)$

which is a parametric form of the curve.

Perhaps the most known of the logarithmic spiral is that any line emanating from the origin the curve under a constant angle $\psi$.  This is seen e.g. by using the vector $\vec{r}$ and its derivative$\frac{d\vec{r}}{d\varphi}=\vec{r}\,^{\prime}$,  the latter of which gives the direction of the tangent line (see vector-valued function):

 $\vec{r}\,^{\prime}\;=\;\left(Ce^{k\varphi}k\cos\varphi-Ce^{k\varphi}\sin% \varphi,\;Ce^{k\varphi}k\sin\varphi+Ce^{k\varphi}\cos\varphi\right).$

One obtains

 $\vec{r}\cdot\vec{r}\,^{\prime}\;=\;kr^{2},\quad|\vec{r}|\;=\;r,\quad|\vec{r}\,% ^{\prime}|\;=\;r\sqrt{1\!+\!k^{2}},$

whence

 $\cos\psi\;=\;\frac{\vec{r}\cdot\vec{r}\,^{\prime}}{|\vec{r}||\vec{r}\,^{\prime% }|}\;=\;\frac{k}{\sqrt{1\!+\!k^{2}}}\;=\;\mbox{constant.}$

It follows that  $k=\cot\psi$.  The angle $\psi$ is called the polar tangential angle.

The logarithmic spiral (1) goes infinitely many times round the origin without to reach it; in the case  $k>0$  one may state that

 $\lim_{\varphi\to-\infty}Ce^{k\varphi}\;=\;0\quad\mbox{but}\quad Ce^{k\varphi}% \;\neq\;0\;\;\forall\varphi\in\mathbb{R}$

The arc length $s$ of the logarithmic spiral is expressible in closed form; if we take it for the interval$[\varphi_{1},\,\varphi_{2}]$,  we can calculate in the case  $k>0$  that

 $s\;=\;\int_{\varphi_{1}}^{\varphi_{2}}\!\sqrt{r^{2}+\left(\frac{dr}{d\varphi}% \right)^{2}}\,d\varphi\;=\;\int_{\varphi_{1}}^{\varphi_{2}}\!\sqrt{C^{2}e^{2k% \varphi}+C^{2}e^{2k\varphi}k^{2}}\,d\varphi\;=\;\frac{\sqrt{1\!+\!k^{2}}}{k}C(% e^{k\varphi_{2}}-e^{k\varphi_{1}}),$

thus

 $s\;=\;\frac{\sqrt{1\!+\!k^{2}}}{k}(r_{2}\!-\!r_{1})\;=\;\frac{r_{2}\!-\!r_{1}}% {\cos\psi}.$

Letting  $\varphi_{1}\to-\infty$  one sees that the arc length from the origin to a point of the spiral is finite.

Other properties

• Any curve with constant polar tangential angle is a logarithmic spiral.

• All logarithmic spirals with equal polar tangential angle are similar.

• A logarithmic spiral rotated about the origin is a spiral homothetic to the original one.

• The inversion$z\mapsto\frac{1}{z}$  causes for the logarithmic spiral a reflexion against the imaginary axis and a rotation around the origin, but the image is congruent to the original one.

• The evolute of the logarithmic spiral is a congruent logarithmic spiral.

• The catacaustic of the logarithmic spiral is a logarithmic spiral.

• The families  $r=C_{1}e^{\varphi}$  and  $r=C_{2}e^{-\varphi}$  are orthogonal curves to each other.

Title logarithmic spiral LogarithmicSpiral 2013-03-22 19:02:26 2013-03-22 19:02:26 pahio (2872) pahio (2872) 26 pahio (2872) Topic msc 14H45 Bernoulli spiral AngleBetweenTwoCurves EvoluteOfCycloid PolarTangentialAngle2 AngleBetweenTwoLines