logarithmic spiral
The equation of the logarithmic spiral^{} in polar coordinates^{} $r,\phi $ is
$r=C{e}^{k\phi}$  (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
$$\overrightarrow{r}=(C{e}^{k\phi}\mathrm{cos}\phi ,C{e}^{k\phi}\mathrm{sin}\phi )$$ 
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 $\overrightarrow{r}$ and its derivative $\frac{d\overrightarrow{r}}{d\phi}={\overrightarrow{r}}^{\prime}$, the latter of which gives the direction of the tangent line^{} (see vectorvalued function):
$${\overrightarrow{r}}^{\prime}=(C{e}^{k\phi}k\mathrm{cos}\phi C{e}^{k\phi}\mathrm{sin}\phi ,C{e}^{k\phi}k\mathrm{sin}\phi +C{e}^{k\phi}\mathrm{cos}\phi ).$$ 
One obtains
$$\overrightarrow{r}\cdot {\overrightarrow{r}}^{\prime}=k{r}^{2},\overrightarrow{r}=r,{\overrightarrow{r}}^{\prime}=r\sqrt{1+{k}^{2}},$$ 
whence
$$\mathrm{cos}\psi =\frac{\overrightarrow{r}\cdot {\overrightarrow{r}}^{\prime}}{\overrightarrow{r}{\overrightarrow{r}}^{\prime}}=\frac{k}{\sqrt{1+{k}^{2}}}=\text{constant.}$$ 
It follows that $k=\mathrm{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
$$\underset{\phi \to \mathrm{\infty}}{lim}C{e}^{k\phi}=\mathrm{\hspace{0.33em}0}\mathit{\hspace{1em}}\text{but}\mathit{\hspace{1em}}C{e}^{k\phi}\ne \mathrm{\hspace{0.33em}0}\forall \phi \in \mathbb{R}$$ 
The arc length^{} $s$ of the logarithmic spiral is expressible in closed form; if we take it for the interval^{} $[{\phi}_{1},{\phi}_{2}]$, we can calculate in the case $k>0$ that
$$s={\int}_{{\phi}_{1}}^{{\phi}_{2}}\sqrt{{r}^{2}+{\left(\frac{dr}{d\phi}\right)}^{2}}\mathit{d}\phi ={\int}_{{\phi}_{1}}^{{\phi}_{2}}\sqrt{{C}^{2}{e}^{2k\phi}+{C}^{2}{e}^{2k\phi}{k}^{2}}\mathit{d}\phi =\frac{\sqrt{1+{k}^{2}}}{k}C({e}^{k{\phi}_{2}}{e}^{k{\phi}_{1}}),$$ 
thus
$$s=\frac{\sqrt{1+{k}^{2}}}{k}({r}_{2}{r}_{1})=\frac{{r}_{2}{r}_{1}}{\mathrm{cos}\psi}.$$ 
Letting ${\phi}_{1}\to \mathrm{\infty}$ 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 $r={C}_{1}{e}^{\phi}$ and $r={C}_{2}{e}^{\phi}$ are orthogonal curves to each other.
Title  logarithmic spiral 

Canonical name  LogarithmicSpiral 
Date of creation  20130322 19:02:26 
Last modified on  20130322 19:02:26 
Owner  pahio (2872) 
Last modified by  pahio (2872) 
Numerical id  26 
Author  pahio (2872) 
Entry type  Topic 
Classification  msc 14H45 
Synonym  Bernoulli spiral 
Related topic  AngleBetweenTwoCurves 
Related topic  EvoluteOfCycloid 
Related topic  PolarTangentialAngle2 
Related topic  AngleBetweenTwoLines 