You are here
Homelogarithmic spiral
Primary tabs
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 characteristic of the logarithmic spiral is that any line emanating from the origin cuts 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 vectorvalued function):
$\vec{r}\,^{{\prime}}\;=\;\left(Ce^{{k\varphi}}k\cos\varphiCe^{{k\varphi}}\sin% \varphi,\;Ce^{{k\varphi}}k\sin\varphi+Ce^{{k\varphi}}\cos\varphi\right).$ 
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 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.
Mathematics Subject Classification
14H45 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
Recent Activity
new question: Prove that for any sets A, B, and C, An(BUC)=(AnB)U(AnC) by St_Louis
Apr 20
new image: informationtheoreticdistributedmeasurementdds.png by rspuzio
new image: informationtheoreticdistributedmeasurement4.2 by rspuzio
new image: informationtheoreticdistributedmeasurement4.1 by rspuzio
new image: informationtheoreticdistributedmeasurement3.2 by rspuzio
new image: informationtheoreticdistributedmeasurement3.1 by rspuzio
new image: informationtheoreticdistributedmeasurement2.1 by rspuzio
Apr 19
new collection: On the InformationTheoretic Structure of Distributed Measurements by rspuzio
Apr 15
new question: Prove a formula is part of the Gentzen System by LadyAnne
Mar 30
new question: A problem about Euler's totient function by mbhatia