logarithmic spiralLogarithmicSpiral2009-09-21 18:59:072009-10-03 18:09:57Topiccurve% this is the default PlanetMath preamble. as your knowledge
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The equation of the \emph{logarithmic spiral} in polar coordinates $r,\,\varphi$ is
\begin{align}
r \;=\; Ce^{k\varphi}
\end{align}
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.\\
\begin{figure}[htp]
\centering
\includegraphics[scale=0.80]{log_spiral.eps}
\end{figure}
Perhaps the most known \PMlinkescapetext{characteristic} of the logarithmic spiral is that any line emanating from the origin \PMlinkescapetext{cuts} the curve under a constant angle $\psi$.\, This is seen e.g. by using the vector $\vec{r}$ and its derivative \,$\frac{d\vec{r}}{d\varphi} = \vec{r}\,'$,\, the latter of which gives the direction of the tangent line (see vector-valued function):
$$\vec{r}\,' \;=\;
\left(Ce^{k\varphi}k\cos\varphi-Ce^{k\varphi}\sin\varphi,\; Ce^{k\varphi}k\sin\varphi+Ce^{k\varphi}\cos\varphi\right).$$
One obtains
$$\vec{r}\cdot\vec{r}\,' \;=\; kr^2, \quad |\vec{r}| \;=\; r, \quad |\vec{r}\,'| \;=\; r\sqrt{1\!+\!k^2},$$
whence
$$\cos\psi \;=\; \frac{\vec{r}\cdot\vec{r}\,'}{|\vec{r}||\vec{r}\,'|} \;=\; \frac{k}{\sqrt{1\!+\!k^2}}
\;=\; \mbox{constant.}$$
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 exponential function never vanishes).\\
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
$$ s \;=\; \int_{\varphi_1}^{\varphi_2}\!\sqrt{r^2+\left(\frac{dr}{d\varphi}\right)^2}\,d\varphi \;=\;
\int_{\varphi_1}^{\varphi_2}\!\sqrt{C^2e^{2k\varphi}+C^2e^{2k\varphi}k^2}\,d\varphi \;=\;
\frac{\sqrt{1\!+\!k^2}}{k}C(e^{k\varphi_2}-e^{k\varphi_1}),$$
thus
$$s \;=\; \frac{\sqrt{1\!+\!k^2}}{k}(r_2\!-\!r_1).$$
Letting\, $\varphi_1 \to -\infty$\, one sees that the arc length from the origin to a point of the spiral is finite.\\
\textbf{Other properties}
\begin{itemize}
\item Any curve with constant polar tangential angle is a logarithmic spiral.
\item All logarithmic spirals with equal polar tangential angle are similar.
\item A logarithmic spiral rotated about the origin is a spiral homothetic to the original one.
\item 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.
\item The evolute of the logarithmic spiral is a congruent logarithmic spiral.
\item The catacaustic of the logarithmic spiral is a logarithmic spiral.
\item The families \,$r = C_1e^{\varphi}$\, and \,$r = C_2e^{-\varphi}$\, are orthogonal curves to each other.
\end{itemize}