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Homelogarithmic spiral
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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.
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