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

$$\overrightarrow{r}=(C{e}^{k\phi }\cos \phi ,C{e}^{k\phi }\sin \phi )$$

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 vector-valued function):

$${\overrightarrow{r}}^{\prime }=(C{e}^{k\phi }k\cos \phi -C{e}^{k\phi }\sin \phi ,C{e}^{k\phi }k\sin \phi +C{e}^{k\phi }\cos \phi ).$$

One obtains

$$\overrightarrow{r}\cdot {\overrightarrow{r}}^{\prime }=k{r}^2,|\overrightarrow{r}|=r,|{\overrightarrow{r}}^{\prime }|=r\sqrt{1+k^2},$$

whence

$$\cos \psi =\frac{\overrightarrow{r}\cdot {\overrightarrow{r}}^{\prime }}{|\overrightarrow{r}||{\overrightarrow{r}}^{\prime }|}=\frac{k}{\sqrt{1+k^2}}=\text{constant.}$$

It follows that  $k=\cot \psi$.  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  $k>0$  onemay state that

$$\underset{\phi \to -\mathrm{\infty }}{lim}C{e}^{k\phi }=\mathrm{\hspace{0.33em}0}\mathit{\hspace{1em}}\text{but}\mathit{\hspace{1em}}C{e}^{k\phi }\ne \mathrm{\hspace{0.33em}0}\forall \phi \in ℝ$$

(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 $[{\phi }_1,{\phi }_2]$,  we can calculate in the case  $k>0$  that

$$s={\int }_{{\phi }_1}^{{\phi }_2}\sqrt{r^2+{\left(\frac{dr}{d\phi }\right)}^2}𝑑\phi ={\int }_{{\phi }_1}^{{\phi }_2}\sqrt{C^2{e}^{2k\phi }+C^2{e}^{2k\phi }{k}^2}𝑑\phi =\frac{\sqrt{1+k^2}}{k}C(e^{k{\phi }_2}-e^{k{\phi }_1}),$$

thus

$$s=\frac{\sqrt{1+k^2}}{k}(r_2-r_1)=\frac{r_2-r_1}{\cos \psi }.$$

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.