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  $k>0$  one may state that

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

(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.

Title	logarithmic spiral
Canonical name	LogarithmicSpiral
Date of creation	2013-03-22 19:02:26
Last modified on	2013-03-22 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