The equation of the logarithmic spiral in polar coordinates $r,\,\varphi$ is

(1)

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 $\frac{d\vec{r}}{d\varphi} = \vec{r}\,'$ , 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.

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 $r = C_1e^{\varphi}$ and $r = C_2e^{-\varphi}$ are orthogonal curves to each other.