The logarithmic spiral r=a e^{k\varphi} \;,\; k\ne 0, has the following properties (see
Spiral): •
Pitch angle: Constant value \tan\alpha=k\quad with pitch angle \alpha (see diagram and animation).(In case of k=0 angle \alpha would be 0 and the curve a circle with radius a.) •
Curvature: \kappa=\frac{1}{r\sqrt{1+k^2}}=\frac{\cos \alpha}{r} •
Arc length: L(\varphi_1,\varphi_2)=\frac{\sqrt{k^2+1}}{k}\big(r(\varphi_2)-r(\varphi_1)\big)= \frac{r(\varphi_2)-r(\varphi_1)}{\sin \alpha}Especially: \ L(-\infty,\varphi_2)=\frac{r(\varphi_2)}{\sin \alpha}\quad\; is finite if k > 0. This property was first realized by
Evangelista Torricelli even before
calculus had been invented. •
Sector area: A=\frac{r(\varphi_2)^2-r(\varphi_1)^2}{4k} •
Inversion: Circle inversion (r\to 1/r) maps the logarithmic spiral r=a e^{k\varphi} onto the logarithmic spiral r=\tfrac{1}{a} e^{-k\varphi} \, . •
Rotating, scaling: Rotating the spiral by angle \varphi_0 yields the spiral r=ae^{-k\varphi_0}e^{k\varphi}, which is the original spiral uniformly scaled (at the origin) by e^{-k\varphi_0}. Scaling by \;e^{kn2\pi}\; , n=\pm 1,\pm2,...,\; gives the
same curve. •
Self-similarity: A result of the previous property: A scaled logarithmic spiral is
congruent (by rotation) to the original curve.
Example: The diagram shows spirals with slope angle \alpha=20^\circ and a=1,2,3,4,5. Hence they are all scaled copies of the red one. But they can also be generated by rotating the red one by angles -109^\circ,-173^\circ,-218^\circ,-253^\circ resp.. All spirals have no points in common (see property on
complex exponential function). •
Relation to other curves: Logarithmic spirals are congruent to their own
involutes,
evolutes, and the
pedal curves based on their centers. •
Complex exponential function: The
exponential function exactly maps all lines not parallel with the real or imaginary axis in the complex plane, to all logarithmic spirals in the complex plane with centre at 0: z(t)=\underbrace{(kt+b)\; +it}_{\text{line}}\quad \to\quad e^{z(t)}=e^{kt+b}\cdot e^{it}= \underbrace{e^b e^{kt}(\cos t+i\sin t)}_{\text{log. spiral}} The pitch angle \alpha of the logarithmic spiral is the angle between the line and the imaginary axis. == Special cases and approximations ==