Logarithmic spiral

A plane transcendental curve whose equation in polar coordinates has the form

$$\rho=a^\phi,\quad a>0.$$

Figure: l060650a

If $a>1$, as $\phi\to+\infty$ the logarithmic spiral evolves anti-clockwise, and as $\phi\to-\infty$ the spiral twists clockwise, tending to its asymptotic point $0$ (see Fig.). If $a<1$, the twisting behaviour is opposite. The angle formed by the tangent at an arbitrary point of the logarithmic spiral and the position vector of that point depends only on the parameter $a$. The length of the arc between two points $M_1(\rho_1,\phi_1)$ and $M_2(\rho_2,\phi_2)$ is:

$$l=\rho_2\frac{\sqrt{1+\ln^2a}}{\ln a}-\rho_1\frac{\sqrt{1+\ln^2a}}{\ln a}.$$

The radius of curvature is $r=\sqrt{1+\ln^2a}$. The natural equation is $s=kr$, where $k=1/\ln a$. Logarithmic spirals go into logarithmic spirals under linear isometries, similarities and inversions of the plane. The logarithmic spiral is related to the so-called pseudo-spirals (see Spirals).

References

[1]	A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian)

References

[a1]	M. Berger, "Geometry" , 1–2 , Springer (1987) (Translated from French)
[a2]	H.S.M. Coxeter, "Introduction to geometry" , Wiley (1963)
[a3]	K. Fladt, "Analytische Geometrie spezieller ebener Kurven" , Akad. Verlagsgesell. (1962)