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A plane [[Transcendental curve|transcendental curve]] whose equation in polar coordinates has the form
 
A plane [[Transcendental curve|transcendental curve]] whose equation in polar coordinates has the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060650/l0606501.png" /></td> </tr></table>
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$$\rho=a^\phi,\quad a>0.$$
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/l060650a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/l060650a.gif" />
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Figure: l060650a
 
Figure: l060650a
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060650/l0606502.png" />, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060650/l0606503.png" /> the logarithmic spiral evolves anti-clockwise, and as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060650/l0606504.png" /> the spiral twists clockwise, tending to its asymptotic point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060650/l0606505.png" /> (see Fig.). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060650/l0606506.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060650/l0606507.png" />. The length of the arc between two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060650/l0606508.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060650/l0606509.png" /> is:
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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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060650/l06065010.png" /></td> </tr></table>
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$$l=\rho_2\frac{\sqrt{1+\ln^2a}}{\ln a}-\rho_1\frac{\sqrt{1+\ln^2a}}{\ln a}.$$
  
The radius of curvature is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060650/l06065011.png" />. The [[Natural equation|natural equation]] is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060650/l06065012.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060650/l06065013.png" />. 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|Spirals]]).
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The radius of curvature is $r=\sqrt{1+\ln^2a}$. The [[Natural equation|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|Spirals]]).
  
 
====References====
 
====References====

Latest revision as of 11:22, 26 July 2014

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)


Comments

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)
How to Cite This Entry:
Logarithmic spiral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Logarithmic_spiral&oldid=32539
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article