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A plane transcendental curve the equation of which in polar coordinates has the form:
 
A plane transcendental curve the equation of which 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/a/a013/a013150/a0131501.png" /></td> </tr></table>
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$$\rho=a\phi.$$
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/a013150a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/a013150a.gif" />
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Figure: a013150a
 
Figure: a013150a
  
It is described by a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013150/a0131502.png" /> moving at a constant rate along a straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013150/a0131503.png" /> that rotates around a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013150/a0131504.png" /> lying on that straight line. At the starting point of the motion, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013150/a0131505.png" /> coincides with the centre of rotation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013150/a0131506.png" /> of the straight line (see Fig.). The length of the arc between the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013150/a0131507.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013150/a0131508.png" /> is
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It is described by a point $M$ moving at a constant rate along a straight line $d$ that rotates around a point $O$ lying on that straight line. At the starting point of the motion, $M$ coincides with the centre of rotation $O$ of the straight line (see Fig.). The length of the arc between the 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/a/a013/a013150/a0131509.png" /></td> </tr></table>
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$$l=\frac a2\left[\phi\sqrt{1+\phi^2}+\ln(\phi+\sqrt{1+\phi^2})\right]_{\phi_1}^{\phi_2}.$$
  
The area of the sector bounded by an arc of the Archimedean spiral and two radius vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013150/a01315010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013150/a01315011.png" />, corresponding to angles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013150/a01315012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013150/a01315013.png" />, is
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The area of the sector bounded by an arc of the Archimedean spiral and two radius vectors $\rho_1$ and $\rho_2$, corresponding to angles $\phi_1$ and $\phi_2$, 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/a/a013/a013150/a01315014.png" /></td> </tr></table>
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$$S=\frac{\rho_2^3-\rho_1^3}{a}.$$
  
 
An Archimedean spiral is a so-called algebraic spiral (cf. [[Spirals|Spirals]]). The generalization of the Archimedean spiral is called a neoid, the equation of which in polar coordinates is
 
An Archimedean spiral is a so-called algebraic spiral (cf. [[Spirals|Spirals]]). The generalization of the Archimedean spiral is called a neoid, the equation of which in polar coordinates 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/a/a013/a013150/a01315015.png" /></td> </tr></table>
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$$\rho=a\phi+l.$$
  
 
The spiral was studied by Archimedes (3rd century B.C.) and was named after him.
 
The spiral was studied by Archimedes (3rd century B.C.) and was named after him.

Revision as of 11:00, 26 July 2014

A plane transcendental curve the equation of which in polar coordinates has the form:

$$\rho=a\phi.$$

Figure: a013150a

It is described by a point $M$ moving at a constant rate along a straight line $d$ that rotates around a point $O$ lying on that straight line. At the starting point of the motion, $M$ coincides with the centre of rotation $O$ of the straight line (see Fig.). The length of the arc between the points $M_1(\rho_1,\phi_1)$ and $M_2(\rho_2,\phi_2)$ is

$$l=\frac a2\left[\phi\sqrt{1+\phi^2}+\ln(\phi+\sqrt{1+\phi^2})\right]_{\phi_1}^{\phi_2}.$$

The area of the sector bounded by an arc of the Archimedean spiral and two radius vectors $\rho_1$ and $\rho_2$, corresponding to angles $\phi_1$ and $\phi_2$, is

$$S=\frac{\rho_2^3-\rho_1^3}{a}.$$

An Archimedean spiral is a so-called algebraic spiral (cf. Spirals). The generalization of the Archimedean spiral is called a neoid, the equation of which in polar coordinates is

$$\rho=a\phi+l.$$

The spiral was studied by Archimedes (3rd century B.C.) and was named after him.

References

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


Comments

References

[a1] E.H. Lockwood, "A book of curves" , Cambridge Univ. Press (1961)
How to Cite This Entry:
Archimedean spiral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Archimedean_spiral&oldid=16350
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article