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Difference between revisions of "Parabolic spiral"

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A transcendental plane 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/p/p071/p071250/p0712501.png" /></td> </tr></table>
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{{TEX|done}}
  
To each value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071250/p0712502.png" /> correspond two values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071250/p0712503.png" />, one positive and one negative.
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A transcendental plane curve whose equation in [[Polar_coordinates | polar coordinates]] has the form
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\begin{equation}
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\rho = a\sqrt{\phi} + l,\quad l>0.
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\end{equation}
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To each value of $\phi$ correspond two values of $\sqrt{\phi}$, one positive and one negative.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p071250a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p071250a.gif" />
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Figure: p071250a
 
Figure: p071250a
  
The curve has infinitely many double points and one point of inflection (see Fig.). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071250/p0712504.png" />, then the curve is called the [[Fermat spiral|Fermat spiral]]. The parabolic spiral is related to the so-called algebraic spirals (see [[Spirals|Spirals]]).
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The curve has infinitely many double points and one [[Point_of_inflection | point of inflection]] (see Fig.). If $l=0$, then the curve is called the [[Fermat spiral|Fermat spiral]]. The parabolic spiral is related to the so-called algebraic spirals (see [[Spirals|Spirals]]).
  
 
====References====
 
====References====

Latest revision as of 11:32, 30 November 2012


A transcendental plane curve whose equation in polar coordinates has the form \begin{equation} \rho = a\sqrt{\phi} + l,\quad l>0. \end{equation} To each value of $\phi$ correspond two values of $\sqrt{\phi}$, one positive and one negative.

Figure: p071250a

The curve has infinitely many double points and one point of inflection (see Fig.). If $l=0$, then the curve is called the Fermat spiral. The parabolic spiral is related to the so-called algebraic spirals (see Spirals).

References

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


Comments

References

[a1] F. Gomez Teixeira, "Traité des courbes" , 1–3 , Chelsea, reprint (1971)
[a2] J.D. Lawrence, "A catalog of special plane curves" , Dover, reprint (1972)
How to Cite This Entry:
Parabolic spiral. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Parabolic_spiral&oldid=28966
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article