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Plane curves which usually go around one point (or around several points), moving either towards or away from it (them). One distinguishes two types: algebraic spirals and pseudo-spirals.
 
Plane curves which usually go around one point (or around several points), moving either towards or away from it (them). One distinguishes two types: algebraic spirals and pseudo-spirals.
  
Algebraic spirals are spirals whose equations in polar coordinates are algebraic with respect to the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086790/s0867901.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086790/s0867902.png" />. These include the [[Hyperbolic spiral|hyperbolic spiral]], the [[Archimedean spiral|Archimedean spiral]], the [[Galilean spiral|Galilean spiral]], the [[Fermat spiral|Fermat spiral]], the [[Parabolic spiral|parabolic spiral]], and the [[Lituus|lituus]].
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Algebraic spirals are spirals whose equations in polar coordinates are algebraic with respect to the variables $\rho$ and $\phi$. These include the [[Hyperbolic spiral|hyperbolic spiral]], the [[Archimedean spiral|Archimedean spiral]], the [[Galilean spiral|Galilean spiral]], the [[Fermat spiral|Fermat spiral]], the [[Parabolic spiral|parabolic spiral]], and the [[Lituus|lituus]].
  
 
Pseudo-spirals are spirals whose natural equations can be written in the form
 
Pseudo-spirals are spirals whose natural equations can be written in 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/s/s086/s086790/s0867903.png" /></td> </tr></table>
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$$r=as^m,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086790/s0867904.png" /> is the radius of curvature and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086790/s0867905.png" /> is the arc length. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086790/s0867906.png" />, this is called the [[Logarithmic spiral|logarithmic spiral]], when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086790/s0867907.png" />, the [[Cornu spiral|Cornu spiral]], and when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086790/s0867908.png" /> it is the evolvent of a circle (cf. [[Evolvent of a plane curve|Evolvent of a plane curve]]).
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where $r$ is the radius of curvature and $s$ is the arc length. When $m=1$, this is called the [[Logarithmic spiral|logarithmic spiral]], when $m=-1$, the [[Cornu spiral|Cornu spiral]], and when $m=1/2$ it is the evolvent of a circle (cf. [[Evolvent of a plane curve|Evolvent of a plane curve]]).
  
 
====References====
 
====References====

Revision as of 11:05, 26 July 2014

Plane curves which usually go around one point (or around several points), moving either towards or away from it (them). One distinguishes two types: algebraic spirals and pseudo-spirals.

Algebraic spirals are spirals whose equations in polar coordinates are algebraic with respect to the variables $\rho$ and $\phi$. These include the hyperbolic spiral, the Archimedean spiral, the Galilean spiral, the Fermat spiral, the parabolic spiral, and the lituus.

Pseudo-spirals are spirals whose natural equations can be written in the form

$$r=as^m,$$

where $r$ is the radius of curvature and $s$ is the arc length. When $m=1$, this is called the logarithmic spiral, when $m=-1$, the Cornu spiral, and when $m=1/2$ it is the evolvent of a circle (cf. Evolvent of a plane curve).

References

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


Comments

References

[a1] J.D. Lawrence, "A catalog of special plane curves" , Dover, reprint (1972)
[a2] M. Berger, "Geometry" , 1–2 , Springer (1987) (Translated from French)
[a3] H.S.M. Coxeter, "Introduction to geometry" , Wiley (1963)
[a4] F. Gomes Teixeira, "Traité des courbes" , 1–3 , Chelsea, reprint (1971)
[a5] K. Fladt, "Analytische Geometrie spezieller ebener Kurven" , Akad. Verlagsgesell. (1962)
How to Cite This Entry:
Spirals. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spirals&oldid=32534
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article