# 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 $\rho$ 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) |

[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=53706