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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.

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


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).


[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:
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article