Difference between revisions of "Spirals"
From Encyclopedia of Mathematics
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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 | + | 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 | ||
| − | + | $$r=as^m,$$ | |
| − | where | + | 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
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