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Hyperbolic spiral

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A plane transcendental curve whose equation in polar coordinates is

It consists of two branches, which are symmetric with respect to a straight line (see Fig.). The pole is an asymptotic point. The asymptote is the straight line parallel to the polar axis at a distance from it. The arc length between two points and is

The area of the sector bounded by an arc of the hyperbolic spiral and the two radius vectors and corresponding to the angles and is

A hyperbolic spiral and an Archimedean spiral may be obtained from each other by inversion with respect to the pole of the hyperbolic spiral.

Figure: h048340a

A hyperbolic spiral is a special case of the so-called algebraic spirals.

References

[1] A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian)
How to Cite This Entry:
Hyperbolic spiral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hyperbolic_spiral&oldid=32543
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article