Hyperbolic Spiral

An Archimedean spiral with polar equation

(1)

The hyperbolic spiral, also called the inverse spiral (Whittaker 1944, p. 83), originated with Pierre Varignon in 1704 and was studied by Johann Bernoulli between 1710 and 1713, as well as by Cotes in 1722 (MacTutor Archive).

It is also a special case of a Cotes' spiral, i.e., the path followed by a particle in a central orbit with power law

(2)

when is a constant and is the specific angular momentum.

The curvature and tangential angle are given by

			(3)
			(4)

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References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 225, 1987.Cotes, R. Harmonia Mensurarum. p. 31 and 98, 1722.Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 91, 1997.Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 186 and 188, 1972.Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, p. 175, 1967.MacTutor History of Mathematics Archive. "Hyperbolic Spiral." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Hyperbolic.html.Newton, I. Book I, §2, Prop. IX in Philosophiae Naturalis Principia Mathematica. 1687.Whittaker, E. T. A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: With an Introduction to the Problem of Three Bodies. New York: Dover, 1944.

Cite this as:

Weisstein, Eric W. "Hyperbolic Spiral." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HyperbolicSpiral.html

Hyperbolic Spiral

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