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Lituus

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Lituus

The lituus is an Archimedean spiral with n=-2, having polar equation

r^2theta=a^2.
(1)

Lituus means a "crook," in the sense of a bishop's crosier. The lituus curve originated with Cotes in 1722. Maclaurin used the term lituus in his book Harmonia Mensurarum in 1722 (MacTutor Archive). The lituus is the locus of the point P moving such that the area of a circular sector remains constant.

The arc length, curvature, and tangential angle are given by

s(theta)=2sqrt(theta)_2F_1(-1/2;-1/4,3/4;-1/4theta^(-2))-2sqrt(theta_0)_2F_1(-1/2;-1/4,3/4;-1/4theta_0^(-2))
(2)
kappa(theta)=(8theta^2-2)(theta/(1+4theta^2))^(3/2)
(3)
phi(theta)=theta-tan^(-1)(2theta),
(4)

where the arc length is measured from theta=theta_0.

See also

Archimedean Spiral

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References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 221, 1987.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. "Lituus." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Lituus.html.Smith, D. E. History of Mathematics, Vol. 2: Special Topics of Elementary Mathematics. New York: Dover, p. 329, 1958.

Cite this as:

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

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