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Spherical Spiral


SphericalSpiral

The spherical curve taken by a ship which travels from the south pole to the north pole of a sphere while keeping a fixed (but not right) angle with respect to the meridians. The curve has an infinite number of loops since the separation of consecutive revolutions gets smaller and smaller near the poles.

It is given by the parametric equations

x=costcosc
(1)
y=sintcosc
(2)
z=-sinc,
(3)

where

 c=tan^(-1)(at)
(4)

and a is a constant. Plugging in therefore gives

x=(cost)/(sqrt(1+a^2t^2))
(5)
y=(sint)/(sqrt(1+a^2t^2))
(6)
z=-(at)/(sqrt(1+a^2t^2)).
(7)

It is a special case of a loxodrome.

The arc length, curvature, and torsion are all slightly complicated expressions.

A series of spherical spirals are illustrated in Escher's woodcuts "Sphere Surface with Fish" (Bool et al. 1982, pp. 96 and 318) and "Sphere Spirals" (Bool et al. 1982, p. 319; Forty 2003, Plate 67).


See also

Helix, Loxodrome, Mercator Projection, Seiffert's Spherical Spiral, Spherical Curve

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References

Bool, F. H.; Kist, J. R.; Locher, J. L.; and Wierda, F. M. C. Escher: His Life and Complete Graphic Work. New York: Abrams, 1982.Escher, M. C. "Sphere Spirals." Woodcut printed from 4 blocks. 1958. http://www.mcescher.com/Gallery/recogn-bmp/LW428.jpg.Escher, M. C. "Sphere Surface with Fish." Woodcut in three colors. 1958. http://www.mcescher.com/Gallery/recogn-bmp/LW427.jpg.Forty, S. M.C. Escher. Cobham, England: TAJ Books, 2003.Gray, A. "Loxodromes on Spheres." §10.6 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 238-240, 1997.Lauwerier, H. "Spherical Spiral." In Fractals: Endlessly Repeated Geometric Figures. Princeton, NJ: Princeton University Press, pp. 64-66, 1991.

Cite this as:

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

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